Advertisement

Complex ΨDOSS and systems of complex differential equations

  • Sabir Umarov
Chapter
  • 1.1k Downloads
Part of the Developments in Mathematics book series (DEVM, volume 41)

Abstract

In Chapters 4–7 we discussed pseudo-differential equations of integer and fractional orders with ψDOSS depending on real variables \(t \in \mathbb{R}\) and \(x \in \mathbb{R}^{n}\). In this section we will discuss differential and pseudo-differential equations depending on complex variables \(t =\tau +i\sigma \in \mathbb{C}\) and \(z = x + iy \in \mathbb{C}^{n}.\) Consider two simple examples with the one-dimensional “spatial” variable:
  1. (i)

    “complex wave” equation, and

     
  2. (ii)

    “complex heat” equation.

     

Keywords

Pseudo-differential Equations Meromorphic Symbols Cauchy-Kowalevsky Theorem General Boundary Value Problems Exponential Functionals 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

9.1 Introduction

In Chapters  4 7 we discussed pseudo-differential equations of integer and fractional orders with ψDOSS depending on real variables \(t \in \mathbb{R}\) and \(x \in \mathbb{R}^{n}\). In this section we will discuss differential and pseudo-differential equations depending on complex variables \(t =\tau +i\sigma \in \mathbb{C}\) and \(z = x + iy \in \mathbb{C}^{n}.\) Consider two simple examples with the one-dimensional “spatial” variable:
  1. (i)

    “complex wave” equation, and

     
  2. (ii)

    “complex heat” equation.

     
The first equation is obtained from the wave equation
$$\displaystyle{ \frac{\partial ^{2}u(\tau,x)} {\partial \tau ^{2}} = -D^{2}u(\tau,x),\quad \tau> 0,\,x \in \mathbb{R}, }$$
(9.1)
where \(D = -id/dx,\) by “complexifying” the variables t and x, that is
$$\displaystyle{ D_{t}^{2}u(t,z) = D_{ z}^{2}u(t,z),\quad t \in \mathbb{C},\,z \in \mathbb{C}, }$$
(9.2)
where \(D_{t} = \frac{\partial } {\partial \tau } + i\frac{\partial } {\partial \sigma }\) and \(D_{z} = \frac{\partial } {\partial x} + i \frac{\partial } {\partial y}.\) The solution to (9.1), satisfying the initial conditions \(u(0,x) =\varphi (x)\) and \(u_{t}(0,x) =\psi (x),\) was obtained in Section  2.2 in the form
$$\displaystyle{ u(\tau,x) = \left [\cos \tau D\right ]\varphi (x) + \left [\frac{\sin \tau D} {D}\right ]\psi (x). }$$
(9.3)
Replacing D in (9.3) by D z one obtains
$$\displaystyle{ u(t,z) = \left [\cos tD_{z}\right ]\varphi (z) + \left [\frac{\sin tD_{z}} {D_{z}} \right ]\psi (z). }$$
(9.4)
Is u(t, z) in equation (9.4) a solution to complex equation (9.2) satisfying the “initial” conditions \(u(0,z) =\varphi (z)\) and \(D_{t}u(0,z) =\psi (z)?\) If yes, in what sense the operators \(\cos tD_{z}\) and \(\frac{\sin tD_{z}} {D_{z}}\) must be understood, and in what spaces these operators act? It is not hard to verify that d’Alembert’s formula in this case takes the form
$$\displaystyle{u(t,z) = \frac{\varphi (ze^{it}) +\varphi (ze^{-it})} {2} + \frac{1} {2}\int _{z-t}^{z+t}\psi (\zeta )d\zeta,}$$
where the integral is the line integral over a smooth curve connecting points zt and z + t on the complex plane.
The second equation is obtained from the heat equation
$$\displaystyle{ \frac{\partial u(\tau,x)} {\partial \tau } = -D^{2}u(\tau,x),\quad \tau> 0,\ x \in \mathbb{R}, }$$
(9.5)
complexifying the variables \(\tau\) and x, that is
$$\displaystyle{ D_{t}u(t,z) = D_{z}^{2}u(t,z),\quad t \in \mathbb{C},\,z \in \mathbb{C}. }$$
(9.6)
Again, replacing D by D z in the solution representation \(u(\tau,x) =\exp (-tD^{2})\varphi (x)\) of equation (9.5), satisfying the initial condition \(u(0,x) =\varphi (x),\) can we state that
$$\displaystyle{u(t,z) = e^{tD_{z}^{2} }\varphi (z)}$$
solves complex equation (9.6) with the “initial” condition \(u(0,z) =\varphi (z)?\) If the answer yes, how should we understand the operator \(\exp (tD_{z}^{2}),\) and in what class of functions it is meaningful?

We note that the complex “wave” equation has a unique solution in the class of analytic functions near \((0,0) \in \mathbb{C}^{2}\) if \(\varphi\) and ψ are analytic in a neighborhood of \(0 \in \mathbb{C},\) while the complex “heat” equation does not possess this property. This is due to the fact that the complex “wave” equation is Kowalevskian (definition is given below), while the complex “heat” equation is not. Hence, in the theory of complex differential and pseudo-differential equations new features appear, making this theory very distinct from its “real” counterpart.

Thus, in this chapter we will discuss the problem of existence and uniqueness of a solution to systems of complex differential and pseudo-differential equations in the complex (n + 1)-dimensional space. These systems in the general form can be represented as
$$\displaystyle\begin{array}{rcl} D_{t}^{p_{j} }u_{j}(t,z) +\sum _{ k=1}^{N}\sum _{ q=0}^{p_{k}-1}A_{ jk}^{q}(t,z,D_{ z})D_{t}^{q}u_{ k}(t,z) = f_{j}(t,z),\quad j = 1,\ldots,N,& &{}\end{array}$$
(9.7)
$$\displaystyle\begin{array}{rcl} \sum _{k=1}^{N}\sum _{ q=0}^{p_{k}-1}B_{ jk}^{mq}(z,D_{ z})D_{t}^{q}u_{ k}(t,z)\Big\vert _{t=t_{0}} =\varphi _{jm}(z),\quad m = 0,\ldots,p_{j} - 1,\,j = 1,\ldots,N,& &{}\end{array}$$
(9.8)
where \(t \in \mathcal{D},\) a connected domain in \(\mathbb{C},\) and \(z \in \mathbb{C}^{n}.\) The operators \(A_{jk}^{q}(t,z,D_{z}),\) \(q = 0,\ldots,p_{j} - 1;\,k,j = 1,\ldots,N,\) and \(B_{jk}^{mq}(D_{z}),\,q,m = 0,\ldots,p_{j} - 1;\,k,j = 1,\ldots,N,\) are, in general, pseudo-differential operators with analytic symbols (see the definition in Section 9.6) in a domain \(G \subset \mathbb{C}^{n}\), and the functions \(f_{j}(t,z),\,j = 1,\ldots,N,\) and \(\varphi _{jm}(z),\,m = 0,\ldots,p_{j} - 1;\,j = 1,\ldots,N,\) satisfy certain conditions clarified in Section 9.7; \(p_{j} \geq 1,\,j = 1,\ldots,N,\) are integers. We note that symbols of ψDO have singularities of finite order at the boundary of G or finite exponential type if \(G = \mathbb{C}^{n}.\)
The Cauchy problem is a particular case, corresponding to \(B_{jk}^{mq}(z,D_{z}) =\delta _{ jk}^{mq}I,\) where I is the identity operator, and
$$\displaystyle{\delta _{jk}^{mq} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &\mbox{ if }q = m,\ \mbox{ and}\ \,k = j, \\ 0,\quad &\mbox{ otherwise}, \end{array} \right.}$$
is the generalized Kronecker symbol. It is not hard to see that boundary conditions (9.8) can be reduced to the Cauchy conditions
$$\displaystyle{ D_{t}^{q}u_{ j}(t,z)\Big\vert _{t=t_{0}} =\psi _{jq}(z),\quad q = 0,\ldots,p_{j} - 1,\,j = 1,\ldots,N, }$$
(9.9)
where the vector function \(\psi _{jq}(z)\) of length \(p_{1} +\ldots +p_{n}\) is a solution to the system of pseudo-differential equations
$$\displaystyle{ \sum _{k=1}^{N}\sum _{ q=0}^{p_{j}-1}B_{ jk}^{mq}(z,D_{ z})\psi _{kq}(z) =\varphi _{jm}(z),\quad m = 0,\ldots,p_{j} - 1,\,j = 1,\ldots,N. }$$
(9.10)
Hence, the general boundary value problem (9.7)–(9.8) splits into two problems:
  1. 1.

    the system of pseudo-differential equations (9.10), and

     
  2. 2.

    the Cauchy problem (9.7), (9.9).

     

A brief history. We start with a brief history, since it casts light on the question: “Where did the conditions for orders of operators \(A_{jk}^{q}(t,z,D_{z})\) and \(B_{jk}^{mq}(z,D_{z})\) appeared in the theorems of this chapter came from?”

The Cauchy problem, the most important and the most studied amongst boundary value problem (9.7)–(9.8), was always a focus of many classics (d’Alembert, Euler, Fourier, Poisson, Cauchy, Hadamard, Holmgren, Petrovskii, Sobolev, etc.). Mizohata in his book [Miz67] emphasized four problems related to the Cauchy problem:
  1. 1.

    existence of a local solution;

     
  2. 2.

    uniqueness of a solution;

     
  3. 3.

    continuous dependence on data;

     
  4. 4.

    existence of a global solution,

     

which to some extent reflect the development of the general theory of the Cauchy problem for partial differential equations in the twenties century.

First result on the existence of a local solution was the Cauchy-Kowalevsky theorem (see, e.g., [Miz67, Hor83]). This theorem in the case of differential operators \(A_{jk}^{q}(t,z,D_{z})\) of finite order \(m_{jk}^{q}\), i.e.
$$\displaystyle{A_{jk}^{q}(t,z,D_{ z}) =\sum _{\vert \alpha \vert \leq m_{jk}^{q}}a_{jk\alpha }^{q}(t,z)D_{ z}^{\alpha },}$$
states that if the coefficients and data of the non-characteristic Cauchy problem are analytic functions, then there exists a unique local solution in the class of analytic functions. An essential contribution to the modern theory of the Cauchy problem was made by Petrovskii [Pet96], Schwartz [Sch51], Hörmander [Hor83], Gårding, Kotake, Leray [LGK67], Mizohata [Miz67, Miz74], Ovsyannikov [Ovs65], Treves [Tre80], Gindikin, Volevich [VG91], Kitagawa [Kit90], etc. The Cauchy problem in the case of infinite order differential operators \(A_{jk}^{q}(D_{z})\) (not depending on z, i.e., with constant coefficients) were studied by Korobeynik [K73], Leont’ev [Leo76], Baouendi, Goulaouic [BG76], Dubinskii [Dub84], Napalkov [Nap82], and others.
We note that yet Cauchy and Kowalevsky had known that if the orders \(m_{jk}^{q}\) of operators \(A_{jk}^{q}(z,D_{z})\) satisfy the condition \(m_{jk}^{q} \leq p_{j} - q,\) then there exists a unique local solution to the Cauchy problem in the class of analytic functions. Kowalevsky [Kow1874] in examples showed that this condition is essential for the analytic solvability, namely, if this condition is not verified then the Cauchy problem may not have a solution in the class of analytic functions. Therefore, systems satisfying this condition are called Kowalevskian; see [Miz74]. In the case of one equation (that is N = 1)
$$\displaystyle{D_{t}^{m}u(t,z) =\sum _{ k=0}^{m-1}A_{ k}(t,z,D_{z})D_{t}^{k}u(t,z) + f(t,z),}$$
this condition takes the form
$$\displaystyle{ m_{k} \leq m - k,\quad k = 0,\ldots,m - 1, }$$
(9.11)
where m k is the order of differential operator \(A_{k}(t,z,D_{z}).\) Obviously, equation (9.2) satisfies condition (9.11), while equation (9.6) does not. In 1974 Mizohata [Miz74] showed that in the case of one equation, condition (9.11) is also necessary for analytic solvability; see also [Kit76].
The sufficient condition for the general system, as was shown by Leray et al. [LGK67] in 1964, is the Leray-Volevich (LV) condition
$$\displaystyle{ m_{jk}^{q} \leq \mu _{ j} -\mu _{k} + p_{j} - q, }$$
where \(\mu _{1},\ldots,\mu _{N}\) are collection of natural numbers (related to the orders of singularities near the boundary). What concerns the necessity of the LV-condition, then as was noted by Dubinskii [Dub90], it depends essentially on the problem setting. Namely, if the singularities of solutions evolve cylindrically, then the necessary condition for existence of a local analytic solution is \(m_{jk}^{q} \leq \mu _{j} -\mu _{k};\) however, if the singularities evolve along the characteristic cone, then the LV-condition becomes necessary [Dub90].

Apart from analytic theory, well posedness in classes of exponential functions were studied. Tikhonov [Tik35] was the first, who in 1935 indicated the exact exponential growth conditions for uniqueness of a solution of the heat equation. For general parabolic systems the uniqueness and well-posedness classes in terms of exponential classes of functions were studied, in particular, in works [Tac36, GS53, Hay78, K81]. In the above-mentioned references [Dub84, Dub90] Dubinskii showed that the analytic and exponential theories are in a dual relationship.

In the case of real \(z \in \mathbb{R}^{n}\) in the system (9.10) and \(p_{j} = 1,\,j = 1,\ldots,N,\) and \(B_{jk}^{00}(z,D_{z}) = B_{jk}(z,D_{z})\) are differential operators of finite order
$$\displaystyle{B_{jk}(z,D_{z}) =\sum _{\vert \alpha \vert \leq \nu _{jk}}b_{jk}^{\alpha }(z)D_{ z}^{\alpha },}$$
the elliptic systems were studied by Bernstein [Ber28], Petrovskii [Pet96], Hörmander [Hor83], Douglis and Nirenberg [DN55], Morrey [Mor58], Oleynik and Radkevich [OR73], etc. An important question of analyticity of a solution was always in the focus of many authors; see, e.g., [Ber04, Ber28, MN57, Mor58, OR73, Pet96] and the references therein. Douglis and Nirenberg [DN55] studied elliptic systems under the following conditions for orders of operators \(B_{jk}(z,D_{z}):\)
$$\displaystyle{ \nu _{jk} \leq \mu _{j} -\nu _{k},\quad k,j = 1,\ldots,N, }$$
(9.12)
where \(\mu _{1},\ldots,\mu _{N}\) and \(\nu _{1},\ldots,\nu _{N}\) are some collection of integers. In the modern literature conditions (9.12) are referred to as the Douglis-Nirenberg, or DN conditions. Mizohata [Miz62] and Suzuki [Suz64] found examples of elliptic equations, smooth solutions to which are not analytic. Therefore, finding necessary and sufficient conditions for analytic solvability of systems is a challenging question. See more on the history and other contributions in Section “Additional notes.”

In this chapter we will present resent results on necessary and sufficient conditions for analytic and exponential solvability of general boundary value problem (9.7)–(9.8) with pseudo-differential operators \(A_{jk}^{q}(t,z,D_{z})\) and \(B_{jk}(z,D_{z})\) with analytic symbols (Section 9.7). For this purpose we construct an algebra of pseudo-differential operators with meromorphic symbols defined on a complex domain (manyfold) (Sections 9.5 and 9.6).

The main tool of the construction of PsDOs in the real case is the Fourier transform (Chapters  2 7). However, in complex analysis there is no primary analog of the Fourier transform. The existing Borel and Fourier-Laplace transforms, introduced in Section  2.7, do not give desired results. In 1984 Dubinskii [Dub84] introduced a complex Fourier transform of f as an analytic functional, defined as an image of the PsDO with the symbol f on the Dirac delta function (see Section  2.7). This transform inherits many properties of the real Fourier transform and can be easily adapted for the construction of PsDO with analytic symbols defined on a complex manyfold \(\varOmega \in \mathbb{C}^{n}\). We note that symbols may have singularities on the boundary of Ω. This construction can be extended to the class of meromorphic symbols, as well [Uma14]. However, in this case the corresponding ψDOSS become multi-valued (Section 9.5).

We have seen in Section  2.7 that the complex Fourier transform is an extension of Borel’s transform to the space of analytic and exponential functionals and the inverse to the Fourier-Laplace transform. The complex Fourier transform in this spirit is adapted to spaces of analytic and exponential functions and functionals, introduced in this chapter and studied in Section 9.4. In contrast to spaces used in [Dub84] (see Section  2.7) we introduce new spaces of analytic and exponential functions and functionals. Under the conditions
$$\displaystyle\begin{array}{rcl} & & m_{jk}^{q} \leq \mu _{ j} -\mu _{k} + p_{j} - q,\quad \nu _{jk}^{qm} \leq \mu _{ j} -\mu _{k} + q - m, {}\\ & & q = 0,\ldots,p_{j} - 1,\ \ \ m = 0,\ldots,p_{k} - 1,\ \ \ k,j = 1,\ldots,N, {}\\ \end{array}$$
to orders of operators \(A_{ij}^{q}\) and \(B_{jk}^{mq}\) we show the existence of a unique local solution of (9.7)–(9.8) in the introduced spaces.

9.2 Some Banach spaces of exponential and analytic functions and functionals

Let \(\xi \in \mathbb{R}^{n}\) and α be a multi-index, that is \(\alpha = (\alpha _{1},\ldots,\alpha _{n}),\) and α j are nonnegative integers. We use notations \(\vert \alpha \vert =\alpha _{1} +\ldots +\alpha _{n},\) \(\vert \xi \vert =\xi _{1} +\ldots +\xi _{n},\) and \(\xi ^{\alpha } =\xi _{ 1}^{\alpha _{1}}\ldots \xi _{n}^{\alpha _{n}}.\) Introduce the function
$$\displaystyle{ G(\xi ) = G_{\mu,r}^{\alpha }(\xi ) = \frac{(1 + \vert \xi \vert )^{\mu }e^{r\vert \xi \vert }} {\xi ^{\alpha }},\quad \xi \in \mathbb{R}_{+}^{n}, }$$
(9.13)
where r > 0 is real and μ is integer fixed numbers and \(\mathbb{R}_{+}^{n} =\{\xi \in \mathbb{R}^{n}:\xi _{1}> 0,\ldots,\xi _{n}> 0\}.\) Obviously, this function is continuous, differentiable, and strictly positive on \(\mathbb{R}_{+}^{n}.\) If one of the components of α is zero, say \(\alpha _{j_{0}} = 0,\) then the domain of \(G(\xi )\) extends to the hyperplane \(\xi _{j_{0}} = 0.\) It follows from the definition of \(G(\xi )\) that if \(\alpha _{j}\neq 0,j = 1,\ldots,n,\) then \(G(\xi ) \rightarrow \infty\) as \(\xi \rightarrow \partial \mathbb{R}_{+}^{n} \cup \{\infty \}.\) If \(\alpha = 0 = (0,\ldots,0),\) then
$$\displaystyle{ \inf _{\mathbb{R}_{+}^{n}}\,G(\xi ) = \left \{\begin{array}{@{}l@{\quad }l@{}} G(0) = 1 \quad &\mbox{ if }\mu \geq -r, \\ e^{-\mu -r}\left ( \frac{r} {-\mu }\right )^{-\mu }\quad &\mbox{ if }\mu <-r. \end{array} \right. }$$
Also it is not hard to see that if μ > −r and \(\alpha _{j_{0}} = 0\) for some \(j_{0} \in \{ 1,\ldots,n\},\) then the infimum is attained on the hyperplane \(\xi _{j_{0}} = 0.\) If all the components of α are not zero, then the following statement on the infimum of \(G(\xi )\) holds.

Proposition 9.1.

For each multi-index \(\alpha,\,\alpha _{j}\neq 0,\,j = 1,\ldots,n,\) there is a unique infimum of \(G(\xi )\) attained at \(\xi ^{{\ast}} \in \mathbb{R}_{+}^{n},\) for which the asymptotic behavior \(\vert \xi ^{{\ast}}\vert \sim O(\vert \alpha \vert )\) for large |α| holds. If μ = 0, then \(\xi _{j}^{{\ast}} =\alpha _{j}/r,\,j = 1,\ldots,n.\)

Proof.

To prove this proposition we consider the system of equations
$$\displaystyle{\frac{\partial G(\xi )} {\partial \xi _{j}} = 0,\quad j = 1,\mathop{\ldots },n,}$$
which reduces to
$$\displaystyle{ r\vert \xi \vert \xi _{j} + (\mu +r)\xi _{j} -\alpha _{j}\vert \xi \vert =\alpha _{j},\quad j = 1,\ldots,n. }$$
(9.14)
Summing the latter over the indices \(j = 1,\ldots,n,\) we obtain the quadratic equation for \(\vert \xi \vert:\)
$$\displaystyle{ r\vert \xi \vert ^{2} - (\vert \alpha \vert -\mu -r)\vert \xi \vert -\vert \alpha \vert = 0. }$$
(9.15)
This equation has one positive and one negative roots if | α | > 0. The point \(\xi ^{{\ast}}\) corresponding to the negative root of (9.15) is out of the domain of \(G(\xi ),\) and hence, the only stationary point \(\xi ^{{\ast}}\) delivering the infimum (minimum) corresponds to the positive root of equation (9.15), i.e., \(\vert \xi ^{{\ast}}\vert> 0.\) The fact that \(\xi ^{{\ast}} \in \mathbb{R}_{+}^{n}\) follows from equations (9.14):
$$\displaystyle{ \xi _{j}^{{\ast}} =\eta \frac{\alpha _{j}} {r}> 0,\quad j = 1,\mathop{\ldots },n, }$$
(9.16)
where
$$\displaystyle{ \eta = \frac{\vert \xi ^{{\ast}}\vert + 1} {\vert \xi ^{{\ast}}\vert + 1 +\mu /r}. }$$
(9.17)
The latter is obviously positive if μ ≥ 0. If μ < 0, then using the root representation of equation (9.15), one can see that \(r(\vert \xi ^{{\ast}}\vert + 1) \leq \vert \alpha \vert -\mu +r,\) which implies \(\eta \geq (\vert \xi ^{{\ast}}\vert + 1)/(\vert \alpha \vert + r)> 0.\) Further, it follows from (9.16) that if μ = 0, then \(\xi _{j}^{{\ast}} =\alpha _{j}/r,\) and \(\vert \xi ^{{\ast}}\vert = \vert \alpha \vert /r,\) and if μ ≠ 0, then \(\vert \xi ^{{\ast}}\vert \sim O(\vert \alpha \vert ),\,\vert \alpha \vert \rightarrow \infty.\) Additionally, equation (9.16) also implies that if μ > −r and \(\alpha _{j_{0}} = 0\) for some j = j0, then \(\xi _{j_{0}}^{{\ast}} = 0.\) Thus, the infimum of \(G(\xi )\) in this case is attained on the hyperplane \(\xi _{j_{0}} = 0.\)
Define a Banach space \(\mathcal{E}_{\mu,r}\) as the set of entire functions \(\varphi (z)\) satisfying the inequality
$$\displaystyle{ \vert \varphi (z)\vert \leq C(1 + \vert z\vert )^{\mu }e^{r\vert z\vert },\quad z \in \mathbb{C}^{n}, }$$
(9.18)
where C ≥ 0 is a constant. The smallest constant \(C = C_{\varphi }\) in (9.18), that is
$$\displaystyle{ \|\varphi \|_{\mu,r} = \sup _{z\in \mathbb{C}^{n}}(1 + \vert z\vert )^{-\mu }e^{-r\vert z\vert }\vert \varphi (z)\vert }$$
is a norm in \(\mathcal{E}_{\mu,r}.\) It follows immediately from (9.18) that if ν > μ and/or s > r, then the embedding
$$\displaystyle{ \mathcal{E}_{\mu,r} \subset \mathcal{E}_{\nu,s} }$$
(9.19)
is continuous.

Let \(\mathcal{K}(\alpha )=\mathcal{K}_{\mu,r}(\alpha ) = G_{\mu,r}^{\alpha }(\xi _{ {\ast}})=\inf _{\xi \in \mathbb{R}_{+}^{n}}G_{\mu,r}^{\alpha }(\xi ).\) Due to Proposition 9.1 \(\mathcal{K}(\alpha )\) is well defined for all multi-indices α. 

Proposition 9.2.

Let \(\varphi \in \mathcal{E}_{\mu,r}.\) Then
$$\displaystyle{ \frac{\vert D^{\alpha }\varphi (z)\vert } {\alpha !} \leq \|\varphi \|_{\mu,r}\mathcal{K}_{\mu,r}(\alpha )(1 + \vert z\vert )^{\mu }e^{r\vert z\vert },\quad \vert \alpha \vert = 0,1,\mathop{\ldots }. }$$
(9.20)

Proof.

In accordance with the Cauchy theorem on integral representation, for arbitrary \(\xi _{j}> 0,\,j = 1,\ldots,n,\) one has
$$\displaystyle{ D^{\alpha }\varphi (z) = \frac{\alpha !} {(2\pi i)^{n}}\int \limits _{\vert \zeta _{1}-z_{1}\vert =\xi _{1}}\ldots \int \limits _{\vert \zeta _{n}-z_{n}\vert =\xi _{n}} \frac{\varphi (\zeta )d\zeta } {(\zeta -z)^{\alpha +(1)}}, }$$
(9.21)
where \(\alpha +(1) = (\alpha _{1} + 1,\ldots,\alpha _{n} + 1).\) The substitution \(\zeta = z +\xi e^{i\theta }\) (i.e., \(\zeta _{j} = z_{j} +\xi _{j}e^{i\theta _{j}},\,j = 1,\ldots,n\)), where \(\theta\) runs over the n-dimensional torus \(T^{n} =\{\theta \in \mathbb{R}^{n}: 0 \leq \theta _{j} <2\pi,\,j = 1,\ldots,n\},\) reduces (9.21) to
$$\displaystyle{ \frac{D^{\alpha }\varphi (z)} {\alpha !} = \frac{1} {(2\pi )^{n}\,\xi ^{\alpha }}\int \limits _{T^{n}}\varphi (z +\xi e^{i\theta })e^{i(\theta _{1}+\ldots +\theta _{n})}d\theta, }$$
(9.22)
Further, multiplying and dividing the integrand in (9.22) by \((1 + \vert z +\xi e^{i\theta }\vert )^{\mu }e^{r\vert z+\xi e^{i\theta }\vert },\) and taking into account \(\varphi \in \mathcal{E}_{\mu,r},\) one has the following estimate:
$$\displaystyle{\frac{\vert D^{\alpha }\varphi (z)\vert } {\alpha !} \leq \|\varphi \|_{\mu,r}G(\xi )(1 + \vert z\vert )^{\mu }e^{r\vert z\vert },}$$
where \(G(\xi )\) is defined in equation (9.13). Minimizing \(G(\xi )\) over all \(\xi \in R_{+}^{n},\) one obtains (9.20).

Corollary 9.1.

  1. 1.
    Let \(\varphi \in \mathcal{E}_{\mu,r}.\) Then
    $$\displaystyle{ \frac{\vert D^{\alpha }\varphi (0)\vert } {\alpha !} \leq \mathcal{K}_{\mu,r}(\alpha )\|\varphi \|_{\mu,r},\quad \vert \alpha \vert = 0,1,\mathop{\ldots }. }$$
    (9.23)
     
  2. 2.
    Let \(\varphi \in \mathcal{E}_{\mu,r}.\) Then
    $$\displaystyle{ \|D^{\alpha }\varphi \|_{\mu,r} \leq \alpha !\mathcal{K}_{\mu,r}(\alpha )\|\varphi \|_{\mu,r},\quad \vert \alpha \vert = 0,1,\mathop{\ldots }. }$$
    (9.24)
     

The converse to the first statement in Corollary 9.1 is also true in the following sense.

Proposition 9.3.

Let an entire function \(\varphi (z)\) satisfy the inequalities
$$\displaystyle{ \frac{\vert D^{\alpha }\varphi (0)\vert } {\alpha !} \leq C\mathcal{K}_{\mu,r}(\alpha ),\quad \vert \alpha \vert \geq N, }$$
(9.25)
where C > 0 is a constant not depending on α, and N is a nonnegative integer. Then \(\varphi \in \mathcal{E}_{\mu,r}.\)

Proof.

Consider the function
$$\displaystyle{ \phi (z) =\sum _{\alpha }\frac{\vert \alpha \vert ^{\mu }(er)^{\vert \alpha \vert }} {\alpha ^{\alpha }} z^{\alpha }. }$$
(9.26)
This function belongs to \(\mathcal{E}_{\mu,r}.\) Indeed, since
$$\displaystyle{{1 \over e}\ \mathop{\lim \sup }\limits_{\vert \alpha \vert \rightarrow \infty }\left (\vert \alpha \vert \root{\vert \alpha \vert }\of{\frac{(er)^{\vert \alpha \vert }} {\alpha ^{\alpha }} }\right ) = r,}$$
it follows from the theory of entire functions (see, e.g., [GR09, Hor90]) that the function \(\phi (z)\) is an exponential function of type r. The function \(\phi (z)\) majorizes \(\varphi (z).\) To verify this we recall that \(\mathcal{K}_{\mu,r}(\alpha ) = G(\xi ^{{\ast}}),\) where \(\xi ^{{\ast}} = \frac{\alpha } {r}\eta,\,\eta = (\vert \xi ^{{\ast}}\vert + 1)(\vert \xi ^{{\ast}}\vert + 1 +\mu /r)^{-1}.\) If μ ≥ 0, then \(0 <\eta \leq 1.\) If μ < 0, then due to the asymptotics \(\vert \xi ^{{\ast}}\vert = O(\vert \alpha \vert ),\) for large | α | one obtains from (9.17) that \(\eta <1+\varepsilon,\) where \(\varepsilon\) is arbitrarily small. Moreover, for large | α | it is easy to see that \(\lim _{\vert \alpha \vert \rightarrow \infty }\eta ^{\vert \alpha \vert } = e^{-\mu }.\) Therefore, there exists an integer N, such that for all | α | ≥ N the inequality \(\eta ^{\vert \alpha \vert }\geq e^{-\mu }-\varepsilon\) holds, where \(0 <\varepsilon <e^{-\mu }.\) Making use of these facts and taking \(\varepsilon\) small enough, one has
$$\displaystyle{ \mathcal{K}_{\mu,r}(\alpha ) = \frac{(1 + \frac{\vert \alpha \vert } {r}\eta )^{\mu }e^{\vert \alpha \vert \eta }r^{\vert \alpha \vert }} {\alpha ^{\alpha }\eta ^{\vert \alpha \vert }} \leq C\frac{\vert \alpha \vert ^{\mu }(er)^{\vert \alpha \vert }} {\alpha ^{\alpha }},\quad C> 0. }$$
(9.27)
The latter together with (9.25) implies that series (9.26) for \(\phi (z)\) is a majorant for \(\varphi (z),\) and hence, \(\varphi \in \mathcal{E}_{\mu,r}\) as well.
We denote the space conjugate to the Banach space \(\mathcal{E}_{\mu,r}\) by \(\mathcal{E}_{\mu,r}^{{\ast}}.\) The space \(\mathcal{E}_{\mu,r}^{{\ast}}\) is a Banach space with the norm
$$\displaystyle{ \|h\|_{\mu,r}^{{\ast}} =\sup _{\varphi \neq 0}\frac{\vert \langle h(z),\varphi (z)\rangle \vert } {\|\varphi \|_{\mu,r}}, }$$
(9.28)
where \(h \in \mathcal{E}_{\mu,r}^{{\ast}},\) \(\varphi \in \mathcal{E}_{\mu,r},\) and the symbol \(\langle \cdot,\cdot \rangle\) stands for the duality pair of the spaces \(\mathcal{E}_{\mu,r}^{{\ast}}\) and \(\mathcal{E}_{\mu,r}.\)

Proposition 9.4.

Let \(h \in \mathcal{E}_{\mu,r}^{{\ast}}.\) Then
$$\displaystyle{ \|h\|_{\mu,r}^{{\ast}} =\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \langle h,z^{\alpha }\rangle \vert. }$$
(9.29)

Proof.

Suppose \(\varphi _{0}(z)\neq 0\) is a function in \(\mathcal{E}_{\mu,r}\) that delivers sup in equation (9.28), that is
$$\displaystyle{\|h\|_{\mu,r}^{{\ast}} = \frac{\vert \langle h(z),\varphi _{0}(z)\rangle \vert } {\|\varphi _{0}\|_{\mu,r}}.}$$
Expanding \(\varphi _{0}(z)\) to Taylor series and using (9.23), we have
$$\displaystyle{\|h\|_{\mu,r}^{{\ast}}\leq \sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \langle h,z^{\alpha }\rangle \vert.}$$
On the other hand, for an arbitrary \(\varphi \in \mathcal{E}_{\mu,r},\)
$$\displaystyle{\|h\|_{\mu,r}^{{\ast}}\geq \frac{\vert \langle h(z),\varphi (z)\rangle \vert } {\|\varphi \|_{\mu,r}}.}$$
We pick the function
$$\displaystyle{ 0\neq \varphi ^{{\ast}}(z) = C_{ 0}\sum _{\vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu r}(\alpha ) \frac{\overline{\langle h,z^{\alpha }\rangle }} {\vert \langle h,z^{\alpha }\rangle \vert }z^{\alpha }, }$$
(9.30)
where C0 is a positive real number. Due to Proposition 9.3, \(\varphi ^{{\ast}} \in \mathcal{E}_{\mu,r}.\) Therefore, \(\|\varphi ^{{\ast}}\|_{\mu,r} <\infty.\) We set \(C_{0} =\|\varphi ^{{\ast}}\|_{\mu,r}\) in (9.30). One can easily see that by definition of \(\varphi ^{{\ast}},\) the expression \(\langle h,\varphi ^{{\ast}}\rangle\) is a real positive number. Hence, we have
$$\displaystyle{\frac{\vert \langle h,\varphi ^{{\ast}}\rangle \vert } {\|\varphi ^{{\ast}}\|_{\mu,r}} \geq \frac{\langle h,\varphi ^{{\ast}}\rangle } {\|\varphi ^{{\ast}}\|_{\mu,r}} =\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \langle h,z^{\alpha }\rangle \vert,}$$
completing the proof.

Proposition 9.5.

Let μ > 0. Linear combinations of quasi-polynomials \(z^{\alpha }e^{\zeta z},\) where |α|≤μ and \(\vert \zeta \vert \leq r,\) form a dense set in \(\mathcal{E}_{\mu,r}.\)

Proof.

Let h be an arbitrary element in \(\mathcal{E}_{\mu,r}^{{\ast}}.\) Assume that \(\langle h,z^{\alpha }e^{\zeta z}\rangle = 0\) for all α, | α ≤ μ | and \(\zeta,\,\vert \zeta \vert = \vert \zeta _{1}\vert +\ldots +\vert \zeta _{n}\vert \leq r.\) To prove the statement we have to show that h = 0. Let, first, μ = 0, i.e., \(\langle h,e^{\zeta z}\rangle = 0\) for all \(\zeta,\,\vert \zeta \vert \leq r.\) Then, in accordance with Proposition 9.4, we have
$$\displaystyle{\|h\|_{0,r}^{{\ast}} =\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{ 0,r}(\alpha )\vert \langle h,z^{\alpha }\rangle \vert =\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{ 0,r}(\alpha )\Big\vert D_{\zeta }^{\alpha }\langle h,e^{\zeta z}\rangle _{ \vert _{\zeta =0}}\Big\vert = 0,}$$
which implies h = 0. If μ > 0, then obviously, \(g_{\alpha } =\bar{ z}^{\alpha }h \in \mathcal{E}_{\mu -\vert \alpha \vert,r}^{{\ast}}.\) In particular, when | α | = μ, the functional \(g_{\alpha } \in \mathcal{E}_{0,r}^{{\ast}}.\) Therefore,
$$\displaystyle{0 =\langle h,z^{\alpha }e^{\zeta z}\rangle =\langle \bar{ z}^{\alpha }h,e^{\zeta z}\rangle =\langle g_{\alpha },e^{\zeta z}\rangle,\,\forall \zeta:\, \vert \zeta \vert \leq r,}$$
implies that g α  = 0. Hence, \(h =\sum _{\vert \alpha \vert \leq \mu }a_{\alpha }D^{\alpha }\delta (z),\) where a α are complex constants and δ is the Dirac delta function. Since, in particular h vanishes at monomials z β , | β | ≤ μ, in fact, a α  = 0, | α | ≤ μ. Thus, h = 0. 

Remark 9.1.

Proposition 9.5 is not valid if either the condition | α | ≤ μ or \(\vert \zeta \vert \leq r\) is replaced by | α | < μ or \(\vert \zeta \vert <r,\) respectively. Indeed, assuming n = 1 and μ = 1, one can prove this claim showing that the function \(\varphi (z) = (1 + z)e^{rz}\) cannot be approximated in \(\mathcal{E}_{\mu,r}\) by linear combinations of quasi-polynomials \(z^{k}e^{\zeta z},k = 0,1,\vert \zeta \vert <r,\) or exponentials \(e^{\zeta z},\,\vert \zeta \vert \leq r.\) We note also that linear combinations \(\{e^{\zeta z}\},\,\vert \zeta \vert \leq r,\) form a dense set in \(\mathcal{E}_{\mu,r}\) if μ ≤ 0 due to embedding (9.19).

Further, introduce a Banach space \(\mathcal{O}_{\mu,r}\) of functions analytic on the polydisc \(U_{r} =\{\zeta \in \mathbb{C}^{n}: \vert \zeta _{j}\vert <r,\,j = 1,\ldots,n\},\) with the norm
$$\displaystyle{[\phi ]_{\mu,r} =\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \phi _{\alpha }\vert,}$$
where \(\phi _{\alpha } = D^{\alpha }\phi (0),\,\vert \alpha \vert = 0,1,\ldots.\) With each function \(\phi \in \mathcal{O}_{\mu,r}\) it is associated a differential operator of infinite order defined as
$$\displaystyle{ \varPhi (D)\varphi (z) =\sum _{ \vert \alpha \vert =0}^{\infty }\frac{\phi _{\alpha }} {\alpha !}D_{z}^{\alpha }\varphi (z). }$$
(9.31)

Proposition 9.6.

The operator \(\varPhi (D)\) associated with the function \(\phi \in \mathcal{O}_{\mu,r}\) maps continuously the space \(\mathcal{E}_{\mu,r}\) into itself. Moreover, for any \(\varphi \in \mathcal{E}_{\mu,r}\) the inequality
$$\displaystyle{ \|\varPhi (D)\varphi (z)\|_{\mu,r} \leq [\phi ]_{\mu,r}\|\varphi \|_{\mu,r} }$$
(9.32)
holds.

Proof.

Let \(\varphi \in \mathcal{E}_{\mu,r}.\) Then using Proposition 9.2, we obtain
$$\displaystyle{\vert \varPhi (D)\varphi (z)\vert \leq \sum _{\vert \alpha \vert =0}^{\infty }\vert \phi _{\alpha }\vert \frac{\vert D_{z}^{\alpha }\varphi (z)\vert } {\alpha !} \leq \|\varphi \|_{\mu,r}(1 + \vert z\vert )^{\mu }e^{r\vert z\vert }\sum _{ \vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \phi _{\alpha }\vert.}$$
This immediately implies inequality (9.32).
By duality, one can define a differential operator of infinite order \(\varPhi (-D)\) associated with the function \(\phi \in \mathcal{O}_{\mu,r}\) in the space \(\mathcal{E}_{\mu,r}^{{\ast}},\) as well. Namely, for \(h \in \mathcal{E}_{\mu,r}^{{\ast}},\) by definition,
$$\displaystyle{\langle \varPhi (-D)h,\varphi \rangle =\langle h,\varPhi (D)\varphi \rangle,\quad \forall \varphi \in \varphi \in \mathcal{E}_{\mu,r}.}$$

Proposition 9.7.

The operator \(\varPhi (D)\) associated with the function \(\phi \in \mathcal{O}_{\mu,r}\) maps continuously the space \(\mathcal{E}_{\mu,r}^{{\ast}}\) into itself. Moreover, for \(h \in \mathcal{E}_{\mu,r}^{{\ast}}\) the inequality
$$\displaystyle{ \|\varPhi (-D)h(z)\|_{\mu,r}^{{\ast}}\leq [\phi ]_{\mu,r}\|h\|_{\mu,r}^{{\ast}} }$$
(9.33)
holds.

Proof.

Let \(\varphi \in \mathcal{E}_{\mu,r}\) be an arbitrary function in \(\mathcal{E}_{\mu,r}.\) Using (9.32), we obtain
$$\displaystyle\begin{array}{rcl} \vert \langle \varPhi (-D)h(z),\varphi (z)\rangle \vert & =& \vert \langle h(z),\varPhi (D)\varphi (z)\rangle \vert \leq \| h\|_{\mu,r}^{{\ast}}\|\varPhi (D)\varphi \| {}\\ & \leq & [\phi ]_{\mu,r}\|h\|_{\mu,r}^{{\ast}}\|\varphi \|_{ \mu,r}. {}\\ \end{array}$$
This immediately implies inequality (9.33).

9.3 Complex Fourier transform

Now we define a complex Fourier transform F for functions of the space \(\mathcal{O}_{\mu,r}.\)

Definition 9.1.

The Fourier transform of a function \(\phi \in \mathcal{O}_{\mu,r}\) is
$$\displaystyle{ F[\phi ](\zeta ) = (2\pi )^{n}\varPhi (-D_{\zeta })\delta (\zeta ). }$$
(9.34)
That is the Fourier transform of \(\phi\) is the value of the differential operator of infinite order \((2\pi )^{n}\varPhi (-D),\) associated with \((2\pi )^{n}\phi,\) at the Dirac delta function. It follows from Definition 9.1 and Proposition 9.7 that the mapping
$$\displaystyle{ F: \mathcal{O}_{\mu,r} \rightarrow \mathcal{E}_{\mu,r}^{{\ast}} }$$
(9.35)
is continuous. Let \(\varphi \in \mathcal{E}_{\mu,r}.\) Using the definition (9.31) of \(\varPhi (D),\) for arbitrary \(\phi \in \mathcal{O}_{\mu,r}\) we have
$$\displaystyle\begin{array}{rcl} \langle F[\phi ](\zeta ),\varphi (\zeta )\rangle & =& (2\pi )^{n}\langle \delta (\zeta ),\varPhi (D)\varphi (\zeta )\rangle = (2\pi )^{n}\sum _{ \vert \alpha \vert =0}^{\infty }\frac{\phi _{\alpha }} {\alpha !}\langle \delta (\zeta ),D_{\zeta }^{\alpha }\varphi (\zeta )\rangle \\ & =& (2\pi )^{n}\sum _{ \vert \alpha \vert =0}^{\infty }\frac{1} {\alpha !} D_{z}^{\alpha }\phi (0)D_{\zeta }^{\alpha }\varphi (0). {}\end{array}$$
(9.36)
One can derive useful implications from this representation. Namely, due to estimate (9.23) (see Corollary 9.1) it follows from (9.36) that
$$\displaystyle{\vert \langle F[\phi ](\zeta ),\varphi (\zeta )\rangle \vert \leq (2\pi )^{n}\|\varphi \|_{ \mu,r}\sum _{\vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \phi _{\alpha }\vert = (2\pi )^{n}[\phi ]_{\mu,r}\|\varphi \|_{\mu,r},}$$
or
$$\displaystyle{ \|F[\phi ]\|_{\mu,r}^{{\ast}}\leq (2\pi )^{n}[\phi ]_{\mu,r}. }$$
Another implication from representation (9.36) is a formula for the inverse Fourier transform F−1. Namely, taking \(\varphi (\zeta ) = e^{z\zeta },\) where \(z\zeta = z_{1}\zeta _{1} +\ldots +z_{n}\zeta _{n},\) we have
$$\displaystyle{\langle F[\phi ](\zeta ),e^{z\zeta }\rangle = (2\pi )^{n}\sum _{ \vert \alpha \vert =0}^{\infty }\frac{D_{z}^{\alpha }\phi (0)} {\alpha !} z^{\alpha } = (2\pi )^{n}\phi (z).}$$
Rewriting the latter in the form
$$\displaystyle{ \phi (z) = F^{-1}[F[\phi ]](z) = \frac{1} {(2\pi )^{n}}\langle F[\phi ](\zeta ),e^{z\zeta }\rangle, }$$
(9.37)
we can see that the inverse Fourier transform coincides with the known Fourier-Laplace transform. This formula implies the following two important formulas:
$$\displaystyle\begin{array}{rcl} F[D_{z}^{\alpha }\phi ](\zeta ) =\zeta ^{\alpha }F[\phi ](\zeta ),& &{}\end{array}$$
(9.38)
$$\displaystyle\begin{array}{rcl} F[(-z)^{\alpha }\phi (z)](\zeta ) = D_{\zeta }^{\alpha }F[\phi ](\zeta ).& &{}\end{array}$$
(9.39)
Further, differentiating (9.37), we have \(D_{z}^{\alpha }\phi (0) =\phi _{\alpha } = (2\pi )^{-n}\langle F[\phi ](\zeta ),\zeta ^{\alpha }\rangle.\) Using this fact and Proposition 9.4, we obtain
$$\displaystyle{[\phi ]_{\mu,r} =\sum _{ \vert \alpha \vert =0}^{\infty }\vert \phi _{\alpha }\vert \mathcal{K}_{\mu,r}(\alpha ) = \frac{1} {(2\pi )^{n}}\sum _{\vert \alpha \vert =0}^{\infty }\mathcal{K}_{\mu,r}(\alpha )\vert \langle F[\phi ](\zeta ),\zeta ^{\alpha }\rangle \vert = \frac{1} {(2\pi )^{n}}\|F[\phi ]\|_{\mu,r}^{{\ast}}.}$$
This equality expresses a complex analog of the Parseval’s equality (Theorem  1.3) of the Fourier transform acting in \(L_{2}(\mathbb{R}^{n}).\) Summarizing, we have proved the following statement.

Theorem 9.1.

The Fourier transform operator \(F: \mathcal{O}_{\mu,r} \rightarrow \mathcal{E}_{\mu,r}^{{\ast}}\) is isometric isomorphism. The inversion formula is given in equation (9.37) .

The representation for the Fourier transform obtained in equation (9.36) is symmetric with respect to \(\phi \in \mathcal{O}_{\mu,r}\) and \(\varphi \in \mathcal{E}_{\mu,r}.\) Therefore, with an appropriate interpretation of the definition of the Fourier transform, similar to Theorem 9.1, one can prove

Theorem 9.2.

The Fourier transform operator \(F: \mathcal{E}_{\mu,r} \rightarrow \mathcal{O}_{\mu,r}^{{\ast}}\) is isometric isomorphism. The inversion formula is again given in equation (9.37) .

Theorems 9.1 and 9.2 imply the following corollary.

Corollary 9.2.

The following commutative diagram holds:
$$\displaystyle{\begin{array}{ccc} \quad \ \mathcal{E}_{\mu,r} &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{E}_{\mu,r}^{{\ast}}\\ & & \\ F^{-1} \uparrow \,\downarrow F & &F^{-1} \downarrow \,\uparrow F\\ & & \\ \quad \ \mathcal{O}_{\mu,r}^{{\ast}} &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{O}_{\mu,r}\\ \end{array} }$$
where symbols “ \(\mathop{\leftrightarrow }\limits^{{\ast}}\) ,” “ \(\mathop{\rightarrow }\limits^{ F^{-1}}\) ,” and “ \(\mathop{\rightarrow }\limits^{ F}\) ” stand for the passage to conjugate, the inverse Fourier transform, and the Fourier transform, respectively.
Now, when the Fourier transform F is defined on \(\mathcal{E}_{\mu,r}\) as well, we note that representation (9.36) can also be interpreted as the Parseval equality
$$\displaystyle{\langle F[\phi ](\zeta ),\varphi (\zeta )\rangle =\langle \phi (\zeta ),F[\varphi ](\zeta )\rangle,\quad \phi \in \mathcal{O}_{\mu,r},\,\varphi \in \mathcal{E}_{\mu,r}.}$$

Proposition 9.8.

Let μ > 0. A function f(z) analytic on the polydisc U r belongs to \(\mathcal{O}_{-\mu,r}\) if and only if it satisfies the inequality
$$\displaystyle{ \vert f(z)\vert \leq \frac{M} {(r -\vert z\vert )^{\mu -1}},\quad z \in U_{r}, }$$
(9.40)
where M > 0 is a constant.

Proof.

For the sake of simplicity we show this fact for n = 1. Let \(f \in \mathcal{O}(U_{r})\) satisfy the estimate
$$\displaystyle{ \vert f(z)\vert \leq \frac{M} {(r -\vert z\vert )^{\nu }},\quad z \in U_{r}, }$$
with a positive integer ν. Without loss of generality one can assume that \(f(z) = \frac{\phi (z)} {(r-z)^{\mu -1}} +\psi (z),\) where \(\phi (z)\) is regular at z = r and a singularity of \(\psi (z)\) at z = r is weaker than the first term of the above representation of f. Then one can easily verify that
$$\displaystyle{ \vert f_{\alpha }\vert = \vert D_{z}^{\alpha }f(0)\vert \sim \frac{(\alpha +\nu - 1)!} {(\nu -1)!} \frac{M} {r^{\alpha +\nu }},\quad \alpha \rightarrow \infty. }$$
(9.41)
Using inequality (9.27) and the Stirling formula, we have
$$\displaystyle\begin{array}{rcl} [f]_{-\mu,r}& =& \sum _{\alpha =0}^{\infty }\mathcal{K}_{ -\mu,r}(\alpha )\vert f_{\alpha }\vert \sim C\sum _{\alpha =0}^{\infty }\mathcal{K}_{ -\mu,r}(\alpha ) \frac{(\alpha +\nu - 1)!} {(\nu -1)!r^{\alpha +1}} {}\\ & \leq & C\sum _{\alpha =0}^{\infty }\frac{\alpha !\alpha ^{\nu -1}(er)^{\alpha }} {\alpha ^{\mu }\alpha ^{\alpha }r^{\alpha }} \leq C\sum _{\alpha =0}^{\infty } \frac{1} {\alpha ^{\mu -\nu + 1 2 }} <\infty, {}\\ \end{array}$$
if ν ≤ μ − 1. Hence, \(f \in \mathcal{O}_{-\mu,r},\) if \(f \in \mathcal{O}(U_{r})\) and satisfies condition (9.40).
Further, if \(\vert f(z)\vert> \frac{M} {(r-\vert z\vert )^{\mu }}\) near the boundary of the disc | z | < r, then using the asymptotic relations
$$\displaystyle{\mathcal{K}_{-\mu,r}(\alpha ) \sim \frac{(er)^{\alpha }} {\alpha ^{\mu }\alpha ^{\alpha }},\quad \alpha ! \sim \left ( \frac{\alpha } {e}\right )^{\alpha }\sqrt{2\pi \alpha },}$$
when \(\alpha \rightarrow \infty,\) and (9.41), one obtains \([f]_{-\mu,r} = \infty,\) that is \(f\notin \mathcal{O}_{-\mu,r}.\)

Remark 9.2.

Dubinskii denoted the class of functions \(f \in \mathcal{O}(U_{r})\) satisfying estimate (9.40) by \(\mathcal{D}_{\mu -1,r};\) see [Dub90]. Proposition 9.8 immediately implies that the space \(\mathcal{D}_{\mu -1,r},\,\mu> 0,\) is isomorphic to the space \(\mathcal{O}_{-\mu,r}.\) Therefore, it follows from Corollary 9.2 that
$$\displaystyle{\begin{array}{ccc} \quad \ \mathcal{D}_{\mu -1,r} &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{D}_{\mu -1,r}^{{\ast}}\\ && \\ F^{-1} \uparrow \,\downarrow F & &F^{-1} \downarrow \,\uparrow F\\ && \\ \quad \ \mathcal{E}_{-\mu,r}^{{\ast}} &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{E}_{-\mu,r},\\ \end{array} }$$
where \(\mathcal{D}_{\mu -1,r}^{{\ast}}\) is the dual space to \(\mathcal{D}_{\mu -1,r}.\)

9.4 Complex Fourier transform in fiber spaces of analytic and exponential functions and functionals

In this section we introduce fiber spaces of exponential and holomorphic functions and locally convex topological vector spaces and extend the Fourier transform introduced in the previous section to these spaces. Note that these spaces will serve as solution spaces for differential and pseudo-differential equations with complex variables.

Suppose Ω is a connected domain (or connected manifold) in \(\mathbb{C}^{n}\) and let \(\zeta \in \varOmega.\) Let μ and r be a nonnegative and positive real numbers, respectively, such that \(r <dist(\zeta, \mathbb{C}^{n}\setminus \varOmega )\). Denote by \(\mathcal{E}_{\mu,r}^{\zeta }\) the set of entire functions \(\varphi (z) = e^{z\zeta }v(z),\,v \in \mathcal{E}_{\mu,r},\) where the Banach space \(\mathcal{E}_{\mu,r}\) was defined in the previous section. The space \(\mathcal{E}_{\mu,r}^{\zeta }\) is also a Banach space with the norm \(\|\varphi \|_{\mu,r,\zeta } =\| e^{-z\zeta }\varphi \|_{\mu,r}.\)

Further, we introduce a fiber bundle \((E,\varOmega,\pi ) \equiv E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) with the base Ω and projection
$$\displaystyle{\pi: (E,\varOmega,\pi ) \rightarrow \varOmega,}$$
where \(\pi ^{-1}(\zeta ) = \mathcal{E}_{\mu,r}^{\zeta },\ \zeta \in \varOmega\). It follows from this definition that the fibers \(E_{\mu,r,\zeta } = exp(z\zeta )\mathcal{E}_{\mu,r}^{\zeta }\) are Banach spaces with the respective norms \(\|\varphi \|_{\mu,r,\zeta } =\| exp(-z\zeta )\varphi \|_{\mu,r}\). It is obvious that \(\pi ^{-1}(\zeta _{1})\) and \(\pi ^{-1}(\zeta _{2})\) are isomorphisms for arbitrary \(\zeta _{1},\zeta _{2} \in \varOmega\). The dual space to \(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) is also a fiber bundle \((E^{{\ast}},-\varOmega,\pi _{{\ast}}) \equiv \Big (E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}}\) with fibers \(E_{\mu,r,\zeta }^{{\ast}}\equiv \left (E_{\mu,r,\zeta }\right )^{{\ast}},\) with the base −Ω, and with the projection
$$\displaystyle{\pi _{{\ast}}^{-1}: (E^{{\ast}},-\varOmega,\pi _{ {\ast}}) \rightarrow -\varOmega.}$$
Schematically the relationship between introduced fiber bundles can be represented as
$$\displaystyle{\begin{array}{ccc} E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})&\mathop{\longleftrightarrow }\limits^{{\ast}}&\Big(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}} \\ & & \\ \pi ^{-1} \nwarrow & & \nearrow \pi _{{\ast}}^{-1}\\ &&\\ &\varOmega &\\ \end{array} }$$
Since fibers in the above constructions are endowed with norms, one can introduce the structure of convergence. Namely, we say that a sequence \(\varphi _{m} \in E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) converges to \(\varphi _{0} \in E_{\mu,r}^{\varOmega }(\mathbb{C}^{n}),\) if for arbitrary \(\zeta \in \varOmega\) we have \(\varphi _{m} \rightarrow \varphi _{0}\) as \(m \rightarrow \infty\) in \(\pi ^{-1}(\zeta ) = E_{\mu,r,\zeta }.\) In the dual space we introduce the weak convergence: a sequence \(\varPhi _{n} \in \left (E_{\mu,r,\zeta }\right )^{{\ast}}\) converges weakly to \(\varPhi _{0} \in \left (E_{\mu,r,\zeta }\right )^{{\ast}},\) if for arbitrary \(\zeta \in \varOmega\) we have \(\varPhi _{m} \rightarrow \varPhi _{0}\) as \(m \rightarrow \infty\) in \(\pi _{{\ast}}^{-1}(\zeta ) = E_{\mu,r,\zeta }^{{\ast}}.\)
Similarly, let \((\mathcal{O},\varOmega,\tau ) \equiv \mathcal{O}_{\mu,r}(\varOmega )\) be a fiber bundle with fibers \(\mathcal{O}_{\mu,r,\zeta }\) with the base Ω and the projection \(\tau: (\mathcal{O},\varOmega,\tau ) \rightarrow \varOmega,\) where \(\tau ^{-1}(\zeta ) = \mathcal{O}_{\mu,r,\zeta },\ \zeta \in \varOmega.\) The space \(\mathcal{O}_{\mu,r,\zeta },\,\zeta \in \varOmega,\) is a Banach space of analytic functions \(\phi (z)\) defined on the polydisc with the center at \(\zeta\) and “poly-radius” \((r,\ldots,r),\) i.e., \(U_{r}(\zeta ) =\{ z \in \mathbb{C}^{n}: \vert z_{j} -\zeta _{j}\vert <r,\,j = 1,\ldots,n\},\) with the norm
$$\displaystyle{[\phi ]_{\mu,r,\zeta } =\sum _{ \vert \alpha \vert =0}^{\infty }\vert \phi _{\alpha }(\zeta )\vert \mathcal{K}_{\mu,r}(\alpha ),\quad \phi _{\alpha }(\zeta ) = D^{\alpha }\phi (\zeta ).}$$
The dual space to \(\mathcal{O}_{\mu,r}(\varOmega )\) is also a fiber bundle \((\mathcal{O}^{{\ast}},-\varOmega,\pi _{{\ast}}) \equiv \mathcal{O}_{\mu,r}^{{\ast}}(\varOmega )\) with fibers \(\mathcal{O}_{\mu,r,\zeta }^{{\ast}}\equiv \left (\mathcal{O}_{\mu,r,\zeta }\right )^{{\ast}},\) with the base −Ω, and with the projection
$$\displaystyle{\tau _{{\ast}}^{-1}: (\mathcal{O}^{{\ast}},-\varOmega,\pi _{ {\ast}}) \rightarrow -\varOmega.}$$
Schematically the relationship between these fiber bundles can be represented as
$$\displaystyle{\begin{array}{ccc} \mathcal{O}_{\mu,r}(\varOmega ) &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{O}_{\mu,r}^{{\ast}}(\varOmega )\\ && \\ \tau ^{-1} \nwarrow & &\nearrow \tau _{{\ast}}^{-1}\\ &&\\ &\varOmega &\\ \end{array} }$$
Since fibers in these constructions are endowed with norms, one can introduce the structure of convergence. Namely, a sequence \(h_{m} \in \mathcal{O}_{\mu,r}(\varOmega )\) converges to \(h_{0} \in \mathcal{O}_{\mu,r}(\varOmega ),\) if for arbitrary \(\zeta \in \varOmega\) we have \(h_{m} \rightarrow h_{0}\) as \(m \rightarrow \infty\) in \(\tau ^{-1}(\zeta ) = \mathcal{O}_{\mu,r,\zeta }.\) In the dual space we introduce the weak convergence: a sequence \(H_{n} \in \mathcal{O}_{\mu,r}^{{\ast}}(\varOmega )\) converges weakly to \(f_{n} \in \mathcal{O}_{\mu,r}^{{\ast}}(\varOmega ),\) if for arbitrary \(\zeta \in \varOmega\) we have \(H_{m} \rightarrow H_{0}\) as \(m \rightarrow \infty\) in \(\tau _{{\ast}}^{-1}(\zeta ) = \mathcal{O}_{\mu,r,\zeta }^{{\ast}}.\)
Finally we introduce the Fourier transform on the spaces \(\mathcal{O}_{\mu,r}(\varOmega )\) and \(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) as the Fourier transform defined fiberwise, i.e., on fibers \(\{\tau ^{-1}(\zeta ),\,\zeta \in \varOmega \}\) and \(\{\pi ^{-1}(\zeta ),\,\zeta \in \varOmega \},\) respectively. For this purpose we need the following isomorphisms:
$$\displaystyle\begin{array}{rcl} & & f_{\zeta }: \mathcal{O}_{\mu,r,\zeta } \rightarrow \mathcal{O}_{\mu,r},\quad f_{\zeta }^{{\ast}}: \mathcal{O}_{\mu,r,\zeta }^{{\ast}}\rightarrow \mathcal{O}_{\mu,r}^{{\ast}}, {}\\ & & g_{\zeta }: E_{\mu,r,\zeta } \rightarrow E_{\mu,r},\quad g_{\zeta }^{{\ast}}: E_{\mu,r,\zeta }^{{\ast}}\rightarrow E_{\mu,r}^{{\ast}}. {}\\ \end{array}$$
By definition, the Fourier transform in the fiber space \(\mathcal{O}_{\mu,r}(\varOmega )\) is the family of mappings \(F_{\zeta _{0}}: \mathcal{O}_{\mu,r,\zeta _{0}} \rightarrow E_{\mu,r,\zeta _{0}}^{{\ast}},\zeta _{0} \in \varOmega,\) defined as
$$\displaystyle{F[h](\zeta ) = g_{\zeta _{0}}^{-1} \circ F[f_{\zeta _{ 0}} \circ h](\zeta ),\quad \zeta _{0} \in \varOmega,}$$
where \(F[f_{\zeta _{0}} \circ h]\) is the Fourier transform given in equation (9.34). Similarly, the Fourier transform in the space \(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) is the family of mappings \(F_{\zeta _{0}}: E_{\mu,r,\zeta _{0}} \rightarrow \mathcal{O}_{\mu,r,\zeta _{0}}^{{\ast}},\) \(\zeta _{0} \in \varOmega,\) defined as
$$\displaystyle{F[\varphi ](\zeta ) = f_{\zeta _{0}}^{-1} \circ F[g_{\zeta _{ 0}}\circ \varphi ](\zeta ),\quad \zeta _{0} \in \varOmega.}$$
For simplicity the Fourier transform in both fiber spaces \(\mathcal{O}_{\mu,r}(\varOmega )\) and \(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\) will also be denoted by the same letter F. The following statement follows immediately from Corollary 9.2.

Theorem 9.3.

The following commutative diagram holds:
$$\displaystyle{\begin{array}{ccc} E_{\mu,r}^{\varOmega }(\mathbb{C}^{n}) &\mathop{\longleftrightarrow }\limits^{{\ast}}&\Big(E_{\mu,r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}}\\ && \\ F^{-1} \uparrow \,\downarrow F & & F^{-1} \downarrow \,\uparrow F\\ && \\ \mathcal{O}_{\mu,r}^{{\ast}}(\varOmega ) &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{O}_{\mu,r}(\varOmega )\\ \end{array} }$$

Remark 9.3.

Dubinskii [Dub84] introduced the space \(Exp_{\varOmega }(\mathbb{C}^{n})\) defined as an inductive limit of Banach spaces composed with the help of \(E_{\mu,r,\zeta }.\) This space and its dual were used as solution spaces for the Cauchy problem for pseudo-differential equations with holomorphic symbols. The Fourier transform in \(Exp_{\varOmega }(\mathbb{C}^{n})\) is defined by the formula \(F[f](\zeta ) = (2\pi )^{n}f(-D_{z})\delta _{0}(\zeta ),\) which maps \(Exp_{\varOmega }(\mathbb{C}^{n})\) onto the space \(\mathcal{O}^{{\ast}}(\varOmega )\) of analytic functionals concentrated on compact sets in Ω. There is a relationship similar to commutative diagram in Theorem 9.3:
$$\displaystyle{\begin{array}{ccc} Exp_{\varOmega }(\mathbb{C}^{n}) &\mathop{\longleftrightarrow }\limits^{{\ast}}&Exp_{\varOmega }^{{\ast}}(\mathbb{C}^{n})\\ && \\ F^{-1} \uparrow \,\downarrow F & & F^{-1} \downarrow \,\uparrow F\\ && \\ \mathcal{O}^{{\ast}}(\varOmega ) &\mathop{\longleftrightarrow }\limits^{{\ast}}& \mathcal{O}(\varOmega ).\\ \end{array} }$$
For details we refer the reader to [Dub84, Dub90].

9.5 An algebra of matrix-symbols with singularities

In Sections 9.6 and 9.7 we will use linear differential operators of the form
$$\displaystyle{L(z,D_{z}) =\sum _{\vert \alpha \vert \leq m}a_{\alpha }(z)D_{z}^{\alpha },}$$
with meromorphic coefficients a α (z), | α | ≤ m, and pseudo-differential operators of the form
$$\displaystyle{L(D_{z},z) =\sum _{\vert \alpha \vert \leq m}z^{\alpha }a_{\alpha }(D_{z}),}$$
with meromorphic symbols a α (z) defined in a domain \(G \subset \mathbb{C}^{n}.\) Therefore, in this section we will study the symbols of the form
$$\displaystyle{a(z,\zeta ) =\sum _{\vert \alpha \vert \leq m}z^{\alpha }a_{\alpha }(\zeta ).}$$

By definition, a symbol of degree m is an ordered collection \(a \equiv \{ a_{\alpha }(\zeta )\}_{\vert \alpha \vert \leq m}\) of functions a α from some space X, which is specified below. The class of symbols of degree m is denoted by S(m, X). Note that if m is a degree of the symbol a, then m + k for an arbitrary nonnegative integer k is also a degree. The least degree m for which there is α, | α | = m, such that \(a_{\alpha }(\zeta )\neq 0,\) but a α  = 0 for all | α | > m is called the exact degree of the symbol a, and denoted \(\deg (a).\) The identity symbol j ∈ S(0, X) is the symbol with \(a_{0}(\zeta ) \equiv 1.\) For the zero-symbol \(\theta\) the functions \(a_{\alpha } \equiv 0\) for all | α | ≤ m. We use the following convention: if \(\deg (a) <0,\) then we accept a = 0 and write \(\deg (a) = -\infty.\) By this convention \(\deg (\theta ) = -\infty,\) and \(\theta \in S(-\infty,X).\)

The sum a + b and product \(a \circ b\) of two symbols a ∈ S(m1, X) and b ∈ S(m2, X) are, respectively, defined by
$$\displaystyle{ a + b \equiv \{ a_{\alpha }(\zeta ) + b_{\alpha }(\zeta )\}_{\vert \alpha \vert \leq \max (m_{1},m_{2})}, }$$
(9.42)
and
$$\displaystyle{ a\circ b \equiv \Big\{\sum _{\begin{array}{c} \vert \gamma \vert \leq m_{1} \\ \gamma \preccurlyeq \alpha \end{array} }\sum _{\begin{array}{c} \vert \beta \vert \leq m_{2} \\ \beta \succcurlyeq \alpha -\gamma \end{array} }{\beta \choose \beta +\gamma -\alpha }\,b_{\beta }(\zeta )\,D_{\zeta }^{\beta +\gamma -\alpha }a_{\gamma }(\zeta )\Big\}_{ \vert \alpha \vert \leq m_{1}+m_{2}}, }$$
(9.43)
where \(\gamma \preccurlyeq \alpha\) means \(\gamma _{j} \leq \alpha _{j},\,j = 1,\ldots,n,\) and \({\sigma \choose \delta }\) for multi-indices \(\sigma\) and δ means
$$\displaystyle{{\sigma \choose \delta } =\prod _{ j=1}^{n} \frac{\sigma _{j}!} {\delta _{j}!(\sigma _{j} -\delta _{j})!}.}$$
The formula (9.43) for composition of two symbols follows from the Leibniz rule for pseudo-differential operators. It is easy to see that, in general, \(a \circ b\neq b \circ a.\) Indeed, for \(a =\{ a_{0}(\zeta ),a_{1}(\zeta )\} =\{ 0,\zeta \}\) with m1 = 1 and \(b =\{ b_{0}(\zeta )\} =\{\zeta \}\) with m2 = 0 formula (9.43) implies that
$$\displaystyle{a \circ b =\{ b_{0}(\zeta )a_{0}(\zeta ),b_{0}(\zeta )a_{1}(\zeta )\} =\{ 0,\zeta ^{2}\},}$$
while
$$\displaystyle{b \circ a =\{ a_{0}(\zeta )b_{0}(\zeta ) + a_{1}(\zeta )b_{0}^{{\prime}}(\zeta ),a_{ 1}(\zeta )b_{0}(\zeta )\} =\{\zeta,\zeta ^{2}\}.}$$
Hence, the product of two symbols is not a commutative operation. The following properties of symbols immediately follow from the above definitions of the sum and the product of symbols.

Proposition 9.9.

Let a ∈ S(m 1 ,X) and b ∈ S(m 2 ,X). Then
  1. 1.

    \(\deg (a + b) =\max \{\deg (a),\deg (b)\};\)

     
  2. 2.

    \(\deg (a \circ b) =\deg (a) +\deg (b);\)

     
  3. 3.

    \(a+\theta =\theta +a = a;\)

     
  4. 4.

    \(a \circ j = j \circ a = a.\)

     

Let \(\varOmega \subset C^{n}\) be an n-dimensional complex domain and \(\mathcal{M}(\varOmega )\) and \(\mathcal{O}(\varOmega )\) be sheaf of germs of meromorphic and holomorphic functions, respectively. We assume that \(f \in \mathcal{M}(\varOmega )\) has a local representation \(f(z) = g(z)/h(z),\) where \(g,h \in \mathcal{O}(\varOmega ),\) in a neighborhood of any point of Ω. We denote by P f and N f the set of poles \(P_{f} =\{ z \in \varOmega: h(z) = 0\},\) and the set of zeros (or null-set) \(N_{f} =\{ z \in \varOmega: g(z) = 0\}\) of the meromorphic function f. 

Let M be an \(N \times N\)-matrix whose entries \(m_{i,j},\,i,j = 1,\ldots,N,\) are allowed degrees of symbols (i.e., nonnegative integers, or \(-\infty\)). We denote by \(S(M,\mathcal{M}(\varOmega ))\) the set of \(N \times N\) matrix-valued symbols with entries
$$\displaystyle{a_{ij}(z,\zeta ) =\sum _{\vert \alpha \vert \leq m_{ij}}z^{\alpha }a_{ij,\alpha }(\zeta ),\quad i,j = 1,\mathop{\ldots },N,}$$
where z ∈ C n , \(\zeta \in \varOmega\), and \(a_{ij,\alpha } \in \mathcal{M}(\varOmega ),\,\vert \alpha \vert \leq m_{ij},\,i,j = 1,\mathop{\ldots },N.\) Introduce the following analytic sets of co-dimension 1:
$$\displaystyle\begin{array}{rcl} P_{j}& =& \bigcup _{i=1}^{N}\left (\bigcup _{ \vert \alpha \vert \leq m_{ij}}P_{a_{{\alpha ij}}}\right ), {}\\ Q_{j}& =& (\bigcup _{k=1}^{j}N_{ a_{{0kk}}}) \cup \left (\bigcup _{1\leq k\leq j-1}\left (\bigcup _{k+1\leq l\leq j}(\bigcup _{\vert \alpha \vert \leq m_{kl}}P_{a_{{\alpha kl}}})\right )\right ). {}\\ \end{array}$$
Let \(A_{0}(\zeta )\) be the constant part of \(A(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\), i.e., the matrix \(A_{0}(\zeta ) = (a_{ij,0}(\zeta ))_{i,j=1}^{N}\) and let \(\varDelta (\zeta )\) be its determinant \(\varDelta (\zeta ) =\det A_{0}(\zeta ).\) Obviously \(\varDelta (\zeta )\) is also meromorphic and let the following local representation hold:
$$\displaystyle{ \varDelta (\zeta ) ={ G(\zeta ) \over H(\zeta )},\quad G,H \in \mathcal{O}(\varOmega ). }$$
(9.44)
We call the set \(P_{A} =\{\zeta \in \varOmega: H(\zeta = 0)\}\) a polar set and the set \(N_{A} =\{\zeta \in \varOmega: G(\zeta ) = 0\}\) a null set of the matrix symbol \(A(z,\zeta ).\) It follows from general theory of determinants that for the inverse matrix \(A_{0}^{-1}(\zeta )\) one has a local representation \(\det (A^{-1}(\zeta )) = H(\zeta )/G(\zeta )\), and therefore, \(P_{A^{-1}} = N_{A}\) and \(N_{A^{-1}} = P_{A}.\) Further, we introduce the sets:
$$\displaystyle{Z(A) = P_{A} \cup N_{A},\quad Z_{reg}(A) = Z(A)\setminus (P_{A} \cap N_{A}),\quad \mbox{ and}\quad Z_{reg,\varOmega }(A) =\varOmega \cap Z_{reg}(A).}$$
It is obvious that these sets are invariant with respect to inversion of the symbol \(A(z,\zeta )\).

Since symbols in \(S(M,\mathcal{M}(\varOmega ))\) have entries \(a_{ij,\alpha } \in S(m_{i,j},\mathcal{M}(\varOmega )),\) one can define the addition, composition, and involution operations in \(S(M,\mathcal{M}(\varOmega ))\) using operations introduced in (9.42) and (9.43).

Theorem 9.4.

A symbol \(A(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) has the inverse \(A^{-1}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) if and only if there exists a collection of integers \(\mu _{1},\ldots,\mu _{N}\) such that the inequalities
$$\displaystyle{ \deg (a_{ij}) \leq \mu _{i} -\mu _{j},\quad i,j = 1,\ldots,N, }$$
(9.45)
hold.

Remark 9.4.

Under the condition of this theorem \(S(M,\mathcal{M}(\varOmega ))\) is a noncommutative involutive algebra.

Proof.

Sufficiency. Let us first assume that the numbers \(\mu _{1},\mathop{\ldots },\mu _{N}\) are strictly ordered in the decreasing order: \(\mu _{1}> \mathop{\ldots }>\mu _{N}.\) Then, due to conditions (9.45), d e g(a i j ) < 0 if i > j, and hence \(a_{ij} =\theta.\) Thus, in this case the symbol \(A(z,\zeta )\) is represented in the form:
$$\displaystyle{A(z,\zeta ) = \left [\begin{array}{*{10}c} a_{11} & a_{12} & \mathop{\ldots }& a_{1N} \\ \theta &a_{22} & \mathop{\ldots }& a_{2N} \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \theta & \theta &\mathop{\ldots }&a_{ NN} \end{array} \right ],}$$
where \(deg(a_{jj}) = 0,\,j = 1,\mathop{\ldots },N,\) and \(a_{ij} \in S(\mu _{i} -\mu _{j},\mathcal{M}(\varOmega ))\) if j > i. Let \(b_{ij},\,i,j = 1,\mathop{\ldots },N,\) be entries of the inverse symbol \(A^{-1}(z,\zeta ).\) The requirement \(A^{-1}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) implies \(b_{ij} =\theta\) if i > j, \(deg(a_{jj}) = 0,\,j = 1,\mathop{\ldots },N,\) and \(b_{ij} \in S(\mu _{i} -\mu _{j},\mathcal{M}(\varOmega ))\) if j > i. This is natural, since the inverse of the right triangular matrix is again a right triangular matrix. The symbols b i j are defined from the system of algebraic equations
$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} a_{jj} \circ b_{jj} = 1, \quad &j = 1,\mathop{\ldots },N, \\ \sum _{k=i}^{j}a_{ik} \circ b_{kj} = 0,\quad &\mbox{ if }i <j, \\ b_{ij} =\theta, \quad &\mbox{ if}\,\,i> j.\end{array} \right. }$$
(9.46)
These equations define all the components of symbols b i j uniquely. Indeed, it follows from (9.46) immediately that
$$\displaystyle{ b_{jj} = 1/a_{jj},\quad j = 1,\mathop{\ldots },N. }$$
Setting \(j = i + 1,\) we have
$$\displaystyle{a_{ii} \circ b_{i\,i+1} + a_{i\,i+1} \circ b_{i+1\,i+1} =\theta,}$$
which implies
$$\displaystyle{ b_{i\,i+1} = - \frac{1} {a_{ii}}[a_{i\,i+1} \circ b_{i+1\,i+1}],\quad i = 1,\mathop{\ldots },N - 1. }$$
Similarly, if all the symbols \(b_{i\,i+\ell-1},\,i = 1,\mathop{\ldots },N -\ell +1,\) are found for some \(1 \leq \ell\leq N - 2,\) then \(b_{i\,i+\ell}\) is defined as
$$\displaystyle{ b_{i\,i+\ell} = - \frac{1} {a_{ii}}\sum _{k=i+1}^{i+\ell}a_{ ik} \circ b_{k\,i+\ell},\quad i = 1,\mathop{\ldots },N -\ell. }$$
Now assume that the numbers \(\mu _{1},\mathop{\ldots },\mu _{N}\) satisfy the ordering \(\mu _{1} =\ldots =\mu _{k_{1}}>\mu _{k_{1}+1} =\ldots =\mu _{k_{2}}>\ldots>\mu _{k_{p}+1} =\ldots =\mu _{N}.\) In this case the symbol \(A(z,\zeta )\) is represented in the block-matrix form:
$$\displaystyle{A(z,\zeta ) = \left [\begin{array}{*{10}c} A_{11} & A_{12} & \mathop{\ldots }&A_{1p} \\ \varTheta &A_{22} & \mathop{\ldots }&A_{2p} \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \varTheta & \varTheta &\mathop{\ldots }&A_{ pp} \end{array} \right ],}$$
where \(k_{1} + \mathop{\ldots } + k_{p} = N,\) \(A_{jj},\,j = 1,\mathop{\ldots },p,\) are \(k_{j} \times k_{j}\)-matrix-symbols in \(S(\varTheta,\mathcal{M}(\varOmega )),\) and A i j , i < j, are \(k_{i} \times k_{j}\)-matrix-symbols that belong to \(S(M,\mathcal{M}(\varOmega ))\) with a degree-matrix M, entries of which are positive numbers. The inverse symbol \(A^{-1}(z,\zeta )\) is of the structure
$$\displaystyle{A^{-1}(z,\zeta ) = \left [\begin{array}{*{10}c} B_{11} & B_{12} & \mathop{\ldots }&B_{1p} \\ \varTheta &B_{22} & \mathop{\ldots }&B_{2p} \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \varTheta & \varTheta &\mathop{\ldots }&B_{ pp} \end{array} \right ],}$$
where the block B i j belongs to the same class of symbols as the corresponding block A i j does. The blocks B i j are defined from the system of algebraic equations
$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} A_{jj} \circ B_{jj} = 1, \quad &j = 1,\mathop{\ldots },p, \\ \sum _{k=i}^{j}A_{ik} \circ B_{kj} = 0,\quad &\mbox{ if }i <j, \\ B_{ij} =\varTheta, \quad &\mbox{ if}\,\,i> j.\end{array} \right. }$$
These equations define all the blocks B i j uniquely. Indeed,
$$\displaystyle{ B_{jj} = A_{jj}^{-1},\quad j = 1,\mathop{\ldots },p, }$$
and if all the blocks \(B_{i\,i+\ell-1},\,i = 1,\mathop{\ldots },p -\ell +1,\) are found for some \(1 \leq \ell\leq p - 2,\) then \(B_{i\,i+\ell}\) is defined as
$$\displaystyle{ B_{i\,i+\ell} = -A_{ii}^{-1}\sum _{ k=i+1}^{i+\ell}A_{ ik} \circ B_{k\,i+\ell},\quad i = 1,\mathop{\ldots },p -\ell. }$$
Finally, if \(\mu _{1},\mathop{\ldots },\mu _{N}\) are arbitrary numbers, then rearranging rows and columns of the matrix-symbol \(A(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) we obtain a matrix-symbol \(\bar{A}(z,\zeta ),\) for which \(\bar{\mu }_{1},\ldots,\bar{\mu }_{N}\) are ordered. Indeed, if for indices i and j, i < j, of the collection \(\mu _{1},\mathop{\ldots },\mu _{N}\) the relation \(\mu _{i} <\mu _{j}\) holds, then switching i-th and j-th columns, and then switching i-th and j-th rows of \(A(z,\zeta ),\) we obtain a collection \(\mu _{1}^{'},\mathop{\ldots },\mu _{N}^{'},\) with \(\mu _{i}^{'} =\mu _{j}>\mu _{i} =\mu _{ j}^{'}.\) These two switchings are equivalent to the multiplication by two matrices C i and R i with determinants \(\det (C_{j}) =\det (R_{j}) = -1.\) Performing these operations finitely many times we arrive to the symbol \(\bar{A}(z,\zeta ) \in S(\bar{M},\mathcal{M}(\varOmega )),\) where \(\bar{M}\) is a matrix of degrees corresponding to the ordered collection \(\bar{\mu }_{1},\ldots,\bar{\mu }_{N}\). Hence, the symbol \(\bar{A}(z,\zeta )\) is connected with \(A(z,\zeta )\) through \(\bar{A}(z,\zeta ) = CA(z,\zeta ),\) where \(C = R_{k}C_{k}\mathop{\ldots }R_{1}C_{1}\) is an invertible \(N \times N\) matrix not depending on z and \(\zeta.\) Therefore, \(A^{-1}(z,\zeta ) =\bar{ A}^{-1}(z,\zeta )C.\) As we have seen above in the ordered case the symbol \(\bar{A}^{-1}(z,\zeta )\) also belongs to the same class \(S(\bar{M},\mathcal{M}(\varOmega )).\) The multiplication of \(\bar{A}^{-1}(z,\zeta )\) by C from the right is equivalent to switching of columns and rows exactly in the reverse order. This implies that \(A^{-1}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega )).\)
Necessity. Assume that the inverse symbol \(A^{-1}(z,\zeta ) = B(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) exists. The relations
$$\displaystyle{\sum _{\ell=1}^{N}a_{ i\ell} \circ b_{\ell j} =\delta _{ij},\quad i,j = 1,\mathop{\ldots },N,}$$
that indicate that the symbols \(A(z,\zeta )\) and \(B(z,\zeta )\) are mutually inverse, contain
$$\displaystyle{ L = \frac{1} {n!}\sum _{i=1}^{N}\sum _{ j=1}^{N}\frac{[\max _{k}\big(\deg (a_{ik}) + m_{kj} + n\big)]!} {[\max _{k}\big(\deg (a_{ik}) + m_{kj}\big)]!} }$$
equations. On the other hand, since each symbol b i j contains \(\frac{(m_{ij}+n)!} {m_{ij}!n!}\) components, then the total number of components of \(B(z,\zeta )\) is
$$\displaystyle{ K = \frac{1} {n!}\sum _{i=1}^{N}\sum _{ j=1}^{N}\frac{(m_{ij} + n)!} {m_{ij}!}. }$$
Due to our assumption on the existence of the inverse symbol, we have L = K. This implies
$$\displaystyle{ \max _{k}\big(\deg (a_{ik}) + m_{kj}\big) = m_{ij},\quad i,j = 1,\mathop{\ldots },N. }$$
(9.47)
It follows from (9.47) that the inequalities
$$\displaystyle{ \deg (a_{ik}) \leq m_{ij} - m_{kj} }$$
(9.48)
are valid for all \(j = 1,\ldots,N.\) Let μ i and ν i are the integer and fractional parts of \((m_{i1} + \mathop{\ldots } + m_{iN})/N,\) respectively. Then, equation (9.48) can be rewritten in the form
$$\displaystyle{ \deg (a_{ik}) \leq \mu _{i} -\mu _{k} + (\nu _{i} -\nu _{k}). }$$
(9.49)
Finally, since \(\deg (a_{ik})\) are integers and \(\vert \nu _{i} -\nu _{k}\vert <1\), it follows from (9.49) that
$$\displaystyle{\deg (a_{ik}) \leq \mu _{i} -\mu _{k},\quad i,k = 1,\mathop{\ldots },N,}$$
proving the necessity of the condition (9.45).

Under the additional condition \(N_{A}\cap \varOmega =\emptyset\) to \(A(z,\zeta )\) the class of symbols \(S(M,\mathcal{O}(\varOmega ))\) becomes an involutive algebra. Namely, the following theorem is valid [Uma91-1]:

Theorem 9.5.

A symbol \(A(z,\zeta ) \in S(M,\mathcal{O}(\varOmega ))\) has the inverse \(A^{-1}(z,\zeta ) \in S(M,\mathcal{O}(\varOmega ))\) if and only if the following two conditions hold:
  1. (i)
    there exists a collection of integers \(\mu _{1},\ldots,\mu _{N}\) such that the inequalities
    $$\displaystyle{\deg (a_{ij}) \leq \mu _{i} -\mu _{j},\quad i,j = 1,\ldots,N,}$$
    are fulfilled, and
     
  2. (ii)

    \(N_{A}\cap \varOmega =\emptyset.\)

     

The proposition below proved in [Vol63] (see also [Miz67]) provides a sufficient condition for the matrix \(m_{ij} =\deg (a_{ij})\) to exist a collection \(\mu _{k},\,k = 1,\ldots,N,\) satisfying the condition (9.45).

Proposition 9.10.

Let a matrix M with rational entries m ij (including \(-\infty\) ) satisfy the condition: \(m_{ii} = 0,\,i = 1,\ldots,N,\) and for any permutation \(\pi\) of the set \(\{1,\ldots,N\}\) the inequality \(\sum _{i=1}^{N}m_{i,\pi (i)} \leq 0\) holds. Then there exists a collection \(\mu _{1},\ldots,\mu _{N}\) of rational numbers satisfying \(m_{ij} \leq \mu _{j} -\mu _{i},\,i,j = 1,\ldots,N.\)

Remark 9.5.

If entries of M are integers, then in the proposition above the numbers \(\mu _{1},\ldots,\mu _{N}\) also can be selected integer. Obviously, if \(m_{ij} \leq \mu _{j} -\mu _{i}\) for all \(i,j = 1,\ldots,N,\) then the transposed matrix satisfies \(m_{ij}^{T} \leq \mu _{i} -\mu _{j}\) for all \(i,j = 1,\ldots,N.\)

9.6 Algebras of pseudo-differential operators with complex symbols with singularities

Let a symbol \(a =\{ a_{\alpha }(\zeta )\}_{\vert \alpha \vert \leq m} \in S(m,X),\) where X is a class of symbols specified below. We define the pseudo-differential operator with the symbol a as
$$\displaystyle{ Af =\sum _{\vert \alpha \vert \leq m}z^{\alpha }F^{-1}[a_{\alpha }(\zeta )F[f](\zeta )](z), }$$
(9.50)
where F is the complex Fourier transform defined in (9.34). The class of pseudo-differential operators with symbols in S(m, X) will be denoted O P S(m, X). We also write d e g(A) having in mind the degree of the corresponding symbol. The sum A + B and composition \(A \circ B\) of operators A ∈ O P S(m1, X) and B ∈ O P S(m2, X) are defined as operators with symbols a + b and \(a \circ b,\) respectively. Hence, O P S(m, X) is an algebra isomorphic to the algebra S(m, X). 

Proposition 9.11.

Let \(A \in OPS(m,\mathcal{O}_{\mu,r,\zeta _{0}}),\,\zeta _{0} \in \varOmega.\) Then the mappings
$$\displaystyle{A \equiv A(z,D_{z}): \mathcal{E}_{\mu,r,\zeta _{0}} \rightarrow \mathcal{E}_{\mu +m,r,\zeta _{0}}\ \ \mbox{ and}\ \ A^{{\ast}}\equiv A(z,-D_{ z}): \mathcal{E}_{\mu +m,r,\zeta _{0}}^{{\ast}}\rightarrow \mathcal{E}_{\mu,r,\zeta _{0}}^{{\ast}}}$$
are continuous. Moreover, for the norms of the operators A and A the estimate
$$\displaystyle{ \|A\| =\| A^{{\ast}}\|\leq \sum _{ \vert \alpha \vert \leq m}[a_{\alpha }]_{\mu,r,\zeta _{0}} }$$
(9.51)
holds.

Proof.

Since the spaces \(\mathcal{E}_{\mu,r,\zeta _{0}}\) for different \(\zeta _{0} \in \varOmega\) are isomorphic, it suffices to consider the case \(\zeta _{0} = 0.\) Let \(\varphi \in \mathcal{E}_{\mu,r}\) be an arbitrary element. It is readily seen that the multiplication operator by a function \(\psi (z) \in \mathcal{E}_{\mu _{0},r_{0}}\) is continuous from \(\mathcal{E}_{\mu,r}\) to \(\mathcal{E}_{\mu +\mu _{0},r+r_{0}}.\) In particular, for \(z^{\alpha } \in \mathcal{E}_{\vert \alpha \vert,0},\) taking into account Proposition 9.6, one has
$$\displaystyle\begin{array}{rcl} \|A\varphi \|_{\mu +m,r}& \leq & \sum _{\vert \alpha \vert \leq m}\|z^{\alpha }a_{\alpha }(D)\varphi \|_{\mu +m,r} \leq \sum _{\vert \alpha \vert \leq m}\|a_{\alpha }(D)\varphi \|_{\mu,r} {}\\ & \leq & \|\varphi \|_{\mu,r}\sum _{\vert \alpha \vert \leq m}[a_{\alpha }]_{\mu,r}. {}\\ \end{array}$$
The second part of the statement now follows by duality.
Introduce the following spaces of direct products with the corresponding direct product topologies:
$$\displaystyle\begin{array}{rcl} & \mathcal{E}_{\bar{\mu },r,\zeta _{0}} =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{E}_{\mu _{j},r,\zeta _{0}},\ \ & \mathcal{E}_{\bar{\mu },r,\zeta _{0}}^{{\ast}} =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{E}_{\mu _{ j},r,\zeta _{0}}^{{\ast}}, {}\\ & \mathcal{O}_{\bar{\mu },r,\zeta _{0}} =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{O}_{\mu _{j},r,\zeta _{0}},\ \ & \mathcal{O}_{\bar{\mu },r,\zeta _{0}}^{{\ast}} =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{O}_{\mu _{ j},r,\zeta _{0}}^{{\ast}}, {}\\ & E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n}) =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}E_{\mu _{j},r}^{\varOmega }(\mathbb{C}^{n}),\ \ & \Big(E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}} =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\Big(E_{\mu _{ j},r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}}, {}\\ & \mathcal{O}_{\bar{\mu },r}(\varOmega ) =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{O}_{\mu _{j},r}(\varOmega ),\ \ & \mathcal{O}_{\bar{\mu },r}^{{\ast}}(\varOmega ) =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{O}_{\mu _{ j},r}^{{\ast}}(\varOmega ), {}\\ \end{array}$$
and
$$\displaystyle{ \mathcal{M}_{\bar{\mu },r}(\varOmega ) =\mathop{\mathop{ \otimes }\limits_{j = 1}}\limits^{N}\mathcal{M}_{\mu _{j},r}(\varOmega ), }$$
where \(\mathcal{M}_{\mu,r}(\varOmega ) = (\mathcal{M},\varOmega,\pi )\) is a fiber space of meromorphic functions with the base Ω, fibers \(\mathcal{M}_{\mu,r,\zeta _{0}},\) and projection
$$\displaystyle{\pi: (\mathcal{M},\varOmega,\pi ) \rightarrow \varOmega,}$$
where \(\pi ^{-1}(\zeta _{0}) = \mathcal{M}_{\mu,r,\zeta _{0}},\ \zeta \in \varOmega\). An element of the fiber \(\mathcal{M}_{\mu,r,\zeta _{0}}\) in a neighborhood of the point \(\zeta _{0} \in \varOmega\) has a local representation \(m(z) = f(z)/g(z) \in \mathcal{O}_{\mu,r}(\varOmega \setminus P_{m}).\) Hence, one can define a dual space \(\mathcal{M}_{\mu,r}^{{\ast}}(\varOmega )\) of meromorphic functionals as well, similar to their analytic and exponential counterparts.

Let A(z, D z ) be a pseudo-differential operator with the matrix-symbol \(\mathcal{A}(z,\zeta ),\) whose entries \(\mathcal{A}_{ij}(z,\zeta ) \in S(m_{ij},\mathcal{O}_{m_{j},r,\zeta _{0}}),\,i,j = 1,\ldots,N.\) We define the adjoint operator \(A^{{\ast}}(z,D_{z})\) as a pseudo-differential operator with the matrix-symbol \(\mathcal{A}^{{\ast}}(z,\zeta ) = \mathcal{A}^{T}(z,-\zeta ),\) that is with entries \(\mathcal{A}_{ij}^{{\ast}}(z,\zeta ) = \mathcal{A}_{ji}(z,-\zeta ) \in S(m_{ji},\mathcal{O}_{m_{j},r,-\zeta _{0}}),\,i,j = 1,\ldots,N.\)

Proposition 9.12.

Let \(\mathcal{A}(z,\zeta )\) be a matrix-symbol with entries \(\mathcal{A}_{ij} \in S(m_{ij},\mathcal{O}_{m_{j},r,\zeta _{0}}),\) and let \(m_{ij} \leq \mu _{i} -\mu _{j}\) for all \(i,j = 1,\mathop{\ldots },N.\) Then the mappings
$$\displaystyle{A(z,D_{z}): \mathcal{E}_{\bar{\mu },r,\zeta _{0}} \rightarrow \mathcal{E}_{\bar{\mu },r,\zeta _{0}};\ \ A^{{\ast}}(z,D_{ z}): \mathcal{E}_{\bar{\mu },r,\zeta _{0}}^{{\ast}}\rightarrow \mathcal{E}_{\bar{\mu },r,\zeta _{0}}^{{\ast}}}$$
are continuous. Moreover, for the norms of these operators the estimate
$$\displaystyle{ \|A\| =\| A^{{\ast}}\|\leq \sum _{ i=1}^{N}\left (\max _{ 1\leq j\leq N}\sum _{\vert \alpha \vert \leq m_{ij}}[a_{\alpha }]_{\mu _{j},r,\zeta _{0}}\right ) }$$
(9.52)
holds.

Proof.

Since the operator \(A_{ij} \in OPS(m_{ij},\mathcal{O}_{\mu _{j},r,\zeta _{0}}),\) it follows from Proposition 9.11 that it maps the space \(\mathcal{E}_{\mu _{j},r,\zeta _{0}}\) continuously onto \(\mathcal{E}_{\mu _{j}+m_{ij},r,\zeta _{0}}.\) The latter is continuously embedded into \(\mathcal{E}_{\mu _{i},r,\zeta _{0}}\) due to inequality \(\mu _{j} + m_{ij} \leq \mu _{i}\) for all \(i,j = 1,\ldots,N.\) These imply the continuity of the operator \(A(z,D_{z}): \mathcal{E}_{\bar{\mu },r,\zeta _{0}} \rightarrow \mathcal{E}_{\bar{\mu },r,\zeta _{0}}.\)

To show (9.52) one can use estimate (9.51) for the operator A i j : 
$$\displaystyle{ \|A_{ij}\varphi _{j}(z)\|_{\mu _{i},r,\zeta _{0}} \leq \sum _{\vert \alpha \vert \leq m_{ij}}[a_{ij\alpha }]_{\mu _{j},r,\zeta _{0}}\|\varphi _{j}\|_{\mu _{j},r,\zeta _{0}}, }$$
where \(\varphi _{j} \in \mathcal{E}_{\mu _{j},r,\zeta _{0}}.\) It follows that
$$\displaystyle{\|(A\varphi )_{i}\|_{\mu _{i},r,\zeta _{0}} \leq \|\varphi \|_{\bar{\mu },r,\zeta _{0}}\max _{1\leq j\leq N}\sum _{\vert \alpha \vert \leq m_{ij}}[a_{ij\alpha }]_{\mu _{j},r,\zeta _{0}}.}$$
Here \((A\varphi )_{i}\) is the i-th component of the vector-function \(A(z,D_{z})\varphi (z).\) Summing the latter inequality over all \(i = 1,\ldots,N,\) one obtains estimate (9.52). The rest of the statement of the theorem follows by duality.

Proposition 9.13.

Let \(\mathcal{A}(z,\zeta )\) be a matrix-symbol with entries \(\mathcal{A}_{ij} \in S(m_{ij},\mathcal{O}_{\mu _{j},r}(\varOmega )).\) Suppose that the collection of integers \(\{\mu _{1},\mathop{\ldots },\mu _{N}\}\) such that \(m_{ij} \leq \mu _{j} -\mu _{i}\) for all \(i,j = 1,\mathop{\ldots },N.\) Then the mappings
$$\displaystyle\begin{array}{rcl} & & A(z,D_{z}):\, E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n}) \rightarrow E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n}), {}\\ & & A^{{\ast}}(z,D_{ z}):\,\Big (E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}}\rightarrow \Big (E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}} {}\\ \end{array}$$
are continuous.

Proof.

Follows easily from Proposition 9.12.

Proposition 9.14.

Let \(\mathcal{A}(z,\zeta )\) be a matrix-symbol with entries \(\mathcal{A}_{ij} \in S(m_{ij},\mathcal{M}_{\mu _{j},r}(\varOmega )).\) Suppose that the collection of integers \(\{\mu _{1},\mathop{\ldots },\mu _{N}\}\) satisfy inequalities \(m_{ij} \leq \mu _{j} -\mu _{i}\) for all \(i,j = 1,\mathop{\ldots },N.\) Then the pseudo-differential operators corresponding to symbols \(\mathcal{A}(z,\zeta )\) and \(\mathcal{A}^{{\ast}}(z,\zeta )\) are continuous as mappings
$$\displaystyle\begin{array}{rcl} A(z,D_{z}): E_{\bar{\mu },r}^{\varOmega \setminus P_{A} }(\mathbb{C}^{n}) \rightarrow E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n});& &{}\end{array}$$
(9.53)
$$\displaystyle\begin{array}{rcl} A^{{\ast}}(z,D_{ z}):\,\Big (E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n})\Big)^{{\ast}}\rightarrow \Big (E_{\bar{\mu },r}^{\varOmega \setminus P_{A} }(\mathbb{C}^{n})\Big)^{{\ast}}.& &{}\end{array}$$
(9.54)
Moreover, the inverse operator \(A^{-1}(z,D_{z})\) exists and is continuous as a mapping
$$\displaystyle{ A^{-1}(z,D_{ z}): E_{\bar{\mu },r}^{\varOmega \setminus N_{j} }(\mathbb{C}^{n}) \rightarrow E_{\bar{\mu },r}^{\varOmega }(\mathbb{C}^{n}). }$$
(9.55)

Proof.

Let \(\varphi _{j} \in \mathcal{E}_{\mu _{j},r,\zeta _{0}}(\mathbb{C}^{n}),\,j = 1,\ldots,N,\) where \(\zeta _{0} \in \varOmega \setminus P_{j}.\) In accordance with the definition of \(\mathcal{M}_{\mu,r}(\varOmega )\) all the functions \(a_{ij\alpha }(\zeta )\) in the symbol \(\mathcal{A}_{ij}(z,\zeta )\) belong to \(\mathcal{O}_{\mu,r,\zeta _{0}}\) with \(r <dist(\zeta _{0},\partial (\varOmega \setminus P_{j})).\) Therefore, \(\mathcal{A}_{ij}(z,\zeta ) \in S(m_{ij},\mathcal{O}_{\mu _{j},r}(\varOmega \setminus P_{j})).\) Now the continuity of mappings (9.53) and (9.55) follow from Proposition 9.13. This fact implies the continuity of the inverse operator \(A^{-1}(z,D_{z})\) in mapping (9.55) too, since due to Theorem 9.4 the inverse symbol \(\mathcal{A}^{-1}(z,\zeta )\) has entries \(\mathcal{A}_{ij}^{-1}(z,\zeta ) \in S(m_{ij},\mathcal{M}_{\mu _{j},r}(\varOmega )).\)

Pseudo-differential operators with meromorphic symbols in \(S(M,\mathcal{M}(\varOmega ))\) behave differently. Unlike the previous cases they act in factor-spaces. To formulate the continuity theorem first we study kernels of pseudo-differential operators with meromorphic symbols.

For an operator \(A \in OPS(M,\mathcal{M}(\varOmega ))\) we denote by \(\kappa _{\pm }\) the dimension of the kernel of \(A^{\pm 1}:\)
$$\displaystyle{\kappa _{\pm } =\kappa _{\pm }(A,\varOmega ) =\dim Ker(A^{\pm 1}).}$$
The meaning of κ+ is obvious. If \(\kappa _{-} = m,\) then the image of the operator A is a factor space factorized by the m-dimensional space K e r A−1. Thus, the operator A in this case is multi-valued. The operator \(A \in OPS(M,\mathcal{M}(\varOmega ))\) is single-valued if and only if \(\kappa _{-} = 0.\)
Let \(\mathcal{P}_{k}^{\pm },\,k = 1,\ldots,K_{\pm },\) be connected irreducible components of \(P_{\mathcal{A}^{\pm 1}} \cap Z_{reg,\varOmega }\) and \(L_{k}^{\pm },k = 1,\ldots,K_{\pm },\) be their respective orders. Denote by \(W_{kl}^{\pm }(\mathcal{A})\) the span of all linear combinations
$$\displaystyle{ f_{k,\ell}(z) = F^{-1}[\delta ^{(\ell)}(\rho _{ k}^{\pm }(\zeta ))](z),\quad \ell = 0,\ldots,L_{ k}^{\pm },\ k = 1,\ldots,K_{ \pm }, }$$
(9.56)
where δ is the Dirac distribution, and \(\rho _{k}^{\pm }(\zeta )\) are holomorphic functions, locally representing \(\mathcal{P}_{k}^{\pm },\) that is \(\mathcal{P}_{k}^{\pm }\equiv \{\zeta:\rho _{ k}^{\pm }(\zeta ) = 0\}.\)

Theorem 9.6.

Let \(A \in OPS(M,\mathcal{M}(\varOmega ))\) and there exists a collection of integers \(\{\mu _{1},\mathop{\ldots },\mu _{N}\}\) such that \(m_{ij} \leq \mu _{j} -\mu _{i}\) for all \(i,j = 1,\mathop{\ldots },N.\) Then
$$\displaystyle{Ker(A^{\pm 1}) =\mathop{\mathop{ \oplus }\limits_{k = 1}}\limits^{K_{ \pm }}\left (\mathop{\mathop{\oplus }\limits_{\ell = 0}}\limits^{L_{k}^{\pm }- 1}W_{ k\ell}^{\mp }(A)\right ).}$$

Proof.

We will show that V ∈ K e r(A−1) if and only if \(V \in Ker(A_{0}^{-1}),\) where A0 is the constant part of the operator A. Indeed, without loss of generality, one can assume that \(\mu _{1},\ldots,\mu _{N}\) are ordered, i.e., \(\mu _{1} =\ldots =\mu _{k_{1}}>\mu _{k_{1}+1} =\ldots =\mu _{k_{2}}>\ldots>\mu _{k_{l-1}+1} =\ldots =\mu _{k_{l}},\) \(k_{1} +\ldots +k_{l} = N.\) Otherwise, with the help of permutations of rows and columns, which correspond to the multiplication of A by a scalar invertible matrices, one gets a desired ordering. Hence, the operator A has the form
$$\displaystyle{ A = \left [\begin{array}{*{10}c} A_{11} & A_{12} & \mathop{\ldots }&A_{1l} \\ \varTheta &A_{22} & \mathop{\ldots }&A_{2l} \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \varTheta & \varTheta &\mathop{\ldots }& A_{ ll} \end{array} \right ], }$$
(9.57)
where \(A_{jj} \in OPS(\theta,\mathcal{M}(\varOmega ))\) form the constant part of the operator A. Due to Theorem 9.4 the inverse matrix A−1 also has the same block-matrix structure as (9.57) with entries \(A_{ij}^{-1}\) of the same size of A i j . Accordingly, one has \(V = (V _{1},\ldots,V _{l}),\) where \(V _{j},\,j = 1,\ldots,l,\) are vector-functions of length k j . Let V ∈ K e r(A−1), that is \(A^{-1}V = 0.\) It follows from matrix structure (9.57) of the inverse operator A−1 immediately that \(A_{ll}^{-1}V _{l} = 0.\) Further, since
$$\displaystyle{A_{l-1l-1}^{-1}V _{ l-1}+A_{l-1,l}^{-1}V _{ l} = 0,\ \ \mbox{ and}\ \ A_{l-1l}^{-1} = -A_{ l-1l-1}^{-1}\circ A_{ l-1l}\circ A_{ll}^{-1},}$$
which also follows from (9.57), one has
$$\displaystyle{A_{l-1l-1}^{-1}V _{ l-1} = -A_{l-1l-1}^{-1} \circ A_{ l-1l} \circ A_{ll}^{-1}V _{ l} = 0.}$$
Consecutively, one obtains \(A_{jj}V _{j} = 0,j = l - 2,\ldots,1.\) This implies \(Ker(A^{-1}) \subset Ker(A_{0}^{-1}).\) Making use of these formulas on reverse order, we conclude that \(Ker(A_{0}^{-1}) \subset Ker(A^{-1}).\) Hence, \(Ker(A^{-1}) = Ker(A_{0}^{-1}).\) Therefore, it suffices to consider the equation \(A_{0}^{-1}(D_{z})V (z) = 0.\) Due to isomorphic property of the Fourier transform, the latter is equivalent to the system of algebraic equations \(\mathcal{A}_{0}^{-1}(\zeta )F[V ](\zeta ) = 0\) with a parameter \(\zeta \in \varOmega.\) Here \(\mathcal{A}_{0}^{-1}(\zeta )\) is the symbol of \(A_{0}^{-1}.\) It is not hard to see that there exists a matrix \(B(\zeta ),\,\det B(\zeta )\neq 0,\) such that
$$\displaystyle{ \mathcal{A}_{0}^{-1}(\zeta )F[V ](\zeta ) = B(\zeta )\Big(H(\zeta )F[V ](\zeta )\Big) = 0, }$$
(9.58)
where \(H(\zeta ) \in \mathcal{O}(\varOmega )\) is defined in a local representation of \(\det (\mathcal{A}_{0})\) given in equation (9.44). Recall a local representation of the meromorphic function \(\det (\mathcal{A}_{0}^{-1}(\zeta )) = H(\zeta )/G(\zeta )\) (see (9.44)). To show (9.58) one can take \(B(\zeta ) = (H(\zeta ))^{-1}\mathcal{A}_{0}^{-1}(\zeta ).\) Then it can be easily verified that \(\det (B(\zeta )) =\det (\mathcal{A}_{0}^{-1}(\zeta ))(G(\zeta ))^{-1}\neq 0,\) \(\zeta \in \varOmega.\) Equation (9.58) means that the problem on description of the kernel of A−1 is reduced to equations
$$\displaystyle{ H(\zeta )F[V _{j}](\zeta ) = 0,\quad j = 1,\ldots,N, }$$
(9.59)
for each component F[V j ] of the vector-function \(F[V ](\zeta ),\) considered on the space of analytic functionals \(\mathcal{O}^{{\ast}}(\varOmega ).\) Now let \(\mathcal{P}_{k}^{-},\,k = 1,\ldots,K_{-},\) be irreducible components of the analytic set \(Z_{reg,\varOmega } \cap P_{A^{-1}}\) with orders \(L_{k}^{-}.\) Then solutions to equation (9.59) have the form \(F[V _{j}](\zeta ) =\delta ^{(\ell)}(\rho _{k}(\zeta )),\,\ell= 0,\ldots,L_{k}^{-},\,k = 1,\ldots,K_{-},\) for each \(j = 1,\ldots,N,\) where \(\rho _{k}(\zeta )\) locally represents \(\mathcal{P}_{k}^{-}.\) Taking the inverse Fourier transform, one has \(V (\zeta ) = f_{k,\ell}(z)\mathbf{v} \in Ker(A^{-1}),\) where \(f_{k,\ell}\) are defined in (9.56), and \(\mathbf{v}\) is an arbitrary scalar vector, obtaining the desired result.

Corollary 9.3.

  1. 1.

    Let \(\mathcal{A}(z,\zeta ) \in S(M,\mathcal{O}(\varOmega ))\) with a Runge domain Ω and a matrix M, entries of which satisfy \(m_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N,\) for some collection \(\mu _{1},\ldots,\mu _{N}.\) Then \(\kappa _{-}(A,\varOmega ) = 0;\)

     
  2. 2.

    Let \(\mathcal{A}(z,\zeta ) \in S(M,\mathcal{O}(\varOmega ))\) with a Runge domain Ω and a matrix M, entries of which satisfy \(m_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N,\) for some collection \(\mu _{1},\ldots,\mu _{N}.\) If \(N_{A}\cap \varOmega =\emptyset,\) then \(\kappa _{+}(A,\varOmega ) = 0.\)

     

It is known [Chi89] that an analytic set in a neighborhood of any regular point represents an analytic submanifold (of co-dimension one in our case). Therefore, for n ≥ 2 it follows from Theorem 9.6 that \(\kappa _{\pm } = \infty,\) as long as \(\varOmega \cap Z_{reg,A^{\pm 1}}\neq \emptyset,\) and \(\kappa _{\pm } = 0,\) otherwise. Hence, if n ≥ 2 only two possibilities may arise. It is not so in the one-dimensional case.

Theorem 9.7.

Let n = 1. Let \(L_{k}^{+}\) and \(L_{k}^{-}\) be orders of poles \(\zeta _{k}^{+} \in P_{A},\,k = 1\ldots,K_{+},\) and zeros \(\zeta _{k}^{-} \in N_{A},\,k = 1,\ldots,K_{-},\) respectively. Then,
$$\displaystyle{ \kappa _{+}(A,\varOmega ) =\sum _{\zeta _{k}^{-}\in \varOmega \cap N_{A}}L_{k}^{-}\ \ \mbox{ and}\ \ \kappa _{ -}(A,\varOmega ) =\sum _{\zeta _{k}^{+}\in \varOmega \cap P_{A}}L_{k}^{+}. }$$
(9.60)

Proof.

In the one-dimensional case solutions of equation (9.59) are \(F[V _{j}](\zeta ) = g_{k,\ell}(\zeta ) =\delta ^{(\ell)}(\zeta -\zeta _{k}^{+}),\,\ell= 0,\ldots,L_{k}^{+},\,k = 1,\ldots,K_{+}.\) Their Fourier inverses are \(V _{j}(z) = f_{k,\ell}(z) = z^{\ell}e^{\zeta _{k}^{+}z },\,\ell= 0,\ldots,L_{k}^{+},\,k = 1,\ldots,K_{+}.\) Obviously, this set of functions is linearly independent. This implies the second formula in (9.60). Since \(\kappa _{+}(A,\varOmega ) =\kappa _{-}(A^{-1},\varOmega ),\) the first formula is also correct.

Theorems 9.6 and 9.7 show that if Ω contains nonempty polar- or null-set of the symbol of a pseudo-differential operator, then the latter has a nontrivial kernel or co-kernel. Therefore, one needs factor-spaces to formulate a continuity statements in this case.

We will use traditional notations: if \(\mathcal{X}\) is a generic topological space and K is its subspace, then \(\mathcal{X}/K\) denotes the factor-space (with the topology of factor-space) of elements \(\phi +\varphi,\) where \(\phi \in \mathcal{X}\) and \(\varphi \in K.\) Elements \(\varPhi =\phi +\varphi\) for all \(\varphi \in K\) are considered identical. The conjugate \((\mathcal{X}/K)^{{\ast}}\) to a factor-space \(\mathcal{X}/K\) consists of elements \(G \in \mathcal{X}^{{\ast}}\) orthogonal to K: \(<G,\varphi>= 0,\,\forall \varphi \in K.\) We will denote the conjugate space \(\mathcal{X}_{K^{\perp }}^{{\ast}}.\)

Proposition 9.15.

Let \(\mathcal{A}(z,\zeta )\) be a matrix-symbol with entries \(\mathcal{A}_{ij} \in S(m_{ij},\mathcal{M}(\varOmega )).\) Suppose that the collection of integers \(\{\mu _{1},\mathop{\ldots },\mu _{N}\}\) satisfy inequalities \(m_{ij} \leq \mu _{j} -\mu _{i}\) for all \(i,j = 1,\mathop{\ldots },N.\) Then the pseudo-differential operators corresponding to symbols \(\mathcal{A}(z,\zeta )\) and \(\mathcal{A}^{{\ast}}(z,\zeta )\) are continuous as mappings
$$\displaystyle\begin{array}{rcl} & & A(z,D_{z}): E_{\bar{\mu },r}^{\varOmega }/Ker(A) \rightarrow E_{\bar{\mu },r}^{\varOmega }/Ker(A^{-1}); {}\\ & & A^{{\ast}}(z,D_{ z}):\,\Big (E_{\bar{\mu },r}^{\varOmega }\Big)_{ Ker(A)^{\perp }}^{{\ast}}\rightarrow \Big (E_{\bar{\mu },r}^{\varOmega }\Big)_{ Ker(A^{-1})^{\perp }}^{{\ast}}. {}\\ \end{array}$$

Proof.

The proof follows from Theorem 9.6 and Proposition 9.14.

Consider the following examples illustrating Theorem 9.6 and 9.7.

Example 9.1.

  1. 1.

    Let a symbol \(a \in S(m,\mathcal{O}(\varOmega )),\) where Ω is an arbitrary Runge domain. Then \(\kappa _{-}(A,\varOmega ) = 0,\) and hence the corresponding operator A is single-valued (uniquely defined).

     
  2. 2.
    Let n = 1 and 0 ∈ Ω. Let the symbol \(a(\zeta ) = 1/\zeta \in S(0,\mathcal{M}(\varOmega )).\) Then, \(\kappa _{-}(A,\varOmega ) = 1,\) and the corresponding operator \(A(D_{z}) = D_{z}^{-1}\) (the primitive) is defined up to an additive constant. Note that if \(0\notin \varOmega,\) then \(D_{z}^{-1}\) is uniquely defined and represents the “natural integral” (see [Dub96]):
    $$\displaystyle{D_{z}^{-1}f(z) = nat\int f(\zeta )d\zeta,\quad f \in Exp_{\varOmega }(\mathbb{C}).}$$

    Now, suppose n = 2 and \(a(\zeta _{1},\zeta _{2}) = 1/\zeta _{1}.\) Assume that Ω is a Runge domain containing (0, 0). Then \(P_{A} =\{ (\zeta _{1},\zeta _{2}) \in \varOmega:\zeta _{1} = 0\}.\) In this case the corresponding operator \(A(D_{z_{1}},D_{z_{2}}) = D_{z_{1}}^{-1}\) represents the integral with respect to the variable z1 and is defined up to an arbitrary function of the variable z2. Hence, \(\kappa _{-}(A,\varOmega ) = \infty.\)

     

9.7 Systems of pseudo-differential equations with meromorphic symbols

In this section we discuss the existence and uniqueness problems for general boundary value problem (9.7)–(9.8). We first consider a system of pseudo-differential equations
$$\displaystyle{ B(z,D_{z})\varPsi (z) =\varPhi (z), }$$
(9.61)
and the Cauchy problem for a system of first order evolution pseudo-differential equations
$$\displaystyle\begin{array}{rcl} D_{t}V (t,z) = A(t,z,D_{z})V (t,z) + H(t,z),& &{}\end{array}$$
(9.62)
$$\displaystyle\begin{array}{rcl} V (0,z) = V _{0}(z),& &{}\end{array}$$
(9.63)
where \(B(z,D_{z}) \in OPS(M,X),\) \(A(t,z,D_{z}) \in OPS(M_{1},X)\) for each fixed t; the space of symbols X, as well as vector-functions (functionals) \(\varPhi (z),\,H(t,z),\) and V0(z) will be specified below.

Theorem 9.8.

Let \(B(z,D_{z}) \in OPS(M,\mathcal{M}(\varOmega ))\) and assume that there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N.\) Then for any vector-function \(\varPhi (z) \in E_{\bar{\mu },r}^{\varOmega }/Ker(B^{-1})\) there exists a unique solution \(\varPsi (z)\) to system (9.61) in the factor-space \(E_{\bar{\mu },r}^{\varOmega }/Ker(B).\)

Proof.

Due to Proposition 9.15 the pseudo-differential operator B = B(z, D z ) with the symbol \(\mathcal{B}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) is well defined in the space \(E_{\bar{\mu },r}^{\varOmega }/Ker(B).\) In accordance with Theorem 9.4 there exists the inverse symbol \(\mathcal{B}^{-1}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega )).\) The corresponding inverse operator \(B^{-1} = B^{-1}(z,D_{z})\) is well defined in the space \(E_{\bar{\mu },r}^{\varOmega }/Ker(B^{-1}).\) Let \(\varPhi (z) \in E_{\bar{\mu },r}^{\varOmega }/Ker(B^{-1}),\) i.e., \(\varPhi (z) =\phi (z) +\varphi (z),\) where \(\phi \in E_{\bar{\mu },r}^{\varOmega },\) and \(\varphi \in Ker(B^{-1}).\) Now one can show that \(\varPsi (z) = B^{-1}(z,Dz)\varPhi (z) +\psi (z),\) for arbitrary \(\psi \in Ker(B),\) solves the system (9.61). Indeed,
$$\displaystyle\begin{array}{rcl} B(z,D_{z})\varPsi (z)& =& B(z,D_{z})\Big(B^{-1}(z,D_{ z})\varPhi (z)+\psi \Big) {}\\ & =& \varPhi (z) + B(z,D_{z})\psi (z) =\varPhi (z). {}\\ \end{array}$$

Theorem 9.9.

Let \(B(z,D_{z}) \in OPS(M,\mathcal{M}(\varOmega ))\) and there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{j} -\mu _{i},\,i,j = 1,\ldots,N.\) Then for any \(\varPhi (z) \in \left (E_{\bar{\mu },r}^{\varOmega }\right )_{Ker(B^{{\ast}})^{\perp }}^{{\ast}}\) there exists a unique weak solution \(\varPsi (z)\) to system (9.61) in the space \(\Big(E_{\bar{\mu },r}^{\varOmega }\Big)_{Ker((B^{{\ast}})^{-1})^{\perp }}^{{\ast}}.\)

Proof.

Let \(\varPhi (z) \in \left (E_{\bar{\mu },r}^{\varOmega }\right )_{Ker(B^{{\ast}})^{\perp }}^{{\ast}}.\) Then for arbitrary \(U \in E_{\bar{\mu },r}^{\varOmega }/Ker(B^{{\ast}})\) one has
$$\displaystyle{\langle B(z,D_{z})\varPsi (z),U(z)\rangle =\langle \varPhi (z),U(z)\rangle,}$$
or
$$\displaystyle{\langle \varPsi (z),B^{{\ast}}(z,D_{ z})U(z)\rangle =\langle \varPhi (z),U(z)\rangle.}$$
Due to Proposition 9.15 the operator \(B^{{\ast}}(z,D_{z})\) is continuous from \(E_{\bar{\mu },r}^{\varOmega }/Ker(B^{{\ast}})\) to the space \(E_{\bar{\mu },r}^{\varOmega }/Ker(B^{{\ast}})^{-1}.\) Note that due to Theorem 9.4 there exists the inverse symbol \(\mathcal{B}^{-1}(z,\zeta ) \in S(M,\mathcal{M}(\varOmega )),\) and hence, the corresponding inverse operator \((B^{{\ast}})^{-1}\) exists and well defined in the space \(E_{\bar{\mu },r}^{\varOmega }/Ker(B^{{\ast}})^{-1}.\) Therefore, if one sets \(B^{{\ast}}(z,D_{z})U(z) = V (z),\) where \(V (z) \in E_{\bar{\mu },r}^{\varOmega }/Ker(B^{{\ast}})^{-1},\) then due to Theorem 9.8 one has \(U(z) = (B^{{\ast}}(z,D_{z}))^{-1}V (z).\) This implies that the functional \(\varPsi (z)\) defined by
$$\displaystyle{ \langle \varPsi (z),V (z)\rangle =\langle \varPhi (z),(B^{{\ast}}(z,D_{ z}))^{-1}V (z)\rangle =\langle \left ((B^{{\ast}}(z,D_{ z}))^{-1}\right )^{{\ast}}\varPhi (z),V (z)\rangle }$$
(9.64)
solves system (9.61) in the weak sense. Representation (9.64) also shows that for the inverse the formula \(B^{-1}(z,D_{z}) = \left ((B^{{\ast}}(z,D_{z}))^{-1}\right )^{{\ast}}\) holds, and \(\langle \varPsi (z),f(z)\rangle = 0\) if \(f \in Ker(B^{{\ast}})^{-1}.\)

If one considers the operator \(A \in OPS(m,\mathcal{M}(\varOmega ))\) in the space \(E_{\bar{\mu },r}^{\varOmega \setminus P_{A}},\) then it follows from the definition (9.50) of a pseudo-differential operator with a meromorphic symbol, that the polar set of the symbol of A does not intersect with \(\varOmega \setminus P_{A}.\) This implies that the symbol belongs to \(S(m,\mathcal{O}_{\mu,r}).\) In this case K e r(A) = { 0}, and therefore, the above theorems take the form:

Theorem 9.10.

Let \(B(z,D_{z}) \in OPS(M,\mathcal{M}(\varOmega ))\) and assume that there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N.\) Then for any vector-function \(\varPhi (z) \in E_{\bar{\mu },r}^{\varOmega \setminus N_{B}}\) there exists a unique solution \(\varPsi (z)\) to system (9.61) in the space \(E_{\bar{\mu },r}^{\varOmega \setminus P_{B}}.\)

Theorem 9.11.

Let \(B(z,D_{z}) \in OPS(M,\mathcal{M}(\varOmega ))\) and assume that there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{j} -\mu _{i},\,i,j = 1,\ldots,N.\) Then for any \(\varPhi (z) \in \left (E_{\bar{\mu },r}^{\varOmega \setminus P_{B}}\right )^{{\ast}}\) there exists a unique weak solution \(\varPsi (z)\) to system (9.61) in the space \(\Big(E_{\bar{\mu },r}^{\varOmega \setminus N_{B}}\Big)^{{\ast}}.\)

Remark 9.6.

  1. 1.

    In Theorems 9.10 and 9.11 one can replace the spaces \(E_{\bar{\mu },r}^{\varOmega \setminus N_{B}},\) \(E_{\bar{\mu },r}^{\varOmega \setminus P_{B}},\) and their conjugates by the spaces \(Exp_{\varOmega \setminus N_{B}}(\mathbb{C}^{n}),\) \(Exp_{\varOmega \setminus P_{B}}(\mathbb{C}^{n})\) defined in Section 9.4 and their respective conjugates.

     
  2. 2.

    Similar to the proof of Theorem 9.4 one can show that the conditions \(m_{ij} \leq \mu _{i} -\mu _{j}\) in the above theorems are also necessary for existence of a solution.

     
Using formulas (9.38) and (9.39) and the scheme (see Theorem 9.3)
$$\displaystyle{\mathcal{O}_{\bar{\mu },r}(\varOmega )\ \ \begin{array}{c} \mathop{\longrightarrow }\limits^{F}\\ \mathop{\longleftarrow }\limits_{F^{-1}} \end{array} \ \ \left (E_{\bar{\mu },r}^{\varOmega }\right )^{{\ast}}}$$
one can obtain dual results in terms of the Fourier transform. Namely, applying the Fourier transform to equation (9.61), one has
$$\displaystyle{B(D_{\zeta },\zeta )H(\zeta ) = G(\zeta ),\quad \zeta \in \varOmega \setminus P_{B},}$$
or, the same
$$\displaystyle{ \sum _{j=1}^{N}\sum _{ \vert \alpha \vert \leq m_{ij}}(-1)^{\alpha }a_{ ij\alpha }(\zeta )D_{\zeta }^{\alpha }h_{ j}(\zeta ) = g_{i}(\zeta ),\quad \zeta \in \varOmega \setminus P_{B},i = 1,\ldots,N, }$$
(9.65)
where \(H(\zeta ) = (h_{1}(\zeta ),\ldots,h_{N}(\zeta ))^{T}\) and \(G(\zeta ) = (g_{1}(\zeta ),\ldots,g_{N}(\zeta ))^{T}.\)

Theorem 9.12.

Let the matrix-symbol \(B(z,\zeta ) \in S(M,\mathcal{M}(\varOmega ))\) and there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{j} -\mu _{i},\,i,j = 1,\ldots,N.\) Then for any vector-function \(G(\zeta ) \in \mathcal{O}_{\bar{\mu },r}(\varOmega \setminus P_{B})\) there exists a solution \(\varPsi (z)\) to system (9.65) in the space \(\mathcal{O}_{\bar{\mu },r}(\varOmega \setminus N_{B}).\)

Proof.

Consider the system (9.65) in the scale of spaces \(\mathcal{O}_{\bar{\mu },r}(\varOmega ).\) Applying the inverse Fourier transform F−1 we have
$$\displaystyle{ B(z,D_{z})F^{-1}[H](z) = F^{-1}[G](z) }$$
(9.66)
in the scale of spaces \(\left (E_{\bar{\mu },r}^{\varOmega }\right )^{{\ast}}.\) In accordance with Theorem 9.11, under the condition of our theorem, for any \(F^{-1}[G] \in \left (E_{\bar{\mu },r}^{\varOmega \setminus P_{B}}\right )^{{\ast}}\) there is a unique solution \(F^{-1}[H] \in \left (E_{\bar{\mu },r}^{\varOmega \setminus N_{B}}\right )^{{\ast}}\) to system (9.66). Now applying the Fourier transform and using isomorphism \(F: \left (E_{\bar{\mu },r}^{\varOmega \setminus N_{B}}\right )^{{\ast}}\rightarrow \mathcal{O}_{\bar{\mu },r}(\varOmega \setminus N_{B})\) one obtains the desired result.
Similarly, using the scheme (see Theorem 9.3)
$$\displaystyle{\mathcal{O}_{\bar{\mu },r}^{{\ast}}(\varOmega )\ \ \begin{array}{c} \mathop{\longrightarrow }\limits^{F^{-1}} \\ \mathop{\longleftarrow }\limits_{F} \end{array} \ \ E_{\bar{\mu },r}^{\varOmega }}$$
we can establish the existence of a solution of the system (9.65) in the space of analytic functionals.

Theorem 9.13.

Let \(B(z,D_{z}) \in OPS(M,\mathcal{M}(\varOmega ))\) and assume that there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N.\) Then for any vector-functional \(\varPhi (z) \in \mathcal{O}_{\bar{\mu },r}^{{\ast}}(\varOmega \setminus N_{B})\) there exists a solution \(\varPsi (z)\) to system (9.65) in the space \(\mathcal{O}_{\bar{\mu },r}^{{\ast}}(\varOmega \setminus P_{B}).\)

Now assume that \(\mathcal{D}\subset \mathbb{C}\) is a domain containing t0 and \(\mathcal{X}\) be a topological vector space. Below we use the spaces of the form \(\mathcal{O}[\mathcal{D};\mathcal{X}],\) elements f(t) of which for each fixed t belong to \(\mathcal{X}\) and analytic in the variable t in the topology of \(\mathcal{X}.\)

Theorem 9.14.

Let \(A = A(t,z,D_{z}) \in \mathcal{O}\big[\mathcal{D};OPS(M,\mathcal{M}(\varOmega ))\big]\) and there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{i} -\mu _{j} + 1,\,i,j = 1,\ldots,N.\) Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functions \(H(t,z) \in \mathcal{O}\left [\mathcal{D};E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ],\) and \(V _{0}(z) \in E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}\) a unique solution V (t,z) to the Cauchy problem (9.62)–(9.63) exists in a δ-neighborhood of t0 and belongs to the factor-space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ].\)

Proof.

The Cauchy problem (9.62)–(9.63) can be written in the equivalent integro-differential form
$$\displaystyle{ V (t,z) = V _{0}(z) +\int _{ t_{0}}^{t}A(\tau,z,D_{ z})V (\tau,z)d\tau +\int _{ t_{0}}^{t}H(\tau,z)d\tau. }$$
Consider the operator
$$\displaystyle{ \mathbf{A}V (t,z) =\int _{ t_{0}}^{t}A(\tau,z,D_{ z})V (\tau,z)d\tau. }$$
For i-th component of this operator one has
$$\displaystyle\begin{array}{rcl} (\mathbf{A}V (t,z))_{i}& =& \sum _{j=1}^{N}\int _{ t_{0}}^{t}A_{ ij}(\tau,z,D_{z})V _{j}(\tau,z)d\tau \\ & =& \sum _{j=1}^{N}\int _{ t_{0}}^{t}\left (\sum _{ \vert \alpha \vert \leq m_{ij}}z^{\alpha }a_{ ij\alpha }(\tau,D_{z})V _{j}(\tau,z)\right )d\tau.{}\end{array}$$
(9.67)
In order to prove the theorem it suffices to show the existence of a unique solution for arbitrary fiber of the space \(E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}.\) Let \(\zeta _{0} \in \varOmega \setminus P_{A}\) be an arbitrary fixed point, and consider equation (9.67) in the fiber \(\mathcal{E}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert,\zeta _{0}}.\) Since \(\zeta _{0}\) is located out of the polar set of the operator A(t, z, D z ), the symbol of this operator belongs to \(\mathcal{O}_{\bar{\mu },r,\zeta _{0}}.\) Therefore, making use of Proposition 9.11 and taking into account the evolution of V (t, z) over the scale \(\mathcal{O}\left [\mathcal{D};\mathcal{E}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert,\zeta _{0}}\right ],\) one obtains the estimate
$$\displaystyle\begin{array}{rcl} & & \vert (\mathbf{A}V (t,z))_{i}\vert \leq \sum _{j=1}^{N}\sum _{ \vert \alpha \vert \leq m_{ij}}\vert z\vert ^{\vert \alpha \vert }\int _{ 0}^{\vert t-t_{0}\vert }\vert a_{ ij\alpha }(\tau,D_{z})V _{j}(\tau,z)\vert \vert d\tau \vert {}\\ & &\qquad \leq I(\vert t - t_{0}\vert )\sum _{j=1}^{N}(1 + \vert z\vert )^{m_{ij}+\mu _{j} }\sup _{t\in \mathcal{D}}\|V _{j}\|_{\mu _{j},r+\sigma \vert t-t_{0}\vert,\zeta _{0}}\sum _{\vert \alpha \vert \leq m_{ij}}\sup _{t\in \mathcal{D}}[a_{ij\alpha }]_{\mu _{j},r,\zeta _{0}}, {}\\ \end{array}$$
where
$$\displaystyle{I(\vert t - t_{0}\vert ) =\int _{ 0}^{\vert t-t_{0}\vert }e^{(r+\sigma \vert \tau -t_{0}\vert )\vert z\vert }\vert d\tau \vert \leq \frac{\vert t - t_{0}\vert } {\sigma \vert z\vert } e^{(r+\sigma \vert t-t_{0}\vert )\vert z\vert }.}$$
Taking this and the inequality \(m_{ij} +\mu _{j} \leq \mu _{i} + 1,i,j = 1,\ldots,N,\) into account, one has
$$\displaystyle\begin{array}{rcl} \vert (\mathbf{A}V (t,z))_{i}\vert \leq \frac{\vert t - t_{0}\vert } {\sigma } (1 + \vert z\vert )^{\mu _{i}}e^{r+\sigma \vert t-\tau \vert }\sum _{j=1}^{N}\sup _{t\in \mathcal{D}}\|V _{j}(t,z)\|_{\mu _{ j},r+\sigma \vert t-t_{0}\vert,\zeta _{0}}\sup _{t\in \mathcal{D}}[\mathcal{A}_{ij}]_{\mu _{j},r,\zeta _{0}}.& & {}\\ \end{array}$$
This implies
$$\displaystyle{\|(\mathbf{A}V (t,z))_{i}\|_{\mu _{i}r+\sigma \vert t-t_{0}\vert,\zeta _{0}} \leq \frac{\vert t - t_{0}\vert } {\sigma } \sup _{t\in \mathcal{D}}\|V (t,z)\|_{\bar{\mu },r+\sigma \vert t-t_{0}\vert,\zeta _{0}}\max _{1\leq j\leq N}\sup _{t\in \mathcal{D}}\|A_{ij}(t,z,D_{z})\|.}$$
Now summing up by index \(i = 1,\ldots,N,\) we have
$$\displaystyle{\|(\mathbf{A}V (t,z))\|_{\bar{\mu },r+\sigma \vert t-t_{0}\vert,\zeta _{0}} \leq \frac{\vert t - t_{0}\vert } {\sigma } \sup _{t\in \mathcal{D}}\|A(t,z,D_{z})\|\sup _{t\in \mathcal{D}}\|V (t,z)\|_{\bar{\mu },r+\sigma \vert t-t_{0}\vert,\zeta _{0}}.}$$
It follows from this estimate that A is a contraction operator if the condition
$$\displaystyle{\vert t - t_{0}\vert \sup _{t\in \mathcal{D}}\|A(t,z,D_{z})\| \leq \sigma }$$
holds. Hence, taking \(\delta <\sigma /\sup _{t\in \mathcal{D}}\|A(t,z,D_{z})\|\) we have that in the δ-neighborhood of t0 a unique solution to the Cauchy problem (9.62)–(9.63) exists.

Theorem 9.15.

Let \(A = A(t,z,D_{z}) \in \mathcal{O}[\mathcal{D};M,\mathcal{M}(\varOmega )]\) and assume that there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{j} -\mu _{i} + 1,\,i,j = 1,\ldots,N.\) Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functionals \(H(t,z) \in \mathcal{O}\left [\mathcal{D};\left (E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}\right )^{{\ast}}\right ]\) and \(V _{0}(z) \in \left (E_{\bar{\mu },r}^{\varOmega \setminus P_{B}}\right )^{{\ast}}\) there exists a unique solution V (t,z) to the Cauchy problem (9.62)–(9.63) in the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;\left (E_{\bar{\mu },r}^{\varOmega \setminus P_{B}}\right )^{{\ast}}\right ]\) with some δ > 0.

Proof.

Since the proof follows from Theorem 9.14 by duality, we only briefly sketch its idea. Let \(V (t,z) \in \in \mathcal{O}\left [\mathcal{D};\left (E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}\right )^{{\ast}}\right ]\) and \(v(t,z) \in \mathcal{O}\left [\mathcal{D};E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ].\) Then the relation
$$\displaystyle{D_{t}\langle V (t,z),v(t,z)\rangle =\langle D_{t}V (t,z),v(t,z)\rangle +\langle V (t,z),D_{t}v(t,z)\rangle }$$
implies
$$\displaystyle\begin{array}{rcl} \langle V (t,z),v(t,z)\rangle & =& \langle V (t_{0},z),v(t_{0},z)\rangle {}\\ & +& \int _{t_{0}}^{t}\langle D_{ s}V (s,z),v(s,z)\rangle ds +\int _{ t_{0}}^{t}\langle V (s,z),D_{ s}v(s,z)\rangle ds. {}\\ \end{array}$$
The latter due to equation (9.62) and the initial condition in (9.63) takes the form
$$\displaystyle\begin{array}{rcl} \langle V (t,z),v(t,z)\rangle & =& \langle \varPhi (z),v(t_{0},z)\rangle +\int _{ t_{0}}^{t}\langle A(s,z,D_{ z})V (s,z) + H(s,z),v(s,z)\rangle ds \\ & +& \int _{t_{0}}^{t}\langle V (s,z),D_{ s}v(s,z)\rangle ds \\ & =& \langle \varPhi (z),v(0,z)\rangle +\int _{ t_{0}}^{t}\langle V (s,z),D_{ s}v(s,z) + A^{{\ast}}(s,z,D_{ z})v(s,z)\rangle ds \\ & +& \int _{t_{0}}^{t}\langle H(s,z),v(s,z)\rangle ds. {}\end{array}$$
(9.68)
Since the latter is valid for arbitrary v(t, z), it is also valid for \(v(t,\tau,z),\) which solves the Cauchy problem
$$\displaystyle\begin{array}{rcl} D_{\tau }v(t,\tau,z) + A^{{\ast}}(\tau,z,D_{ z})v(\tau,z) = 0,\quad t_{0} <\tau <t,& &{}\end{array}$$
(9.69)
$$\displaystyle\begin{array}{rcl} v(t,\tau,z)_{\vert \tau =t} = v(t,z).& &{}\end{array}$$
(9.70)
For the symbol of the adjoint operator \(A^{{\ast}}(t,z,D_{z})\) the order-matrix \(m_{ij}^{{\ast}}\) satisfies the inequality \(m_{ij}^{{\ast}}\leq \mu _{i} -\mu _{j} + 1,\,i,j = 1,\ldots,N.\) Therefore, in accordance with Theorem 9.14 the Cauchy problem (9.69)–(9.70) has a unique solution in the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ]\) for any fixed v(t, z), if | tt0 | < δ, where δ > 0 small enough. Substituting v(t, z) in equation (9.68) by \(v(t,\tau,z),\) we have
$$\displaystyle{ \langle V (t,z),v(t,z)\rangle =\langle \varPhi (z),v(t,t_{0},z)\rangle +\int _{ t_{0}}^{t}\langle H(s,z),v(t,\tau,z)\rangle d\tau,\ \ \vert t - t_{ 0}\vert <\delta. }$$
(9.71)
The functional V (t, z) defined by (9.71) is a unique solution to the Cauchy problem (9.62)–(9.63). It can be readily seen that \(V (t,z) \in \mathcal{O}\left [\vert t - t_{0}\vert <\delta;\left (E_{\bar{\mu },r}^{\varOmega \setminus N_{B}}\right )^{{\ast}}\right ],\) and hence is a desired solution.
Now consider general boundary value problems for the first order systems
$$\displaystyle\begin{array}{rcl} D_{t}V (t,z) = A(t,z,D_{z})V (t,z) + H(t,z),& &{}\end{array}$$
(9.72)
$$\displaystyle\begin{array}{rcl} B(z,D_{z})V (t,z)_{\vert t=0} =\varPhi (z),& &{}\end{array}$$
(9.73)
This problem can be reduced to the equivalent Cauchy problem for system (9.72) with the initial condition
$$\displaystyle{ V (0,z) =\varPsi (z), }$$
where \(\varPsi (z)\) is a solution to the system of pseudo-differential equations
$$\displaystyle{ B(z,D_{z})\varPsi (z) =\varPhi (z). }$$

Combining the above proved Theorems 9.8 and 9.14 (in the dual case Theorems 9.9 and 9.15) one can prove the following statements.

Theorem 9.16.

Let operators \(A = A(t,z,D_{z}) \in \mathcal{O}\big[\mathcal{D};OPS(M,\mathcal{M}(\varOmega ))\big]\) and \(B(z,D_{z}) \in OPS(\mathcal{N},\mathcal{M}(\varOmega )).\) Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)

    the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{i} -\mu _{j} + 1,\,i,j = 1,\ldots,N;\)

     
  2. ii)

    the entries of the matrix \(\mathcal{N}\) satisfy the inequalities \(n_{ij} \leq \mu _{i} -\mu _{j},\,i,j = 1,\ldots,N.\)

     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functions \(H(t,z) \in \mathcal{O}\left [\mathcal{D};E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ],\) and \(\varPhi (z) \in E_{\bar{\mu },r}^{\varOmega }/Ker(B^{-1})\) a unique solution V (t,z) to the Cauchy problem (9.72)–(9.73) exists in a δ-neighborhood of t0 and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ].\) Moreover, the kernel of this problem is isomorphic to the kernel of the operator B(z,Dz).

Theorem 9.17.

Let operators \(A = A(t,z,D_{z}) \in \mathcal{O}\big[\mathcal{D};OPS(M,\mathcal{M}(\varOmega ))\big]\) and \(B(z,D_{z}) \in OPS(\mathcal{N},\mathcal{M}(\varOmega )).\) Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)

    the entries of the matrix M satisfy the inequalities \(m_{ij} \leq \mu _{j} -\mu _{i} + 1,\,i,j = 1,\ldots,N;\)

     
  2. ii)

    the entries of the matrix \(\mathcal{N}\) satisfy the inequalities \(n_{ij} \leq \mu _{j} -\mu _{i},\,i,j = 1,\ldots,N.\)

     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functionals \(H(t,z) \in \mathcal{O}\left [\mathcal{D};\left (E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right )^{{\ast}}\right ],\) and \(\varPhi (z) \in \left (E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}\right )_{Ker(B^{{\ast} })^{\perp }}^{{\ast}}\) a unique solution V (t,z) to the Cauchy problem (9.72)–(9.73) exists in a δ-neighborhood of t0 and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;\left (E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right )_{Ker((B^{{\ast} })^{-1})^{\perp }}^{{\ast}}\right ].\)

As an example of application of these theorems consider the following boundary value problem for a pseudo-differential equation of higher order
$$\displaystyle\begin{array}{rcl} D_{t}^{m}u(t,z) +\sum _{ k=0}^{m-1}A_{ k}(t,z,D_{z})D_{t}^{k}u(t,z) = h(t,z),\quad t \in \mathcal{D},\,z \in \mathbb{C}^{n},& &{}\end{array}$$
(9.74)
$$\displaystyle\begin{array}{rcl} \sum _{j=0}^{m-1}B_{ ij}(z,D_{z})D_{t}^{j}u(t,z)\Big\vert _{ t=0} =\varphi _{i}(z),\quad z \in \mathbb{C}^{n},\,i = 0,\ldots,m - 1,& &{}\end{array}$$
(9.75)
where \(A_{k}(t,z,D_{z}),\,k = 0,\ldots,m - 1,\) are pseudo-differential operators with symbols
$$\displaystyle{ \mathcal{A}_{k}(t,z,\zeta ) =\sum _{\vert \alpha \vert \leq m_{k}}a_{k\alpha }(t,\zeta )z^{\alpha }. }$$
This problem is equivalent to the following system:
$$\displaystyle\begin{array}{rcl} & & D_{t}v(t,z) +\tilde{ A}(t,z,D_{z})v(t,z) = H(t,z), {}\\ & & B(z,D_{z})v(t,z)\Big\vert _{t=0} =\phi (z), {}\\ \end{array}$$
where the vector-functions \(v(t,z) = (u(t,z),\ldots,u_{t}^{(m-1)}(t,z))^{T},\) \(H(t,z) = (0,\ldots,h(z))^{T},\) \(\phi (z) = (\varphi _{0}(z),\ldots,\varphi _{m-1}(z))^{T},\) and the operator \(\tilde{A}(t,z,D_{z})\) has the matrix-symbol with entries
$$\displaystyle{\tilde{A}_{ij}(t,z,\zeta ) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1, \quad &\mbox{ if }j = i + 1,i = 0,\ldots,m - 2, \\ A_{j}(t,z,\zeta ),\quad &\mbox{ if }i = m,\,j = 0\ldots,m - 1, \\ \theta, \quad &\mbox{ otherwise}. \end{array} \right.}$$
In the matrix form
$$\displaystyle{ \tilde{A}(t,z,\zeta ) = \left [\begin{array}{*{10}c} \theta & 1 &\mathop{\ldots }& \theta & \theta \\ \theta & \theta &\mathop{\ldots } & \theta &\theta \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \theta & \theta &\mathop{\ldots }& \theta & 1 \\ A_{0} & A_{1} & \mathop{\ldots }&A_{m-2} & A_{m-1} \end{array} \right ], }$$
Applying Theorem 9.16 one has \(\mu _{j} = j,j = 0,\ldots,m - 1.\) Therefore, boundary value problem (9.74)–(9.75) have a local solution in the scale \(E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}},\) \(\bar{\mu }= (0,\ldots,m - 1),\) if the polynomial degrees m k and m i j of symbols \(\mathcal{A}_{k}(t,z,\zeta )\) and \(\mathcal{B}_{ij}(z,\zeta ),\) satisfy, respectively, the following inequalities:
$$\displaystyle{ m_{k} \leq m - k,\quad k = 0,\ldots,m - 1, }$$
and
$$\displaystyle{ m_{ij} \leq i - j,\quad i,j = 0,\ldots,m - 1. }$$

9.8 Reduction to a system of first order

The general system of pseudo-differential equations (9.7) can be reduced to a system of first order of the form (9.72). Boundary condition (9.8) in this process also changes to the form (9.73). We prove the following statement:

Lemma 9.1.

Let a vector-function \(u(t,x) = (u_{1}(t,x),\ldots,u_{N}(t,x))\) solve the general problem (9.7)–(9.8) . Then the vector-function
$$\displaystyle{V (t,x) = (u_{1}(t,x),\ldots,D_{t}^{p_{1}-1}u_{ 1}(t,x),\ldots,u_{N}(t,x),\ldots,D_{t}^{p_{N}-1}u_{ N}(t,x))}$$
of length \(p_{1} +\ldots +p_{N}\) solves a problem of the form (9.72)–(9.73) , with vector-functions H(t,z) and \(\varPhi (z)\)
$$\displaystyle\begin{array}{rcl} H(t,z) = (\mathbf{h}_{1}(t,z),\ldots,\mathbf{h}_{N}(t,z)),\,\mathbf{h}_{j}(t,z) = (0,0,\ldots,f_{j}(t,z)),& &{}\end{array}$$
(9.76)
$$\displaystyle\begin{array}{rcl} \varPhi (z) = ({\boldsymbol \phi }_{1}(z),\ldots,{\boldsymbol \phi }_{N}(z)),\,{\boldsymbol \phi }_{j}(z) = (\varphi _{j0}(z),\ldots,\varphi _{jp_{j}-1}),& &{}\end{array}$$
(9.77)
where \(\mathbf{h}_{j}(t,z)\) is a vector of length pj with only nonzero pj-th component fj(t,z); and the matrix-operators \(A(t,z,D_{z}) = \mathbf{A}_{ij}(t,z,D_{z})\) and \(B(z,D_{z}) = \mathbf{B}_{ij(z,D_{z})},\,i,j = 1,\ldots,N,\) are block-matrices with respective blocks of sizes \(p_{i} \times p_{j}:\)
$$\displaystyle\begin{array}{rcl} \mathbf{A}_{ij}(t,z,D_{z}) = \left \{\begin{array}{@{}l@{\quad }l@{}} \left [\begin{array}{*{10}c} \theta & 1 &\mathop{\ldots }& \theta & \theta \\ \theta & \theta &\mathop{\ldots } & \theta &\theta \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \theta & \theta &\mathop{\ldots }& \theta & 1 \\ A_{jj}^{0}&A_{jj}^{1}&\mathop{\ldots }&A_{jj}^{p_{j}-2}&A_{jj}^{p_{j}-1} \end{array} \right ],\quad &\mbox{ if }\,\,i = j, \\ \quad \\ \left [\begin{array}{*{10}c} \theta & \theta &\mathop{\ldots }& \theta & \theta \\ \theta & \theta &\mathop{\ldots } & \theta &\theta \\ & \mathop{\ldots } &\mathop{\ldots }&\\ \theta & \theta &\mathop{\ldots }& \theta & \theta \\ A_{ij}^{0}&A_{ij}^{1}&\mathop{\ldots }&A_{ij}^{p_{j}-2}&A_{ij}^{p_{j}-1} \end{array} \right ],\quad &\mbox{ if }\,\,i\neq j. \end{array} \right.& &{}\end{array}$$
(9.78)
and
$$\displaystyle\begin{array}{rcl} \mathbf{B}_{i,j}(z,D_{z})& =& \left [\begin{array}{*{10}c} B_{ij}^{00} & B_{ij}^{01} & \mathop{\ldots }& B_{ij}^{0p_{j}-1} \\ B_{ij}^{10} & B_{ij}^{11} & \mathop{\ldots }& B_{ij}^{1p_{j}-1} \\ & \mathop{\ldots } &\mathop{\ldots }& \\ B_{ij}^{p_{i}-10} & B_{ij}^{p_{i}-11} & \mathop{\ldots }&B_{ij}^{p_{i}-1p_{j}-1} \end{array} \right ].{}\end{array}$$
(9.79)

Proof.

In accordance with the definition of the vector-function V (t, z), it can be represented in the form \(V (t,z) = (\mathbf{v}_{1}(t,z),\ldots,\mathbf{v}_{N}(t,z)),\) where
$$\displaystyle\begin{array}{rcl} \mathbf{v}_{1}(t,z)& =& (v_{1},\ldots,v_{p_{1}}) \equiv \left (u_{1}(t,z),\ldots,D_{t}^{p_{1}-1}u_{ 1}(t,z)\right ), {}\\ \mathbf{v}_{2}(t,z)& =& (v_{p_{1}+1},\ldots,v_{p_{1}+p_{2}}) \equiv \left (u_{2}(t,z),\ldots,D_{t}^{p_{2}-1}u_{ 2}(t,z)\right ), {}\\ & \ldots & {}\\ \mathbf{v}_{N}(t,z)& =& (v_{p_{1}+\ldots +p_{N-1}+1},\ldots,v_{p_{1}+\ldots +p_{N}}) \equiv \left (u_{N}(t,z),\ldots,D_{t}^{p_{N}-1}u_{ N}(t,z)\right ). {}\\ \end{array}$$
This together with equation (9.7) implies that
$$\displaystyle\begin{array}{rcl} & & \qquad D_{t}v_{1}(t,z) = v_{2}(t,z), {}\\ & & \qquad D_{t}v_{2}(t,z) = v_{3}(t,z), {}\\ & & \qquad \qquad \qquad \ldots {}\\ & & \quad D_{t}v_{p_{1}-1}(t,z) = v_{p_{1}}(t,z), {}\\ & & D_{t}v_{p_{1}}(t,z) =\sum _{ j=1}^{N}\left [A_{ 1j}^{0}v_{ p_{1}+\ldots +p_{j-1}+1}(t,z) +\ldots +A_{1j}^{p_{j}-1}v_{ p_{1}+\ldots +p_{j}}(t,z)\right ] + f_{1}(t,z), {}\\ & & \qquad \qquad \qquad \mathop{\ldots }\mathop{\ldots }\mathop{\ldots }\mathop{\ldots } {}\\ \end{array}$$
$$\displaystyle\begin{array}{rcl} & & D_{t}v_{p_{1}+\ldots +p_{N-1}+1}(t,z) = v_{p_{1}+\ldots +p_{N-1}+2}(t,z), {}\\ & & D_{t}v_{p_{1}+\ldots +p_{N-1}+2}(t,z) = v_{p_{1}+\ldots +p_{N-1}+3}(t,z), {}\\ & & \qquad \qquad \qquad \qquad \qquad \ldots {}\\ & & \quad D_{t}v_{p_{1}+\ldots +p_{N}-1}(t,z) = v_{p_{1}+\ldots +p_{N}}(t,z), {}\\ & & D_{t}v_{p_{1}+\ldots +p_{N}}(t,z) =\sum _{ j=1}^{N}\left [A_{ Nj}^{0}v_{ p_{1}+\ldots +p_{j-1}+1}(t,z) +\ldots +A_{Nj}^{p_{j}-1}v_{ p_{1}+\ldots +p_{j}}(t,z)\right ] + f_{N}(t,z).{}\\ \end{array}$$

These equations show that the vector-function V (t, z) satisfies equation (9.72) with the operator A(t, z, D z ) in (9.78) and H(t, z) in (9.76). Similarly, one can show that V (t, z) satisfies boundary conditions (9.73) with the operator B(z, D z ) in (9.79) and vector-function \(\varPhi (z)\) in (9.77).

9.9 Existence theorems for general boundary value problems

Theorem 9.18.

Let operators \(A \equiv \{ A_{jk}^{q}(t,z,D_{z})\} \in \mathcal{O}\big[\mathcal{D};OPS(M^{q},\mathcal{M}(\varOmega ))\big]\) and \(B \equiv \{ B_{jk}^{mq}(z,D_{z})\} \in OPS(\mathcal{N}^{mq},\mathcal{M}(\varOmega )).\) Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)
    the entries of the matrices \(M^{q},q = 0,\ldots,p_{j} - 1,\) satisfy the inequalities
    $$\displaystyle{m_{jk}^{q} \leq \mu _{ j} -\mu _{k} + p_{j} - q,\quad j,k = 1,\ldots,N,\ q = 0,\ldots,p_{j} - 1;}$$
     
  2. ii)
    the entries of the matrix \(\mathcal{N}^{mq},\,m = 0,\ldots,p_{j} - 1,q = 0,\ldots,p_{k} - 1,\) satisfy the inequalities
    $$\displaystyle{n_{jk}^{mq} \leq \mu _{ j} -\mu _{k} + m - q,\quad j,k = 1,\ldots,N,\ m = 0,\ldots,p_{j} - 1,\ q = 0,\ldots,p_{k} - 1.}$$
     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functions \(H(t,z) \in \mathcal{O}\left [\mathcal{D};E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ],\) and \(\varPhi (z) \in E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}/Ker(B^{-1})\) a solution V (t,z) to boundary value problem (9.7)–(9.8) exists in a δ-neighborhood of t0 and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right ].\) Moreover, the kernel of this problem is isomorphic to the kernel of the operator B.

Proof.

Applying Lemma 9.1 we can reduce problem (9.7)–(9.8) to the first order system of the form (9.72)–(9.73). Now the proof follows immediately due to Theorem 9.16.

The theorem below follows from the previous by duality.

Theorem 9.19.

Let operators \(A \equiv \{ A_{jk}^{q}(t,z,D_{z})\} \in \mathcal{O}\big[\mathcal{D};OPS(M^{q},\mathcal{M}(\varOmega ))\big]\) and \(B \equiv \{ B_{jk}^{mq}(z,D_{z})\} \in OPS(\mathcal{N}^{mq},\mathcal{M}(\varOmega )).\) Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)
    the entries of the matrices \(M^{q},q = 0,\ldots,p_{j} - 1,\) satisfy the inequalities
    $$\displaystyle{m_{jk}^{q} \leq \mu _{ k} -\mu _{j} + p_{j} - q,\quad j,k = 1,\ldots,N,\ q = 0,\ldots,p_{j} - 1;}$$
     
  2. ii)
    the entries of the matrix \(\mathcal{N}^{mq},\,m = 0,\ldots,p_{j} - 1,q = 0,\ldots,p_{k} - 1,\) satisfy the inequalities
    $$\displaystyle{n_{jk}^{mq} \leq \mu _{ k} -\mu _{j} + m - q,\quad j,k = 1,\ldots,N,\ m = 0,\ldots,p_{j} - 1,\ q = 0,\ldots,p_{k} - 1.}$$
     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functionals \(H(t,z) \in \mathcal{O}\left [\mathcal{D};\left (E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right )^{{\ast}}\right ],\) and \(\varPhi (z) \in \left (E_{\bar{\mu },r}^{\varOmega \setminus P_{A}}\right )_{Ker(B^{{\ast} })^{\perp }}^{{\ast}}\) a solution V (t,z) to the Cauchy problem (9.7)–(9.8) exists in a δ-neighborhood of t0 and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;\left (E_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{\varOmega \setminus P_{A}}\right )_{Ker((B^{{\ast} })^{-1})^{\perp }}^{{\ast}}\right ].\)

Finally, using the duality relations between exponential and analytic functions and functionals through the Fourier transform established in Theorem 9.3, we can prove the existence results for general boundary value problems for systems of differential equations of the form
$$\displaystyle\begin{array}{rcl} & & D_{t}^{p_{j} }U_{j}(t,\zeta ) +\sum _{ k=1}^{N}\sum _{ q=0}^{p_{k}-1}A_{ jk}^{q}(t,D_{\zeta },\zeta )D_{ t}^{q}U_{ k}(t,\zeta ) = G_{j}(t,\zeta ), \\ & & \qquad \quad \qquad \qquad \qquad \qquad \qquad t \in \mathcal{D},\ \ \zeta \in \varOmega,\ \ j = 1,\ldots,N, {}\end{array}$$
(9.80)
$$\displaystyle\begin{array}{rcl} & & \sum _{k=1}^{N}\sum _{ q=0}^{p_{k}-1}B_{ jk}^{mq}(D_{\zeta },\zeta )D_{ t}^{q}U_{ k}(t,\zeta )\Big\vert _{t=t_{0}} =\varPsi _{jm}(\zeta ), \\ & & \qquad \quad \quad \zeta \in \varOmega,\ \ m = 0,\ldots,p_{j} - 1,\ \ j = 1,\ldots,N, {}\end{array}$$
(9.81)
where \(\mathcal{D}\subset \mathbb{C}\) is a connected domain containing t0; \(\varOmega \subset \mathbb{C}^{n}\) does not contain polar sets P A and P B associated with operators \(A_{jk}^{q}\) and \(B_{jk}^{mq},\) whose symbols are
$$\displaystyle\begin{array}{rcl} \mathcal{A}_{jk}^{q}(t,z,\zeta ) =\sum _{ \vert \alpha \vert \leq m_{jk}^{q}}a_{jk\alpha }(t,\zeta )z^{\alpha } \in \mathcal{O}[\mathcal{D};S(M^{q},\mathcal{M}(\varOmega ))],& &{}\end{array}$$
(9.82)
$$\displaystyle\begin{array}{rcl} \mathcal{B}_{jk}^{mq}(z,\zeta ) =\sum _{ \vert \beta \vert \leq n_{jk}^{mq}}b_{jk\beta }(\zeta )z^{\beta } \in S(\mathcal{N}^{qm},\mathcal{M}(\varOmega )),& &{}\end{array}$$
(9.83)
$$\displaystyle\begin{array}{rcl} q = 0,\ldots,p_{j} - 1,\ \ m = 0,\ldots,p_{k} - 1,\ \ j,k = 1,\ldots,N.& & {}\\ \end{array}$$
Due to formulas (9.38) and (9.39), applying the Fourier transform, one can reduce boundary value problem (9.80)–(9.81) to the problem of the form (9.7)–(9.8). Hence, by duality, Theorems 9.18 and 9.19 imply the following statements.

Theorem 9.20.

Let the symbols of differential operators \(A \equiv \{ A_{jk}^{q}(t,D_{\zeta },\zeta )\}\) and \(B \equiv \{ B_{jk}^{mq}(D_{\zeta },\zeta )\}\) satisfy conditions (9.82) and (9.83) , respectively. Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)
    the entries of the matrices \(M^{q},q = 0,\ldots,p_{j} - 1,\) satisfy the inequalities
    $$\displaystyle{m_{jk}^{q} \leq \mu _{ k} -\mu _{j} + p_{j} - q,\,j,k = 1,\ldots,N,\,q = 0,\ldots,p_{j} - 1;}$$
     
  2. ii)
    the entries of the matrix \(\mathcal{N}^{mq},\,m = 0,\ldots,p_{j} - 1,q = 0,\ldots,p_{k} - 1,\) satisfy the inequalities
    $$\displaystyle{n_{jk}^{mq} \leq \mu _{ k} -\mu _{j} + q - m,\quad j,k = 1,\ldots,N,\,m = 0,\ldots,p_{j} - 1,\,q = 0,\ldots,p_{k} - 1.}$$
     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functions \(H(t,z) \in \mathcal{O}\left [\mathcal{D};\mathcal{O}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }(\varOmega )\right ],\) and \(\varPhi (z) \in \mathcal{O}_{\bar{\mu },r}(\varOmega \setminus N_{B})\) a solution U(t,z) to boundary value problem (9.80)–(9.81) exists in a δ-neighborhood of t0, where \(\delta <r/\sigma,\) and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;\mathcal{O}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }(\varOmega )\right ].\)

Theorem 9.21.

Let the symbols of differential operators \(A \equiv \{ A_{jk}^{q}(t,D_{\zeta },\zeta )\}\) and \(B \equiv \{ B_{jk}^{mq}(D_{\zeta },\zeta )\}\) satisfy conditions (9.82) and (9.83) , respectively. Suppose there exists a collection \(\bar{\mu }=\mu _{1},\ldots,\mu _{N},\) such that
  1. i)
    the entries of the matrices \(M^{q},q = 0,\ldots,p_{j} - 1,\) satisfy the inequalities
    $$\displaystyle{m_{jk}^{q} \leq \mu _{ j} -\mu _{k} + p_{j} - q,\,j,k = 1,\ldots,N,\,q = 0,\ldots,p_{j} - 1;}$$
     
  2. ii)
    the entries of the matrix \(\mathcal{N}^{mq},\,m = 0,\ldots,p_{j} - 1,q = 0,\ldots,p_{k} - 1,\) satisfy the inequalities
    $$\displaystyle{n_{jk}^{mq} \leq \mu _{ j} -\mu _{k} + m - q,\quad j,k = 1,\ldots,N,\,m = 0,\ldots,p_{j} - 1,\,q = 0,\ldots,p_{k} - 1.}$$
     

Then there exist numbers r > 0 and \(\sigma> 0\) such that for any vector-functions \(H(t,z) \in \mathcal{O}\left [\mathcal{D};\mathcal{O}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{{\ast}}(\varOmega )\right ],\) and \(\varPhi (z) \in \mathcal{O}_{\bar{\mu },r}^{{\ast}}(\varOmega )\) a solution U(t,z) to boundary value problem (9.80)–(9.81) exists in a δ-neighborhood of t0, where \(\delta <r/\sigma,\) and belongs to the space \(\mathcal{O}\left [\vert t - t_{0}\vert <\delta;\mathcal{O}_{\bar{\mu },r+\sigma \vert t-t_{0}\vert }^{{\ast}}(\varOmega \setminus N_{B})\right ].\)

9.10 Additional notes

  1. 1.
    The Cauchy problem. The Cauchy problem has a long and rich history. We refer the reader to survey papers [Miz67, VG91, S88, Dub90] on the history and modern state of this theory. In the general form the Cauchy problem was first posed by Augustin Louis Cauchy and the existence of a unique local solution of this problem was proved in his paper [Cau42] in 1842. Sophie von Kowalevsky1 was not aware of Cauchy’s result and reproved [Kow1874] this theorem in 1875. The theorem was later named the Cauchy-Kowalevsky theorem. Kowalevsky showed the importance of the condition \(m_{k} \leq m -k,\,k = 0,\ldots,m - 1,\) for existence of an analytic solution of the Cauchy problem for equation (9.74) in the following example:
    $$\displaystyle\begin{array}{rcl} D_{t}u(t,z)& =& D_{z}^{2}u(t,z),\quad \vert t\vert <1,\,\vert z\vert \leq 1, {}\\ u(0,z)& =& \varphi (z),\quad \vert z\vert <1, {}\\ \end{array}$$
    with analeptic function \(\varphi (z)\) in the unit disc | z | < 1. The solution of this problem has the representation
    $$\displaystyle{u(t,z) = e^{tD_{z}^{2}}\varphi (z) =\sum _{ n=0}^{\infty }\frac{D_{z}^{2n}\varphi (z)} {n!} t^{n}.}$$
    Now taking \(\varphi (z) = (1 -z)^{-1},\) one can see that \(D_{z}^{n}\varphi (z) = n!(1 -z)^{-n-1},\) one obtains a power series
    $$\displaystyle{u(t,z) =\sum _{ n=0}^{\infty }\frac{(2n)!} {n!} \frac{t^{n}} {(1 -z)^{n+1}},}$$
    divergent for all t and z in any neighborhood of the origin (except t = 0).
     
  2. 2.

    On necessary conditions for existence of a solution. Mizohata [Miz74] (see also [Kit76]) showed that the condition \(m_{k} \leq m -k,\,k = 0,\ldots,m - 1,\) is necessary for the existence of an analytic solution of the Cauchy problem for equation (9.74). More precisely, he proved the following statement.

     

Theorem 9.22.

(Mizohata [Miz74]) In order that the Cauchy-Kowalevsky theorem for the Cauchy problem for equation (9.74) hold at the origin, it is necessary that
$$\displaystyle{ m_{k} \leq m -k,\,k = 0,\ldots,m - 1, }$$
(9.84)
Let \(p =\max _{k,\alpha }\{\vert \alpha \vert /(k + n(k,\alpha ))\},\) where \(n(k,\alpha ) =\min \{\mu: a_{k,\alpha }^{\mu }\not\equiv 0\},\) and \(a_{k,\alpha }^{\mu (x)}\) are coefficients of the operator \(A_{k}(t,z,D_{z}) =\sum _{\alpha,\mu }a_{k,\alpha }t^{\mu }D_{z}^{\alpha }.\) Mizohata showed that p ≤ 1, which is equivalent to condition (9.84). Kitagawa [Kit90] introduced weights p k and p v by
$$\displaystyle{p_{{\ast}} =\max _{k,\alpha }\{\vert \alpha \vert /(k + n(k,\alpha )),\,\vert \alpha \vert \leq k\}\,\,\text{and}\,\,p^{{\ast}} =\max _{ k,\alpha }\{\vert \alpha \vert /(k + n(k,\alpha )),\,\vert \alpha \vert> k\},}$$
and proved that in order that the Cauchy-Kowalevsky theorem for the Cauchy problem for equation (9.74) hold at the origin, it is necessary that \(p^{{\ast}}<p_{{\ast}}.\) The latter again implies condition(9.84).
Leray-Volevich’s (LV) condition (9.1), that is
$$\displaystyle{m_{kj}^{q} \leq \mu _{ k} -\mu _{j} + p_{k} -q,\quad k,j = 1,\ldots,N,}$$
first appeared in Volevich [Vol63] in 1963, and in the context of the Cauchy problem for systems of differential equations in Gårding-Kotake-Leray [LGK67], in 1964. Mizohata [Miz74] called systems satisfying LV conditions (9.1) Kowalevskian in the sense of Volevich. The case μ k  = k was used by Leray in 1953 [Ler53]. Usual Kowalevskian systems correspond to the case \(\mu _{k} = 0,\,k = 1,\ldots,N.\)
  1. 3.

    Infinite order differential operators. Differential operators of infinite order obviously do not satisfy LV conditions, and therefore, the corresponding system with such operators are not Kowalevskian in the sense of Volevich. The Cauchy problem for equations and systems with differential operators of infinite order was studied by Korobeynik [K73], Leont’ev [Leo76], Baouendi and Goulaouic [BG76], Dubinskii [Dub84], Napalkov [Nap82], and others. The related theory of analytic pseudo-differential operators is in the focus of many researchers; see survey paper [S88] on results up to 1988, and in works [Dub96, Ren10] on its current state. The analytic solutions of differential equations with the real time variable and complex spatial variables are studied in [Gal08] in model cases.

     
  2. 4.

    On uniqueness of a solution. Holmgren [Hol01] in 1901 showed that the Cauchy problem for equations with analytic coefficients, but not necessarily analytic data, cannot have more than one solution. However, if coefficients of the equation are \(C^{\infty }\) functions, then the Cauchy problem may not have a unique solution. Namely, Plis [Pl54] in 1954 constructed an example of fourth order equation with \(C^{\infty }\)-coefficients for which the uniqueness does not hold. Later other examples were constructed; see [Met93]. Calderon [Cal58] in 1958 proved the uniqueness theorem, which played a key role for further development of the Cauchy theory. Later, other variations or weaker versions of uniqueness conditions were found. In particular, the uniqueness of a solution to the Cauchy problem for differential equations with partially holomorphic coefficients is obtained in works [Hor83, Uch04] and for systems of such equations in [Tam06].

     

Footnotes

  1. 1.

    Under this name she published her paper [Kow1874]. Her original Russian full name is Sofia Vasilyevna Kovalevskaya.

References

  1. [BG76]
    Baouendi M.S., Goulaouic, C. Cauchy problem for analytic pseudo-differential operators. Comm. in Part. Diff. Equat., 1 (2), 135–189 (1976)MathSciNetCrossRefGoogle Scholar
  2. [Ber04]
    Bernstein, S.N.: Sur la natur analytique des solutions des équations aux dérivées partielles du second ordre. Math. Ann. 59, 20–76 (1904)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Ber28]
    Bernstein, S.N.: Demonstration du theoreme de M. Hilbert sur la nature analytique des solutions des equations du type elliptique sans l’emploi des series normales. Math. Ztschr. 28, 330–348 (1928)zbMATHGoogle Scholar
  4. [Cal58]
    Calderon, A.P.: Uniqueness in the Cauchy problem for partial differential equations. Amer. J. of Math., 80, 16–35 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [Cau42]
    Cauchy, A.-L.: Mémoire sur l’emploi du calcul des limites dans l’intégration des équations aux dérivées partielles. Comptes rendus 15, 44–59 (1842)Google Scholar
  6. [Chi89]
    Chirka, E.M.: Complex analytic sets. Kluwer Academic Publishers (1989)Google Scholar
  7. [DN55]
    Douglis, A., Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Comm. Pure. & Appl. Math. 8, 503–538 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Dub84]
    Dubinskii, Yu.A.: Fourier transformation of analytic functions. The complex Fourier method. Dokl. Akad. Nauk SSSR., 275 (3), 533–536 (1984) (in Russian)MathSciNetGoogle Scholar
  9. [Dub90]
    Dubinskii, Yu. A.: The Cauchy problem and pseudo-differential operators in the complex domain. Russ. Math. Surv., 45 (2), 95–128 (1990)MathSciNetCrossRefGoogle Scholar
  10. [Dub96]
    Dubinskii Yu. A.: Cauchy problem in complex domains. Moscow (1996) (In Russian)Google Scholar
  11. [Gal08]
    Gal, C.G., Gal, S.G., Goldstein, G.A.: Evolution equations with real time variable and complex spatial variables. Compl. Var. Elliptic Equ. 53, 753–774; Higher-order heat and Laplace-type equations with real time variable and complex spatial variable. 55, 357–373; Wave and telegraph equations with real time variable and complex spatial variables. 57, 91–109 (2008–2010–2012)Google Scholar
  12. [GR09]
    Ganning, R., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing (2009)Google Scholar
  13. [LGK67]
    Gårding, L., Leray, J., Kotake, T.: Probleme de Cauchy. Moscou, Mir (1967) (in Russian)zbMATHGoogle Scholar
  14. [GS53]
    Gel’fand I.M., Shilov, G.E.: Fourier transforms of rapidly growing functions and questions of uniqueness of the solution of Cauchy’s problem. Usp. Mat. Nauk. 8 (6), 3–54 (1953)zbMATHGoogle Scholar
  15. [VG91]
    Gindikin S.G., Volevich, L.R.: The Cauchy problem. In Partial Differential Equations III. Egorov, Yu.V., Shubin M.A. (eds), Springer-Verlag, Berlin, 1–87 (1991)Google Scholar
  16. [Hay78]
    Hayne, R.M.:. Uniqueness in the Cauchy problem for parabolic equations. Trans. Am. Math. Soc., 241, 373–399 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Hol01]
    Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58, 91–103 (1901)Google Scholar
  18. [Hor83]
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, I - IV. Springer-Verlag, Berlin-Heidelberg-New-York (1983)Google Scholar
  19. [Hor90]
    Hörmander, L.: An introduction to complex analysis in several variables. North-Holland (1990)Google Scholar
  20. [K81]
    Kamynin, L.I., Khimchenko, B.N.: Tikhonov-Petrovskii problem for second-order parabolic equations. Siberian Mathematical Journal, 22 (5), 709–734 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Kit76]
    Kitagawa, K.: A remark on a necessary condition of the Cauchy-Kowalevsky. Publ. RIMS, Kyoto Univ. 11, 523–534 (1976)Google Scholar
  22. [Kit90]
    Kitagawa, K.: Sur le téorème de Cauchy-Kowalevski. J. Math. Kyoto Univ. 30 (1), 1–32 (1990)MathSciNetzbMATHGoogle Scholar
  23. [K73]
    Korobeinik, Yu.F., Kubrak, V.K.: The existence of particular solutions of a differential equation of infinite order with prescribed growth. Izv. Vys. Matematika. 9 (136), 36–45 (1973) (in Russian)MathSciNetGoogle Scholar
  24. [Kow1874]
    Kowalevsky, S.: Zür theorie der partiellen differentialgleichungen. Journal für die reine und angevandte mathematik, 1–32 (1874)Google Scholar
  25. [Leo76]
    Leont’ev, A.F.: Series of Exponents, Nauka, Moscow (1976) (in Russian)Google Scholar
  26. [Ler53]
    Leray, J.: Hyperbolic differential equations. Princeton (1953)Google Scholar
  27. [Met93]
    Métivier, G.: Counterexamples to Hölmgren’s uniqueness for analytic non linear Cauchy problems. Invent. Math., 112, 217–222 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Miz62]
    Mizohata, S.: Solutiones nulles et solutiones non analytiques. J. Math. Kioto. Univ. 2 (1), 271–302 (1962)Google Scholar
  29. [Miz67]
    Mizohata, S.: The Theory of Partial Differential Equations. Cambridge University Press (1979)Google Scholar
  30. [Miz74]
    Mizohata, S.: On Kowalewskian systems. Russ. Math. Surv. 29 (2), 223–235 (1974) doi:10.1070/RM1974v029n02ABEH003837MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Miz74]
    Mizohata, S.: On Cauchy-Kowalevski’s Theorem: A Necessary Condition. Publ. RIMS, Kyoto Univ. 10, 509–519 (1974)Google Scholar
  32. [Mor58]
    Morrey, C.: On the analyticity of the solutions of analytic nonlinear elliptic systems of partial differential equations. Amer. J. Math. 80 (1), 198–277 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [MN57]
    Morrey, C., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10, 271–290 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [Nap82]
    Napalkov, V.V.: Convolution equations in multidimensional spaces. Mathematical Notes of the Acad. Sci. USSR, 25 (5), 393 pp., Springer (1979)Google Scholar
  35. [OR73]
    Oleynik, O.A., Radkevich, E.V.: On the analyticity of solutions of linear partial differential equations. Mathematics of the USSR-Sbornik, 19 (4), 581–596 (1973)CrossRefGoogle Scholar
  36. [Ovs65]
    Ovsyannikov, L.V.: A singular operator in a scale of Banach space. Dokl. Akad. Nauk SSSR, 163, 819–822 (1965) (English transl. Soviet Math. Dokl., 6 (1965))Google Scholar
  37. [Pet96]
    Petrovskii, I.G.: Selected works: Systems of partial differential equations and algebraic geometry, Part I. Gordon and Breach Publishers (1996)Google Scholar
  38. [Pl54]
    Plis, A.: The problem of uniqueness for the solution of a system of partial differential equations. Bull. Acad. Polon. Sci. 2, 55–57 (1954)MathSciNetzbMATHGoogle Scholar
  39. [Ren10]
    Render, H.: Goursat and Dirichlet problems for holomorphic partial differential equations. Comput. Meth. Funct. Theory. 10, 519–554 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [Sch51]
    Schwartz, L.: Théorie des distributions I, II. Hermann, Paris (1951)Google Scholar
  41. [S88]
    Sternin, B.Y., Shatalov, V.E.: Differential equations on complex manifolds and Maslov’s canonical operators. Sov. Math. Survs. 43, 99–124 (1988)MathSciNetzbMATHGoogle Scholar
  42. [Suz64]
    Suzuki, H.: Analytic hypoelliptic differential operators of first order in two independent variables. J. Math. Society Japan. 16 (4), 367–374 (1964)CrossRefzbMATHGoogle Scholar
  43. [Tac36]
    Täcklind, S.: Sur les classes quasianalitiques des solutions des l’equations aux derivées partielles du type parabolique. Nord. Acta. Regial. Sociatis schientiarum Uppsaliensis. Ser. 4, 10 (3), 3–55 (1936)Google Scholar
  44. [Tam06]
    Tamura, M.: On uniqueness in the Cauchy problem for systems for systems with partial analytic coefficients. Osaka J. Math. 43, 751–769 (2006)MathSciNetzbMATHGoogle Scholar
  45. [Tik35]
    Tikhonov, A.N.: Uniqueness theorems for heat equations, Mat. Sb., 42 (2), 199–216 (1935)MathSciNetzbMATHGoogle Scholar
  46. [Tre80]
    Treves, F.: Introduction to Pseudo-Differential and Fourier Integral Operators. Plenum Publishing Co., New York (1980)CrossRefGoogle Scholar
  47. [Uma91-1]
    Umarov, S.R.: Algebra of pseudo-differential operators with variable analytic symbols and well-posedness of corresponding equations. Differential Equations. 27 (6), 1056–1063 (1991)MathSciNetGoogle Scholar
  48. [Uma14]
    Umarov, S.R.: Pseudo-differential operators with meromorphic symbols and systems of complex differential equations. Complex variables and elliptic equations: An International Journal. 60(6), 829–863 (2015) DOI: 10.1080/17476933.2014.979812MathSciNetCrossRefGoogle Scholar
  49. [Uch04]
    Uchida, M.: Hörmander form and uniqueness for the Cauchy problem. Advances in Mathematics, 189, 237–245 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  50. [Vol63]
    Volevich, L.R.: A problem in linear programming stemming from differential equations. Uspekhi Mat. Nauk, 18, 3 (111), 155–162 (1963)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sabir Umarov
    • 1
  1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

Personalised recommendations