Regularity and Solvability of Pseudo-differential Operators with Double Characteristics

Abstract

This paper deals with several classes of pseudo-differential operators with double involutive characteristics as well as operators with double involutive–symplectic characteristics. In the first case, microlocal non-solvability is shown, while in the second case, the conditions imposed on the subprincipal symbol are the same as the conditions guaranteeing subellipticity of the operators of principal type. The corresponding loss of regularity is $1+ \frac{k}{k+1}$, $k \in {\mathbf{N}}$. At the end of the paper, an inverse operator of a model one with involutive characteristics and non-elliptic subprincipal symbol is constructed into explicit form.

Appendix: Subelliptic Estimates for Classes of Differential Operators via Newton Polyhedron

To begin with consider the polynomial

$$\begin{aligned} P(\xi ) = \sum _{\gamma \in M_{P}} a_{\gamma } \xi ^{\gamma }, \xi = (\xi _{1}, \xi _{2}, \ldots , \xi _{n}) \in {\mathbf{R}}^{n} \end{aligned}$$

(3.1)

with real coefficients $a_{\gamma }$, some of them being eventually zero. The symbol $M_{P}$ stands for the finite set of vectors having non-negative integer coordinates, i.e., $\gamma = (\gamma _{1}, \ldots , \gamma _{n}) \in {\mathbf{R}}_{+}^{n},$ $\gamma _{j} \in \mathbf{Z}_{+}$ $1 \le j \le n$. Evidently, the monomial $\xi ^{\gamma } = \xi _{1}^{\gamma _{1}} \ldots \xi _{n}^{\gamma _{n}}$. Moreover:

$$\begin{aligned} |P(\xi )| \le c \sum _{\gamma \in M_{P}} |\xi ^{\gamma }|, c = \mathrm{const} > 0. \end{aligned}$$

(3.2)

We are interested to find the polynomials of the form (3.1) for which $|P(\xi )|$ can be estimated in a similar way as in (3.2) but from below.

Assume that $\gamma \in {\mathbf{R}}_{+}^{n}$ ($\gamma$ is not obliged to belong to $\mathbf{Z}_{+}^{n}$) belongs to the convex hull of the points $\gamma ^{1}, \dots , \gamma ^{N}$, i.e., $\gamma = \sum _{j=1}^{N} \alpha _{j} \gamma ^{j}$, $\alpha _{j} \ge 0$, $\sum _{j=1}^{N} \alpha _{j} = 1$.

Applying Iensen’s inequality, we get that

$$\begin{aligned} |\xi ^{\gamma }| \le \sum _{j=1}^{N} |\xi ^{\gamma ^{j}}| \ \ \ \mathrm{for} \ \ \mathrm{each} \ \ \ \xi \in {\mathbf{R}}^{n}. \end{aligned}$$

(3.3)

Denote by $\Gamma _{P}$ the convex polyhedron generated by the points of $M_{P}$ ( $\mathrm{dim} \Gamma _{P} \le n$ and it is possible $\mathrm{dim} \Gamma _{P} < n$) and by $V_{P}$ the set of the vertexes of $\Gamma _{P}$. $\Gamma _{P}$ is called Newton polyhedron of the polynomial P. Put

$$\begin{aligned} N_{P}(\xi ) = \sum _{\gamma \in V_{P}} |\xi ^{\gamma }|, \xi \in {\mathbf{R}}^{n}. \end{aligned}$$

One can see that $N_{P} (\xi ) \le \sum _{\gamma \in M_{P}} |\xi ^{\gamma }|$ and according to (3.3):

$$\begin{aligned} \sum _{\gamma \in \Gamma _{P}} |\xi ^{\gamma }| \le c N_{P} (\xi ) \end{aligned}$$

(3.4)

for some constant $c > 0$.

This way, we come to the following question: when does $|P(\xi )|$ admit the estimate from below

$$\begin{aligned} c N_{P}(\xi ) \le |P(\xi )|, \forall \xi \in {\mathbf{R}}^{n} \end{aligned}$$

(3.5)

for some constant $c > 0$?

The answer of this question is given in [10] and [11]. As we shall use it in obtaining of some subelliptic estimates, we shall formulate the corresponding results. The Newton polyhedrons can be described by their supporting hyperplanes. Thus, for each fixed $\kappa \in {\mathbf{R}}^{n} {\setminus } 0$, we consider the supporting hyperplane $(\kappa , \gamma ) = q(\kappa )$, such that $(\kappa , \gamma ) \le q(\kappa )$ for $\gamma \in \Gamma _{P}$. $q(\kappa )$ is uniquely determined, while $\Gamma _{P}$ is defined by infinitely many inequalities $(\kappa , \gamma ) \le q(\kappa )$, $\kappa \in {\mathbf{R}}^{n} {\setminus } 0$. One can select finitely many among them that imply the rest of the inequalities. Denote by $\Gamma _{P}^{\kappa }$ the set of all points $\gamma \in \Gamma _{P}$ (i.e., $\Gamma _{P}^{\kappa } \subset \Gamma _{P}$) for which $(\kappa , \gamma ) = q(\kappa )$. Evidently, $\Gamma _{P}^{\kappa }$ is a face of the polyhedron $\Gamma _{P}$ and if $\mathrm{dim} \Gamma _{P} = n$, the faces of dimension $(n-1)$ are determined uniquely by the vector $\kappa$ up to a positive factor. The description of the faces of dimension $\le n-2$ is more complicated and we avoid it. We can deal with vectors $\kappa$ with rational and even integer components only but having arbitrary signs.

Put

$$\begin{aligned} P^{\kappa } (\xi ) = \sum _{\gamma \in \Gamma _{P}^{\kappa }} a_{\gamma } \xi ^{\gamma }, \end{aligned}$$

i.e., in the sum participate the monoms corresponding to the faces $\Gamma _{P}^{\kappa }$.

The results which we are proposing can be found in [10, 11].

Proposition 3.1

$P(\rho ^{\kappa } \xi ) = \rho ^{q(\kappa )} P^{\kappa }(\xi ) + o(\rho ^{q(\kappa )})$, $\rho \rightarrow + \infty$, $\rho > 0$; $P^{\kappa }(\rho ^{\kappa } \xi ) = \rho ^{q(\kappa )} P^{\kappa } (\xi )$, where $\rho ^{\kappa } \xi = (\rho ^{\kappa _{1}} \xi _{1}, \ldots , \rho ^{\kappa _{n}} \xi _{n})$.

We shall say that the polynomial $P^{\kappa }(\xi )$ is the principal $\kappa$-homogeneous part of $P(\xi )$. Certainly, $P^{\kappa }$ is quasihomogeneous polynomial with respect to the weight vector $\kappa$. We define $\mathrm{deg}\,_{\kappa } P = q(\kappa )$ and point out that $(\rho ^{\kappa } \xi )^{\gamma } = \rho ^{(\kappa , \gamma )} \xi ^{\gamma }$. As a corollary of Proposition 3.1, we get that

$$\begin{aligned} (\mathrm{PQ})^{\kappa } = P^{\kappa } Q^{\kappa }, \mathrm{deg}\,_{\kappa } (\mathrm{PQ}) = \mathrm{deg}\,_{\kappa } P + \deg _{\kappa } Q. \end{aligned}$$

This is the main result.

Theorem 3.2

The inequality

$$\begin{aligned} c N_{P}(\xi ) \le |P(\xi )|, \forall \xi \in {\mathbf{R}}^{n} \end{aligned}$$

holds with some constant $c > 0$ if and only if

$$\begin{aligned} P(\xi ) \ne 0, P^{\kappa }(\xi ) \ne 0 \ \ \ \mathrm{for} \ \ \ \xi ^{(1)} = \xi _{1} \ldots \xi _{n} \ne 0, \end{aligned}$$

i.e., outside the coordinate hyperplanes.

There are other conditions guaranteeing the fulfillment of (3.5).

Theorem 3.3

The inequality (3.5) holds if and only if

• (i) One can find $\varepsilon > 0$ and such that all polynomials $P_{\delta }(\xi ) = \sum (a_{\gamma } + \delta _{\gamma }) \xi ^{\gamma }$, $|\delta _{\gamma }| < \varepsilon$, $\gamma \in \Gamma _{P}$ do not vanish at $\xi ^{(1)} \ne 0$.

• (ii) There exists $c > 0$, such that for each $\xi \in {\mathbf{R}}^{n}$, the polynomial $P(\zeta ) \ne 0$ in the polycylinder:

  $$\begin{aligned} | \zeta _{j} - \xi _{j}| \le c |\xi _{j}|, 1 \le j \le n, \zeta \in \mathbf{C}^{n}. \end{aligned}$$

Remark 3.4

Assume that the polynomial $P(\xi )$ verifies the conditions of Theorem 3.2. Then, $Q = P^{2}$ satisfies the same condition. In fact, $Q(\xi ) \ge 0$ for each $\xi \in {\mathbf{R}}^{n}$, $Q(\xi ^{(1)}) > 0$, $Q^{\kappa } = (P^{\kappa })^{2} \Rightarrow Q^{\kappa }(\xi ^{(1)}) > 0$.

Consider the ordinary differential operator $\mathrm{\,\mathrm{Ru}\,} = u^{\prime }(t) + Q(\lambda , t) u$ with $\lambda = (\lambda _{1}, \dots , \lambda _{n-1})$ and $\xi = (\lambda , t) \in {\mathbf{R}}^{n}$. As we know, $Q \ge 0$, $Q(\lambda ^{(1)}, t^{(1)}) > 0$ for $\lambda _{1} \ldots \lambda _{n-1} t \ne 0$, $\lambda ^{(1)} = \lambda _{1} \ldots \lambda _{n-1}$, $t = t^{(1)}$ and $Q^{\kappa }(\lambda ^{(1)}, t^{(1)}) > 0$ imply that

$$\begin{aligned} Q(\lambda ,t) \ge c N_{Q}(\lambda ,t), \forall (\lambda ,t) \in {\mathbf{R}}^{n}. \end{aligned}$$

We shall suppose further on that $(\lambda _{1}, \ldots , \lambda _{n-1}) \in {\mathbf{R}}_{+}^{n-1}$, $\lambda$ being a “large parameter”; $\lambda _{1} \ge 1$, $\ldots , \lambda _{n-1} \ge 1$, $t \in {\mathbf{R}}^{1}$.

Repeating the proof of estimate (1.9), we have the following:

$$\begin{aligned} 2 \mathrm{Re} \int _{-\infty }^{+ \infty } [u^{\prime }(s) + Q(\lambda ,s) u]\bar{u}(s) \mathrm{d}s= |u|^{2}(t) + 2 \int _{-\infty }^{t} Q(\lambda , s) |u|^{2} \mathrm{d}s, \end{aligned}$$

where $u \in C_{0}^{\infty } ({\mathbf{R}}^{1})$. Therefore:

$$\begin{aligned} 2 \int _{-\infty }^{\infty } |\,\mathrm{Ru}\,(s)| |u(s)\ \mathrm{d}s\ge & {} \frac{1}{2} \mathrm{max}_{t} |u(t)|^{2} + \int _{-\infty }^{\infty } Q(\lambda ,s) |u|^{2} \mathrm{d}s \ge \frac{1}{2} \mathrm{max}_{t} |u(t)|^{2} \nonumber \\&+\,c \int _{-\infty }^{\infty } N_{Q}(\lambda ,s) |u|^{2} \mathrm{d}s \ge \frac{1}{2} \mathrm{max}_{t} |u(t)|^{2}\nonumber \\&+\, c \int _{-\infty }^{\infty } \sum _{\gamma \in V_{Q}} |\xi ^{\gamma }| |u(s)|^{2} \mathrm{d}s, \end{aligned}$$

(3.6)

$\xi = (\lambda ,\rho )$ and if $\gamma = (\gamma ^{\prime }, \gamma _{n}) \in V_{Q}$, $|\xi ^{\gamma }| = \lambda ^{\gamma ^{\prime }} |\rho |^{\gamma _{n}}$.

As in Theorem 3.2, $V_{Q} \subset {\mathbf{R}}_{+}^{n}$ is the Newton polyhedron of the polynomial $Q(\xi )$. We split ${\mathbf{R}}_{s}^{1}$ into two parts: $M = \{ s \in {\mathbf{R}}^{1}: |s| \ge \lambda ^{\alpha } \}$ and ${\mathbf{R}}^{1} {\setminus } M$. Here, the vector $\alpha = (\alpha _{1}, \ldots , \alpha _{n-1})$ will be found; later on, $\alpha = \alpha (\gamma )$.

Easy computations show that

$$\begin{aligned} \lambda ^{\gamma ^{\prime }} \int _{-\infty }^{\infty } |\rho |^{\gamma _{n}} |u(s)|^{2} \mathrm{d}s \ge \lambda ^{\gamma ^{\prime }+\alpha \gamma _{n}} \int _{M} |u|^{2} \mathrm{d}s, \int _{{\mathbf{R}}^{1} {\setminus } M} |u|^{2}\mathrm{d}s \le 2 \lambda ^{\alpha } \mathrm{max}_{s} |u|^{2}. \end{aligned}$$

The right-hand side of (3.6) is estimated from above by $||\,\mathrm{Ru}\,||_{L^{2}({\mathbf{R}}^{1})} ||u||_{L^{2}({\mathbf{R}}^{1})}$, while the left-hand side is estimated from below by the sum $c_{1} \sum _{\gamma \in V_{Q}} (\lambda ^{-\alpha } \int _{{\mathbf{R}}^{1} {\setminus } M} |u|^{2} + \lambda ^{\gamma ^{\prime } + \alpha \gamma _{n}} \int _{M} |u|^{2})$, where $\alpha = \alpha (\gamma )$.

We solve the equation:

$$\begin{aligned} - \alpha = \gamma ^{\prime } + \alpha \gamma _{n} \iff \alpha = - \frac{\gamma ^{\prime }}{1+ \gamma _{n}}. \end{aligned}$$

Consequently:

$$\begin{aligned} ||u^{\prime }(t) + Q(\lambda ,t) u(t)||_{0} \ge c_{2} \sum _{(\gamma ^{\prime }, \gamma _{n}) \in V_{Q}} \lambda ^{\frac{\gamma ^{\prime }}{1+\gamma _{n}}} ||u||_{0}, \end{aligned}$$

(3.7)

$\forall u \in C_{0}^{\infty }({\mathbf{R}}^{1})$ and $c_{2} = \mathrm{const} > 0$.

The result is interesting if there exists at least one vertex $(\gamma ^{' 0}, \gamma _{n}^{0}) \in V_{Q}$ for which $\gamma ^{'0} \ne 0$.

Define the pseudo-differential operator $D_{1+}$ with symbol $\lambda _{1}$ for $\lambda _{1} \ge 1$ and 0 for $\lambda _{1} \le 1/2$, respectively, $D_{+}^{\overrightarrow{1}} = D_{1+} \ldots D_{n-1+}$, $\lambda = (\lambda _{1}, \ldots , \lambda _{n-1})$, i.e., $F(D_{+}^{\overrightarrow{1}}) = \lambda _{1} \lambda _{2} \ldots \lambda _{n-1}$, for $\lambda _{j} \ge 1$, $j =1,2,\ldots , n-1$, $\overrightarrow{1}=(1,\ldots ,1)$. The above proposed considerations lead us to Proposition 3.5.

Thus, define the differential operator $M = \partial _{t} + Q(D_{x},t)$, $x \in {\mathbf{R}}^{n-1}$, $t \in {\mathbf{R}}$, where the dual variable of x is $\lambda$, $\xi = (\lambda ,t)$, the polynomial $Q(\lambda ,t) \ge 0$ and $Q(\lambda ,t) \ge c N_{Q}(\lambda ,t)$, $c > 0$, $N_{Q}(\xi ) = \sum _{\gamma \in V_{Q}} |\xi ^{\gamma }|$. As we mentioned, $V_{Q}$ is the set of all vertexes of some convex polyhedron $\Gamma _{Q} \subset {\mathbf{R}}_{+}^{n}$. Put $\hat{\hat{u}} (\lambda ,t) = F_{x \rightarrow \lambda } (u(x,t))$ for the partial Fourier transform of $u \in C_{0}^{\infty }({\mathbf{R}}^{n})$.

Evidently, $\hat{\hat{f}} = \widehat{\widehat{Mu}} = \partial _{t} \hat{\hat{u}} + Q(\lambda ,t) \hat{\hat{u}}$. After an integration with respect to $\lambda$, one gets from (3.7) and Parseval’s formula:

$$\begin{aligned} ||Mu||^{2}_{L^{2}({\mathbf{R}}_{x}^{n-1} \times {\mathbf{R}}_{t}^{1})}\ge & {} c_{3}^{2} \sum _{(\gamma ^{\prime },\gamma _{n}) \in V_{Q}, \gamma ^{\prime } \ne 0} \int _{{\mathbf{R}}^{n-1}} \int _{-\infty }^{\infty } \lambda ^{\frac{2 \gamma ^{\prime }}{1+\gamma _{n}}} |\hat{u}(\lambda , \tau )|^{2} d \lambda d \tau \nonumber \\= & {} c_{3}^{2} \sum _{\gamma \in V_{Q}, \gamma ^{\prime } \ne 0} ||D_{x+}^{\frac{\gamma ^{\prime }}{1+\gamma _{n}}} u||^{2}_{L^{2}({\mathbf{R}}_{x,t}^{n})}. \end{aligned}$$

(3.8)

Certainly, $D_{+}^{\frac{\gamma ^{\prime }}{1+\gamma _{n}}}= D_{1+}^{\frac{\gamma _{1}}{1+\gamma _{n}}} \ldots D_{n-1+}^{\frac{\gamma _{n-1}}{1+\gamma _{n}}}$, $c_{3} > 0$.

Proposition 3.5

Let $M_{\pm } = \partial _{t} \pm Q(D_{x},t)$ be a differential operator with polynomial symbol $Q(\lambda ,t) \ge 0$ for which condition (3.5) holds. Then, $M_{\pm }$ satisfies the estimate (3.8).

This way, we proved subelliptic type estimate (3.8) for special classes of degenerate parabolic (backward parabolic) operators with polynomial coefficients in the time variable t.