Abstract
This paper is devoted to a simpler derivation of energy estimates and a proof of the well-posedness, compared to previously existing ones, for effectively hyperbolic Cauchy problem. One difference is that instead of using the general Fourier integral operator, we only use a change of local coordinates x (of the configuration space) leaving the time variable invariant. Another difference is an efficient application of the Weyl-Hörmander calculus of pseudodifferential operators associated with several different metrics.
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1 Introduction
Consider
where \(A_j(t, x, D)\) are differential operators of order j depending smoothly on t, having the principal symbol
where \(a(t, x, \xi )\) is positively homogeneous of degree 2 in \(\xi \) and nonnegative for any \((t, x, \xi )\in U\times {{\mathbb {R}}}^d\) with some neighborhood U of \((0, 0)\in {{\mathbb {R}}}^{d+1}\).
In [6], Ivrii and Petkov proved that if the Cauchy problem for P is \(C^{\infty }\) well posed for any lower order term then every critical point of \(p=0\) is effectively hyperbolic, namely the Hamilton map has a pair of non-zero real eigenvalues there. In [7], Ivrii has proved that if every critical point is effectively hyperbolic and p admits a decomposition \(p=q_1q_2\) nearby with real smooth symbols \(q_i\) vanishing at the reference point, then the Cauchy problem is \(C^{\infty }\) well-posed for every lower order term, transforming the original P by operator powers of operator to one with a suitable lower order term for which a standard energy method can be applied, and has conjectured that this is true without any restriction.
If a critical point \((t, x, \tau , \xi )\) is effectively hyperbolic then \(\tau \) is a characteristic root of multiplicity at most 3 ([6, Lemma 8.1]) and if every multiple characteristic root is at most double, the conjecture has been proved in [9,10,11, 16, 17]. In [9, 10] the proof is based on the reduction of the original P to an operator for which an improved version of the method of [7] can be applied, where the reduction is made applying the Nash-Moser implicit function theorem. On the other hand, in [16] (see also [19]) the proof is based on energy estimates with pseudodifferential weights of which symbol comes from a geometric characterization of effectively hyperbolic characteristic points, after some preliminary transformations by Fourier integral operators, while in [20] another way to obtain microlocal energy estimates without the use of Fourier integral operators was given, where the original P is transformed by Gevrey pseudodifferential operators on the (t, x)-space to one with symbol extended in the complex directions, to which one can apply the classical separating operator method.
In this paper, we propose a simpler derivation of energy estimates and proof of the well-posedness of the Cauchy problem for effectively hyperbolic operators. Although we follow [19] mainly, one difference is that instead of using the general Fourier integral operator when transforming the operator, we only use a change of local coordinates x (of the configuration space which extends as a linear transformation outside a compact set) leaving the time variable invariant. This allows us to simplify the analysis of deducing the result for the original operator from that obtained for the transformed operator. Another difference is the application of Weyl-Hörmander calculus of pseudodifferential operators associated with several different metrics. The method has been used in a naive way in [19], but here we aim to organize the approach thoroughly. As a result, the argument to derive energy estimates for localized operators is made simpler and clearer and so is the proof of the local existence and uniqueness of the solution to the original Cauchy problem.
For the Cauchy problem for operators with triple effectively hyperbolic characteristics, where p cannot be smoothly factorized, see [22] and the references given there.
2 Geometric characterization of effectively hyperbolic characteristics
In this section, we prove the following proposition, which provides a geometric characterization of effectively hyperbolic characteristics ([18, Lemmas 3.1, 3.2], [19, Section 2.1]).
Proposition 2.1
Assume that \((0, 0, 0, {\bar{\xi }})\) is effectively hyperbolic. One can choose a local coordinates x around \(x=0\) such that \({\bar{\xi }}=e_d=(0,\ldots , 0, 1)\) and smooth function \(\psi (x, \xi )\), positively homogeneous of degree 0 vanishing at \((0,e_d)\), such that either \(d\psi =d\xi _1\) or \(d\psi =\varepsilon dx_1+cdx_d\) at \((0, e_d)\) where \(c\in {{\mathbb {R}}}\) and \(\varepsilon =0\) or 1, and smooth \(\ell (t, x, \xi )\), \(q(t, x, \xi )\ge 0\) vanishing at \((0, 0, e_d)\), positively homogeneous of degree 1, 2 respectively such that
with some \({{\bar{c}}}>0\) on a conic neighborhood of \((0, 0, e_d)\) where
The change of coordinates \(x\mapsto \chi (x)\) can be extended to a diffeomorphism on \({{\mathbb {R}}}^d\) such that \(\chi (x)\) is a linear transformation outside a neighborhood of \(x=0\).
The coordinates change is called (a) or (b) according to the resulting form \(d\psi =d\xi _1\) or \(d\psi =\varepsilon x_1+cx_d\), in each case one can write
where \(dr(0, 0, e_d)=0\). Note that \(\{\psi , \{\psi , q\}\}(0, 0, e_d)=0\) implies that
according to the case (a) or (b) since \(\partial _{\xi _j\xi _d}^2q(0, 0, d_d)=0\) by the Euler’s identity for homogeneous functions.
2.1 A key lemma
In this subsection, for typographical reason, we write \(x_0\) for t and \(\xi _0\) for \(\tau \) and denote \(x=(x_0, x')=(x_0, x_1,\ldots , x_d)\) and \(\xi =(\xi _0, \xi ')=(\xi _0, \xi _1,\ldots , \xi _d)\) so that \( p(x, \xi )=-\xi _0^2+a(x, \xi ')\). We also write \(z=(x, \xi )\), \(v=(y, \eta )\in {{\mathbb {R}}}^{d+1}\times {{\mathbb {R}}}^{d+1}=V\). Let \(\rho =(0, {\bar{\xi }})\) be a critical point of \(p=0\) and hence \({\bar{\xi }}_0=0\) and \(p(\rho )=\nabla p(\rho )=(\partial p(\rho )/\partial x, \partial p(\rho )/\partial \xi )=0\). Consider the Hamilton equation
then it is clear that the linearized equation at \(\rho \) is given by
where the half of the coefficient matrix is denoted by \(F_p(\rho )\) and called the Hamilton map (matrix) of p at \(\rho \). Denoting the quadratic (polarized) form associated with the Hesse matrix of p at \(\rho \) by Q(z, v) it is clear that
where \(\sigma (z, v)=\langle {\xi , y}\rangle -\langle {x, \eta }\rangle \), \(z=(x, \xi )\), \(v=(y, \eta )\) is the symplectic two form on V. From the definition we see \(p(\rho +\epsilon z)=\epsilon ^2Q(z)/2+O(\epsilon ^3)\) as \(\epsilon \rightarrow 0\) and Q has the signature (r, 1) with some \(r\in {{\mathbb {N}}}\) since \(a(x, \xi ')\) is nonnegative near \(\rho '=(0, {\bar{\xi }}')\in {{\mathbb {R}}}^{d+1}\times {{\mathbb {R}}}^d\). Moreover, it follows from the Morse lemma (see, e.g. [4, Lemma C.6.2]) that one can find \(\phi _1, \ldots , \phi _r\) and g vanishing at \(\rho '\), homogeneous of degree 1, 2 in \(\xi '\) respectively, \(C^{\infty }\) in a conic neighborhood of \(\rho '\) such that \(\nabla \phi _1, \ldots , \nabla \phi _r\) are linearly independent at \(\rho '\) and \(g\ge 0\), \(\nabla ^2\,g(\rho ')=O\) and
With \(\phi _0=\xi _0\) it is clear \(Q(z, v)=-\langle {\nabla \phi _0, z}\rangle \langle {\nabla \phi _0, v}\rangle +\sum _{j=1}^r\langle {\nabla \phi _j, z}\rangle \langle {\nabla \phi _j, v}\rangle \). Then noticing \(\langle {\nabla \phi _j, z}\rangle =\sigma (z, H_{\phi _j})\) we see that
and hence \(F_p v=-\sigma (v, H_{\phi _0})H_{\phi _0}+\sum _{j=1}^r\sigma (v, H_{\phi _j})H_{\phi _j}\). Therefore the kernel and the image of \(F_p\) are given by
Consider the following open convex cone in V
which is the connected component of \(\{z\in V\mid Q(z)\ne 0\}\) containing the positive \(\xi _0\) axis. Recall [3, Corollary 1.4.7] for which we give a more direct proof here.
Lemma 2.1
If \(F_p(\rho )\) has a nonzero real eigenvalue then \(\Gamma \cap \textrm{Im}F_p\ne \{0\}\).
Proof
Let \(\lambda \ne 0\) be a real eigenvalue and \(F_pz=\lambda z\) with \(0\ne z\in V\). Then from \(0=\sigma ((F_p-\lambda )z, v)=\sigma (z, (-F_p-\lambda )v)\) for all \(v\in V\) we see that \(F_p+\lambda \) is not surjective which proves that \(-\lambda \) is also an eigenvalue. Let \(F_pz_{\pm }=\pm \lambda z_{\pm }\), \(z_{\pm }\ne 0\) then \(z_{\pm }\in \textrm{Im}F_p\) for \(\lambda \ne 0\). Note that the signature of Q is (r, 1) with \(r\ge 1\) otherwise Q(z) would be \(-\xi _0^2\) and hence \(F_p\) has no nonzero eigenvalues. The quadratic form Q induces a quadratic form \({{\bar{Q}}}\) in \(V_0=V/\textrm{Ker}F_p\) which is non-degenerate and of Lorenz signature. If \(\sigma (z_{+}, z_{-})=0\) then \({{\bar{Q}}}\) would vanish on the 2 dimensional linear subspace of \(V_0\) spanned by \([z_{+}], [z_{-}]\) which is a contradiction. Thus with \(z=\alpha z_{+}+\beta z_{-}\in \textrm{Im}F_p\) we have
Choosing \(\alpha \), \(\beta \) such that \(\alpha \beta \lambda \sigma (z_{+}, z_{-})>0\) we get \(Q(z)<0\) hence either \(z\in \Gamma \) or \(-z\in \Gamma \). \(\square \)
For a linear subspace \(S\subset V\) we denote \(S^{\sigma }=\{z\in V\mid \sigma (z, S)=0\}\) hence \((S^{\sigma })^{\sigma }=S\) and for \(0\ne z\in V\), \(\langle {z}\rangle \) stands for the line \({{\mathbb {R}}}z\). Introduce the dual cone of \(\Gamma \) with respect to \(\sigma \) defined by
The next lemma [19, Lemma 1.1.3] is the key to the geometric characterization of effectively hyperbolic characteristics.
Lemma 2.2
Let \(\theta \) be the unit vector directed to positive \(\xi _0\) axis. The following three conditions are equivalent;
-
(i)
\(\Gamma \cap \textrm{Im}F_p\ne \{0\}\),
-
(ii)
there is a linear subspace \(H\subset V\) of codimension 1 such that \( H\cap C=\{ 0\}\) and \(\textrm{Ker}F_p+\langle {\theta }\rangle \subset H\),
-
(iii)
\(\Gamma \cap \textrm{Im}F_p\cap \langle {\theta }\rangle ^{\sigma }\ne \{0\}\).
Proof
First note that
In fact if there were \(0\ne v\in \langle {z}\rangle ^{\sigma }\cap C\) we would have \(\sigma (v, z+w)=\sigma (v, w)\le 0\) for any small w since \(\Gamma \) is open leads to a contradiction.
\(\mathrm{(i)}\Longrightarrow \mathrm{(ii)}\). We first assume \(\theta \in \textrm{Ker}F_p+\textrm{Im}F_p\) so that \(\theta =z_1+z_2\) with \(z_1\in \textrm{Ker}F_p\) and \(z_2\in \textrm{Im}F_p\). Then \(0\ne z_2\in \Gamma \) since \(\theta \in \Gamma \) and \(\Gamma +\textrm{Ker}F_p\subset \Gamma \) and \(\Gamma \cap \textrm{Ker}F_p=\emptyset \). It is clear that \(\theta \in \langle {z_2}\rangle ^{\sigma }\) because \(\textrm{Ker}F_p\subset \langle {z_2}\rangle ^{\sigma }\) and \(z_2\in \langle {z_2}\rangle ^{\sigma }\) therefore \(H=\langle {z_2}\rangle ^{\sigma }\) is a desired subspace by (2.8).
Next consider the case \(\theta \not \in \textrm{Ker}F_p+\textrm{Im}F_p\) and hence \((\textrm{Ker}F_p+\textrm{Im}F_p)\cap \langle {\theta }\rangle =\{0\}\). Take \(0\ne w\in \Gamma \cap \textrm{Im}F_p\) then \(\textrm{Ker}F_p=(\textrm{Im}F_p)^{\sigma }\subset \langle {w}\rangle ^{\sigma }\) and \(\langle {w}\rangle ^{\sigma }\cap C=\{0\}\) by (2.8), while \(C\subset \textrm{Im}F_p\) for \(\Gamma +\textrm{Ker}F_p\subset \Gamma \) one concludes \(\textrm{Ker}F_p+\textrm{Im}F_p\not \subset \langle {w}\rangle ^{\sigma }\). Therefore we have \(\langle {w}\rangle ^{\sigma }+(\textrm{Ker}F_p+\textrm{Im}F_p)=V\) and hence \(\langle {w}\rangle ^{\sigma }\cap (\textrm{Ker}F_p+\textrm{Im}F_p)\) is of codimension 1 in \(\textrm{Ker}F_p+\textrm{Im}F_p\). Now writing \(V=(\textrm{Ker}F_p+\textrm{Im}F_p)\oplus \langle {\theta }\rangle \oplus W\) (direct sum) it is clear that \(H=(\langle {w}\rangle ^{\sigma }\cap (\textrm{Ker}F_p+\textrm{Im}F_p))\oplus \langle {\theta }\rangle \oplus W\) is a desired subspace.
\(\mathrm{(ii)}\Longrightarrow \mathrm{(iii)}\). Choose \(0\ne v\in V\) such that \(\langle {v}\rangle =H^{\sigma }\). It is clear that \(\langle {v}\rangle \subset \textrm{Im}F_p\cap \langle {\theta }\rangle ^{\sigma }\) for \(\textrm{Ker}F_p+\langle {\theta }\rangle \subset H\). Show that v or \(-v\) belongs to \(\Gamma \). If not we would have \(\langle {v}\rangle \cap \Gamma =\emptyset \) and by the Hahn-Banach theorem there were \(0\ne w\in V\) such that \(\sigma (w, z)\le 0, \forall w\in C\) and \(w\in \langle {v}\rangle ^{\sigma }=H\) which contradicts with (ii).
\(\mathrm{(iii)}\Longrightarrow \mathrm{(i)}\) is trivial. \(\square \)
2.2 Proof of Proposition 2.1
In this subsection we return to the original notation and write t for \(x_0\) and \(\tau \) for \(\xi _0\) and denote \(x=(x_1,\ldots , x_d)\), \(\xi =(\xi _1,\ldots , \xi _d)\). After a suitable linear change of local coordinates x we may assume that \({\bar{\xi }}=(0, \ldots , 0, 1)=e_d\). We write \(\rho '=(0, 0, e_d)\in {{\mathbb {R}}}^{d+1}\times {{\mathbb {R}}}^d\) and \(\rho ''=(0, e_d)\in {{\mathbb {R}}}^d\times {{\mathbb {R}}}^d\). Thanks to Lemma 2.2 one can take \(0\ne z\in \Gamma \cap \textrm{Im}F_p\cap \langle {\theta }\rangle ^{\sigma }\) where \(z=\sum _{j=1}^r\alpha _jH_{\phi _j}(\rho )+\alpha _0H_{\phi _0}(\rho )\) in view of (2.6), where we see \(\alpha _0=-\sigma (z, \theta )=0\) for \(z\in \langle {\theta }\rangle ^{\sigma }\). Let
Since \(H_{f}(\rho ')=z\in \Gamma \) it is clear that \(\partial f/\partial t< 0\) at \(\rho '\) in view of (2.7) then one can write \( f(t, x, \xi )=e(t, x, \xi )(t-\psi (x, \xi ))\) where \(e(\rho ')< 0\). It is clear from (2.5)
with some \(c_1>0\). Since \(-H_{x_0-\psi }(\rho ')\in \Gamma \) we see from (2.7) that
from which, taking (2.5) and \(\nabla ^2g(\rho ')=O\) into account, we conclude that
The next lemma is well known.
Lemma 2.3
Assume \(d\psi \ne 0\) and not proportional to \(dx_d\) at \(\rho ''\). Then one can find a system of local coordinates \(x=(x_1,\ldots , x_d)\) such that either \(d\psi =d\xi _1\) or \(d\psi =dx_1+cdx_d\) with some \(c\in {{\mathbb {R}}}\) at \(\rho ''\).
Proof
Since \(\partial _{\xi _d}\psi (\rho '')=0\) by the Euler’s identity one can write \(\psi (x, \xi )=\langle {{{\tilde{a}}}, {\tilde{\xi }}}\rangle +\langle {{{\tilde{b}}}, {{\tilde{x}}}}\rangle +b_dx_d+r(x, \xi )\) where \({\tilde{\xi }}=(\xi _1, \ldots , \xi _{d-1})\), \({{\tilde{x}}}=(x_1,\ldots , x_{d-1})\) and r vanishes at \(\rho ''\) of order 2. If \({{\tilde{a}}}=0\) hence \({{\tilde{b}}}\ne 0\) a linear change of coordinates \({{\tilde{x}}}\) gives a desired form. If \({{\tilde{a}}}\ne 0\) one can assume \(\langle {{{\tilde{a}}}, {\tilde{\xi }}}\rangle =\xi _1+\cdots +\xi _k\) renumbering and dilating \(x_j\), \(1\le j\le d-1\). Changing the coordinate \(x_d\) to \(x_d-\sum _{j=1}^kb_jx_j^2/2\) yields \(\langle {{{\tilde{b}}}, {\tilde{x}}}\rangle +b_dx_d=\sum _{j=k+1}^db_jx_j\). Changing again the coordinate \(x_d\) to \(x_d-x_1\sum _{j=k+1}^d b_jx_j\) yields \(b_{k+1}=\cdots =b_d=0\) hence after a linear change of coordinates \((x_1,\ldots , x_k)\) one has \(d\psi =d\xi _1\) at \(\rho ''\). \(\square \)
Proof of Proposition 2.1
Let \(\psi \) be the one given in (2.9). If \(d\psi =0\) or proportional to \(dx_d\) at \(\rho ''\) it suffices to take \(\ell =0\) and \(q=a\) because \(\partial _{\xi _d}^2a(\rho ')=0\) by the Euler’s identity. Assume \(d\psi (\rho '')\ne 0\) and not proportional to \(dx_d\). From Lemma 2.3 we may assume \(d\psi =d\xi _1\) or \(d\psi =dx_1+cdx_d\). Assume \(d\psi =d\xi _1\) at \(\rho ''\). If \(\partial _{x_1}^2a(\rho ')=0\) it suffices to take \(\ell =0\) and \(q=a\). Otherwise, thanks to the Malgrange preparation theorem (e.g. [5, Theorem 7.5.5]) one can write
where \(e(\rho ')>0\) and h, g, vanishing at \(\rho '\), are of homogeneous of degree 0. Choose
and set \(\psi _1(t, x',\xi )=\psi (h(t, x', \xi ), x', \xi )\) then \(d\psi _1=d\psi \) at \(\rho '\). From (2.9) there is \(c_2>0\) such that
Since \(\partial \psi _1/\partial t=0\) at \(\rho '\) one can write \( t-\psi _1(t, x',\xi )=e'(t, x', \xi )(t-\psi _2(x', \xi ))\). Since \(d\psi _2=d\psi _1=d\xi _1\) at \(\rho '\) then \(\{{\psi _2},\{{\psi _2}, q\}\}(\rho ')=0\) hence it follows from (2.10) that \(\{\ell , \psi _2\}^2(\rho ')<1\). Thus \(\psi _2\) is a desired one. When \(d\psi =dx_1+cdx_d\) the proof is similar. In Lemma 2.3 we used coordinates changes such that \(y=A x+q(x)\) where A is a non-singular matrix and q(x) is a quadratic form in x, thus cutting q(x) off outside a neighborhood of \(x=0\) it is clear that the resulting change of coordinates satisfies the requirements in Proposition 2.1. \(\square \)
3 Quantitative expression of Proposition 2.1 by localized symbols
In this section, we localize the symbols obtained in Proposition 2.1 around \((0, e_d)\) with a positive parameter M and we will use this M to quantitatively express the condition (2.2). We first localize coordinates functions. Let \(\chi (s)\in C^{\infty }({{\mathbb {R}}})\) be such that \(\chi (s)=s\) on \(|s|\le 1\), \(|\chi (s)|=2\) on \(|s|\ge 2\) and \(0\le d\chi (s)/ds=\chi ^{(1)}(s)\le 1\) everywhere. Define \(y(x)=(y_1(x),\ldots , y_d(x))\) and \({\eta }(\xi )=(\eta _1(\xi ),\ldots , \eta _d(\xi ))\) by
where \(\langle {\xi }\rangle _{\gamma }=(\gamma ^2+|\xi |^2)^{1/2}\) and \(\delta _{ij}\) is the Kronecker’s delta. Here M and \(\gamma \) are positive parameters constrained by
Clearly there is \(C>0\) such that
so that \((y(x), \eta (\xi )+e_d)\) is contained in a neighborhood of \((0, e_d)\) which shrinks with M. Note that \((y(x), (\eta +e_d)\langle {\xi }\rangle _{\gamma })=( x,\xi )\) in a “conic like” neighborhood \(W_{M, \gamma }\) of \((0, e_d)\) given by
because if \((x,\xi )\in W_{M, \gamma }\) then
From now on, fixing a \(T_0>0\), we assume that the range of t is also constrained by
Definition 3.1
For a smooth function \(f(t, x, \xi )\) near \((0, 0, e_d)\) the localization \(f_M\) is defined to be \(f(t, y(x), \eta (\xi )+e_d)\). When f is defined in a conic neighborhood of \((0, 0, e_d)\) and of homogeneous of degree m in \(\xi \) we define \(f_M=f(t, y(x), \eta (\xi )+e_d)\langle {\xi }\rangle _{\gamma }^m=f(t, y(x), (\eta (\xi )+e_d)\langle {\xi }\rangle _{\gamma })\).
Throughout the paper, \(A\lesssim B\) means \(A\le CB\) with some constant C independent of all involved parameters (\(M, \gamma \) here) if otherwise stated. We denote \(A_1\approx A_2\) if \(A_1\lesssim A_2\) and \(A_2\lesssim A_1\). To express (2.2) quantitatively introduce a preliminary metric
It is clear that \(y_j\in S(M^{-1},G)\) and
shows \(\eta _j\in S(M^{-1}, G)\).
Lemma 3.1
Let \(f(t, x, \xi )\) be a smooth function in a neighborhood of \((0, 0, e_d)\) such that \(\partial _t^k\partial _x^{\alpha }\partial _{\xi }^{\beta }f(0, 0, e_d)=0\) for \(k+|\alpha +\beta |<r\). Then \(f_M\in S(M^{-r}, G)\) and
and \(\partial _tf_M\in S(M^{-r+1}, G)\). If the term \(\sum _{k+|\alpha +\beta |=r}\cdots \) contains no \(y_l\) then \(\partial _{x_l}f_M\in S(M^{-r}, G)\) and contains no \(\eta _d\) then \(\partial _{\xi _d}f_M\in S(M^{-r}\langle {\xi }\rangle _{\gamma }^{-1}, G)\). Moreover if the term contains neither \(\eta _d\) nor \(\eta _l\) \((1\le l\le d-1)\) then we have \(\partial _{\xi _l}f_M\in S(M^{-r}\langle {\xi }\rangle _{\gamma }^{-1}, G)\).
Proof
Noting
for \(1\le j\le d-1\), \(1\le k\le d\) the proof follows from the Taylor formula
where the integral belongs to S(1, G) since \(|(t, y, \eta )|\le CM^{-1}\). \(\square \)
Let \(x\mapsto \chi (x)\) be the diffeomorphism on \({{\mathbb {R}}}^d\) obtained in Proposition 2.1 and denoting \((T u)(t, x)=u(t, \kappa (x))\) the localized symbol of \(T^{-1}PT\) is given by
where \(\ell _M\in S(M^{-1}\langle {\xi }\rangle _{\gamma }, G)\), \(q_M\in S(M^{-2}\langle {\xi }\rangle _{\gamma }^2, G)\) and \(a_j\in S(\langle {\xi }\rangle _{\gamma }^j, G)\). Noting \(|\eta (\xi )+e_d|\ge (1-CM^{-1})\) from (2.1) one finds \(M_1>0\), \({\underline{c}}>0\) such that
The following two propositions are quantitative expressions of (2.2).
Proposition 3.1
We have \(\{\psi _M, q_M\}\in S(M^{-2}\langle {\xi }\rangle _{\gamma }, G)\) and that \(|\{\psi _M, q_M\}|\le CM^{-1/2}\sqrt{q_M}\).
Proof
Choose \(f=q\) and \(r=2\) in (3.6) then the quadratic form in \((t, y, \eta )\) is nonnegative definite since \(q(t, y,\eta +e_d)\) is nonnegative. In the case (a) this quadratic form contains no \(y_1\) because of (2.4) hence \(\partial _{x_1}^2q_M(t, x, \xi )\in S(M^{-1}\langle {\xi }\rangle _{\gamma }^2, G)\) and \(\partial _{x_j}^2q_M(t, x, \xi )\in S(\langle {\xi }\rangle _{\gamma }^2, G)\) by Lemma 3.1 then from the Glaeser inequality one obtains
In the case (b), thanks to Euler’s identity and (2.4) we have \(\partial _{\xi _d}^2q(0, 0, e_d)=0\) and \(\varepsilon \partial _{\xi _1}^2q(0, 0, e_d)=0\) hence repeating the same arguments as above one obtains
Next study \(\psi _M\). In the case (a) since \( |\eta (\xi )+e_d|^2=\sum _{j=1}^{d-1}\eta _j^2+(\eta _d+1)^2=1+k\) with \(k\in S(M^{-1}, G)\) hence \(1/|\eta (\xi )+e_d|=1+{{\tilde{k}}}\) with \({{\tilde{k}}}\in S(M^{-1}, G)\) one sees \(\eta _1(\xi )/|\eta (\xi )+e_d|-\eta _1(\xi )\in S(M^{-2}, G)\). Then noting (2.3) it follows from Lemma 3.1 that
In the case (b) we have similarly that
Now proceed to the proof of the proposition. In the case (a), noting \(\partial _{x_1}q_M\in S(M^{-2}\langle {\xi }\rangle _{\gamma }^2, G)\), the first assertion follows from (3.10) and Lemma 3.1. The second assertion follows from (3.8) and (3.10). The proof for the case (b) is similar. \(\square \)
Proposition 3.2
We have \(\{{\ell _M}, {\psi _M}\}\in S(1, G)\) and \(\sup |\{\ell _M, \psi _M\}|\le |\kappa |+CM^{-1}\) where \(|\kappa |<1\).
Proof
Note that \(\partial _{\xi _d}\ell _M \in S(M^{-1}, G)\) for \(\partial _{\xi _d}\ell (0, 0, e_d)=0\) by Euler’s identity. According to the case (a) or (b) we have \(\partial _x^{\alpha }\psi _M\in S(M^{-1}, G)\) or \(\partial _{\xi }^{\alpha }\psi _M(M^{-1}\langle {\xi }\rangle _{\gamma }^{-1}, G)\) for \(|\alpha |=1\) in view of (2.3) then it follows from (3.10) and (3.11) that
where \(\kappa =\partial _{x_1}\ell (0, e_d)\) or \(\kappa =-\varepsilon \partial _{\xi _1}\ell (0, e_d)\) and \(|\kappa |<1\) by (2.2). Noting that \(\chi ^{(1)}(Mx_1)\chi ^{(1)}(M\xi _1\langle {\xi }\rangle _{\gamma }^{-1})\in S(1, G)\) and whose modulus is at most 1 the proof is complete. \(\square \)
From now on, for notational simplicity we simply write \(\psi \), \(\ell \) and q instead of \(\psi _M\), \(\ell _M\) and \(q_M\).
4 Energy estimates for localized operators
In this section, we utilize \(t-\psi (x, \xi )\) obtained from the geometric characterization of effectively hyperbolic characteristic points to derive the weighted energy estimate for the localized operator \({{\hat{P}}}=\textrm{op}({{{\hat{P}}}(t, x, \tau , \xi )})\).
4.1 Metrics and weights related to energy estimates
In this paper the following simple metrics are used;
where \(g_{\epsilon }\) is related to the coordinates change (a) or (b), namely \(\epsilon \) is either a or b and \(\delta _{\epsilon \epsilon '}=1\) if \(\epsilon =\epsilon '\) and 0 otherwise. The properties of pseudodifferential operators associated with metrics (4.1) are summarized in the Appendix. It is clear that
such that \(g_{\epsilon }\) satisfies (6.31). Noting that \(a\in S(m, g_{\epsilon })\) if and only if
and \(M^{|\alpha +\beta |}\langle {\xi }\rangle _{\gamma }^{-|\beta |}\le (M^4\langle {\xi }\rangle _{\gamma }^{-1})^{|\alpha +\beta |/2}M^{-\epsilon (\alpha , \beta )}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}\) it is clear that \(S(m, G)\subset S(m, g_{\epsilon })\). Following Sect. 6.2 we set
then there exists \({\bar{\lambda }}\) such that for \(\lambda \ge {\bar{\lambda }}\) both Proposition 6.1 and Lemma 6.8 hold. From now on we fix such a \(\lambda ={\bar{\lambda }}\), while M and \(\gamma \) remain to be free under the constraints (3.1) and (6.21). Introducing
and taking (3.7) and \(\langle {\xi }\rangle _{\gamma }^{-1/2}\le \omega \) into account one sees that b satisfies (\({\bar{\lambda }}\ge {\underline{c}}\) can be assumed)
Lemma 4.1
We have \(\partial _x^{\alpha }\partial _{\xi }^{\beta }q\in S(\langle {\xi }\rangle _{\gamma }^{1-|\beta |} b, {{\bar{g}}})\) for \(|\alpha +\beta |=1\), \(\partial _tq\in S(\langle {\xi }\rangle _{\gamma }b, {{\bar{g}}})\) and \(\{q, \psi \}\in S(M^{-1/2}b, {{\bar{g}}})\).
Proof
The first two assertions are immediate consequences of Lemma 6.7. The third assertion follows from Proposition 3.1 and (6.30). \(\square \)
The following weight is a key to energy estimates
where it is clear that \(\phi \) verifies
Lemma 4.2
We have \( \partial _x^{\alpha }\partial _{\xi }^{\beta }\psi \in S(\langle {\xi }\rangle _{\gamma }^{-1/2}M^{-\epsilon (\alpha , \beta )}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}, g_{\epsilon })\) for \(|\alpha +\beta |\ge 1\).
Proof
Recall that \(\psi =\eta _1(\xi )+r\) or \(\psi =\varepsilon y_1(x)+cy_d(x)+r\) with \(r\in S(M^{-2}, G)\) according to the coordinates change (a) or (b). For \(\nu =\beta '+\beta \), \(|\beta |\ge 1\) we have
For \(\mu =\alpha '+\alpha \), \(|\alpha |\ge 1\) one has
Let \(\mu =\alpha '+\alpha \), \(|\alpha |\ge 1\) and \(\nu =\beta '+\beta \), \(|\beta |\ge 1\) then noting \(|\mu +\nu |\le 2|\mu +\nu |-\epsilon (\mu , \nu )\) one has
Since \(M^{2+2\delta _{\epsilon \epsilon '}}\langle {\xi }\rangle _{\gamma }^{-1}\le M^4\langle {\xi }\rangle _{\gamma }^{-1}\le 1\) by (3.1) the assertion is proved. \(\square \)
Lemma 4.3
We have \( \partial _x^{\alpha }\partial _{\xi }^{\beta }\omega ^s\in S( \omega ^{s-1}\langle {\xi }\rangle _{\gamma }^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}, g_{\epsilon })\) for \(|\alpha +\beta |\ge 1\) and \(s\in {{\mathbb {R}}}\). In particular \(\omega ^s\in S(\omega ^s, g_{\epsilon })\).
Proof
We first show the assertion for \(s=2\). Since \(\omega ^2=(t-\psi )^2+\langle {\xi }\rangle _{\gamma }^{-1}\) noting \(\omega \langle {\xi }\rangle _{\gamma }^{1/2}\ge 1\) and \(|t-\psi |\le \omega \) one sees for \(\nu =\beta '+\beta \), \(|\beta |\ge 1\) that
where it should be understood that the second term on the right-hand side is absent when \(|\nu |=1\). To estimate the last term it suffices to note \(\langle {\xi }\rangle _{\gamma }^{-|\nu |}\le (M^2\langle {\xi }\rangle _{\gamma }^{-1})^{-|\nu |/2}M^{-\epsilon (0,\nu )}\langle {\xi }\rangle _{\gamma }^{-|\nu |/2}\). When \(\mu =\alpha '+\alpha \), \(|\alpha |\ge 1\) we see
where if \(|\alpha |=1\) the second term on the right-hand side is absent as above. When \(\mu =\alpha '+\alpha \), \(\nu =\beta '+\beta \), \(|\alpha +\beta |\ge 1\) and \(|\mu |\ge 1\), \(|\nu |\ge 1\) noting that \(\partial _x^{\mu }\partial _{\xi }^{\nu }\psi =\partial _x^{\mu }\partial _{\xi }^{\nu } r\) and \(\partial _x^{\mu }\partial _{\xi }^{\nu }\psi \in S(M^{-3+|\mu +\nu |}\langle {\xi }\rangle _{\gamma }^{-|\nu |}, G)\) we have
where \(1-|\mu +\nu |\le -\epsilon (\alpha ', \beta ')\) and \(\langle {\xi }\rangle _{\gamma }^{-1}\le \omega \langle {\xi }\rangle _{\gamma }^{-1/2}\) are used. Thus the case \(s=2\) is proved. Since \(\langle {\xi }\rangle _{\gamma }^{-1/2}\le \omega \) it is obvious \(\omega ^2\in S(\omega ^2, g_{\epsilon })\). The estimates for general \(\omega ^s=(\omega ^2)^{s/2}\) follows from those of \(\omega ^2\). \(\square \)
Lemma 4.4
We have \(\phi \in S(\phi , g_{\epsilon })\).
Proof
Using (4.5) we write
Since \(\omega ^{-1}\in S(\omega ^{-1}, g_{\epsilon })\) by Lemma 4.3 then
in view of \(\langle {\xi }\rangle _{\gamma }^{-1/2}\le M^{-1}\) and (4.4). On the other hand Lemma 4.2 shows
Hence differentiating (4.6) the assertion is proved by induction on \(|\alpha +\beta |\) noting (4.7) and (4.8). \(\square \)
Proposition 4.1
We have \(\omega ^s\in S(\omega ^s, g_{\epsilon })\) and \(\phi ^s\in S(\phi ^s, g_{\epsilon })\). For \(|\alpha +\beta |\ge 1\)
Proof
It remains to prove the assertion for \(\phi \). Let \(\phi _{\alpha \beta }\), \(\psi _{\alpha \beta }\) be those in (4.6). Note \(\phi _{\alpha \beta }\in S(\omega ^{-1}\langle {\xi }\rangle _{\gamma }^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}, g_{\epsilon })\) for \(|\alpha +\beta |\ge 1\) by Lemma 4.2, while \(\psi _{\alpha \beta }\in S(\omega ^{-1}\langle {\xi }\rangle _{\gamma }^{-1/2}\phi \langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}, g_{\epsilon })\) for \(|\alpha +\beta |\ge 1\) because of (4.7) and (4.4). Hence the assertion for \(s=1\) follows from (4.6). The estimate for general \(s\in {{\mathbb {R}}}\) follows from the estimate for the case \(s=1\). \(\square \)
Proposition 4.2
\(\omega \) and \(\phi \) are \(g_{\epsilon }\) admissible weights (Definition 6.1).
Proof
It suffices to show
If \(|\eta |\ge \langle {\xi }\rangle _{\gamma }/2\) noting \( \langle {\xi }\rangle _{\gamma }^{-1/2}\le \omega \le CM^{-1}\) one has
Thus in view of (4.4) one sees
Assume \(|\eta |<\langle {\xi }\rangle _{\gamma }/2\). Set \(f=t-\psi \) and \(h=\langle {\xi }\rangle _{\gamma }^{-1/2}\) so that \(\omega ^2=f^2+h^2\). Since \(|f(z+w)+f(z)|/|\omega (z+w)+\omega (z)|\) and \(|h(z+w)+h(z)|/|\omega (z+w)+\omega (z)|\) are bounded by 2 we have
Noting \(|f(z+w)-f(z)|=|\psi (z+w)-\psi (z)|\) the estimate
follows from Lemma 4.2 and (6.28). Similarly noting \(g_{\epsilon , z}^{1/2}(w)\ge M^{-1}\langle {\xi }\rangle _{\gamma }^{-1/2}|\eta |\) one has \( |h(z+w)-h(z)|\le C\langle {\xi }\rangle _{\gamma }^{-3/2}|\eta |\le CM\langle {\xi }\rangle _{\gamma }^{-1}g_{\epsilon , z}^{1/2}(w)\le C\omega (z)g_{\epsilon , z}^{1/2}(w)\) hence (4.11) gives
Together with (4.10) one concludes that \(\omega \) is \(g_{\epsilon }\) admissible weight. Turn to \(\phi \). Since \(\phi =\omega +f\) one can write \(\phi (z+w)-\phi (z)\) as
where \(|f(z+w)-f(z)|\le C\langle {\xi }\rangle _{\gamma }^{-1/2}g_{\epsilon , z}^{1/2}(w)\) by (4.12) and \(|h^2(z+w)-h^2(z)|\le CM\langle {\xi }\rangle _{\gamma }^{-3/2 }g_{\epsilon , z}^{1/2}(w)\) is easy. The insertion of these estimates into (4.14) yields
From \(\phi (z)\ge M\langle {\xi }\rangle _{\gamma }^{-1}/C\) by (4.4) it follows that
If \(C\langle {\xi }\rangle _{\gamma }^{-1/2}\,g^{1/2}_{\epsilon , z}(w)\big /(\omega (z+w)+\omega (z))<1/3\) then \(\big |\phi (z+w)/\phi (z)-1\big |\le (\phi (z+w)/\phi (z)+2)/3\) and hence
If \( C\langle {\xi }\rangle _{\gamma }^{-1/2}\,g^{1/2}_{\epsilon , z}(w)\big /(\omega (z+w)+\omega (z))\ge 1/3\) then \(C^2g_{\epsilon , z}(w)\ge \langle {\xi }\rangle _{\gamma }(\omega (z+w)+\omega (z))^2/9\ge 2\langle {\xi }\rangle _{\gamma }\omega (z+w)\omega (z)/9\) hence noting \(\phi (z)\ge \langle {\xi }\rangle _{\gamma }^{-1}/(2\omega (z))\) and using an obvious inequality \( 2\,\omega (z+w)\ge \phi (z+w)\) one obtains
which together with (4.10) proves that \(\phi \) is \(g_{\epsilon }\) admissible weight. \(\square \)
4.2 Weighted energy estimates
With \({\bar{\lambda }}\) which we have fixed in the previous section we write \({{\hat{P}}}(t, x, \tau , \xi )\) as
Let us denote
In what follows \({{\hat{P}}}\) and \({{\hat{P}}}(t, x, \tau ,\xi )\) stands for operator and its symbol respectively. Since \(\ell \in S(M^{-1}\langle {\xi }\rangle _{\gamma }, G)\) hence \(\partial _x^{\alpha }\partial _{\xi }^{\beta }\ell \in S(M\langle {\xi }\rangle _{\gamma }^{1-|\beta |}, g_{\epsilon })\) for \(|\alpha +\beta |=2\), Theorem 6.1 shows \(\ell \#\ell -\ell ^2\in S(M^2, g_{\epsilon })\) so that \( \textrm{op}({\ell ^2})=L^2+\textrm{op}({r})\) with \(r\in S(M^2, g_{\epsilon })\). Thus \({{\hat{P}}}\) can be written
for \(M^2\le \langle {\xi }\rangle _{\gamma }^{1/2}\). Let \(\theta >0\) be a parameter we consider \({{\hat{P}}_{\theta }}=e^{-\theta t}{{\hat{P}}}e^{\theta t}\). Noting \((D_t-i\theta )=e^{-\theta t}D_t e^{\theta t}\) one can write \({{\hat{P}}_{\theta }}\) as
Here we define several weights for energy estimates.
Definition 4.1
Define \(\varPhi ^{k\sharp }_n=\textrm{op}({\omega ^{-k/2}\phi ^{-n}})\), \(\varPsi ^{k\sharp }_n=\textrm{op}({\omega ^{1-k/2}\langle {\xi }\rangle _{\gamma }\phi ^{-n}})\), \(k=0,1,2, 3\). We denote \(\varPhi _n^{0\sharp }\), \( \varPhi _n^{1\sharp }\) simply by \(\varPhi _n\), \(\varPhi _n^{\sharp }\). We apply the same abbreviation for \(\varPsi _n^{k\sharp }\). For simplicity we will write \(\varPhi ^{k\sharp }\), \(\varPsi ^{k\sharp }\) dropping the parameter n, but it should be reminded that they include parameters n, M and \(\gamma \).
Throughout the section, small letters such as c, \({{\hat{c}}}\), \({{\bar{c}}}\), \(c_i\) denote constants independent of n, M, \(\gamma \) and \(\theta \), while capital letter C, may change from line to line, denotes constants which may depend on n but independent of M, \(\gamma \) and \(\theta \).
Lemma 4.5
If \(K^*=K\) then
and we have
Proof
To see the first equality it is enough to write
and note \( 2{\textsf{Im}}(K \varPhi u, A\varPhi u)=\partial _t(K\varPhi u, \varPhi u) +2\theta (K \varPhi u, \varPhi u)-{\textsf{Re}}((\partial _tK)\varPhi u, \varPhi u)\) for \(\partial _t=iA-\theta \). To see the second equality write
where the twice of the imaginary part of the first 4 terms on the right-hand side coincide with the last 4 terms on the right-hand side of (4.20). Thus it suffices to show \(2{\textsf{Im}}(\varPhi Ku, A\varPhi Ku)=\partial _t\Vert \varPhi Ku\Vert ^2+2\theta \Vert \varPhi Ku\Vert ^2\) which is clear. \(\square \)
We aim to estimate \(2{\textsf{Im}}(\varPhi {{\hat{P}}_{\theta }}u, \varPhi Au)\). Start with \(2{\textsf{Im}}(\varPhi L^2 u, \varPhi A u)\). Consider \(2{\textsf{Im}}([A, \varPhi ]Lu, \varPhi L u)\). Since \(\partial _t\phi =\omega ^{-1}\phi \) then \([A, \varPhi ]=in\,\textrm{op}({\omega ^{-1}\phi ^{-n}})\) hence
Noting \(\phi ^{-n}\#(\omega ^{-1} \phi ^{-n})-\omega ^{-1}\phi ^{-2n}\in S(M^{-1}\omega ^{-1}\phi ^{-2n}, g_{\epsilon })\) we have from Corollary 6.4 and Lemma 6.11 that
Next estimate \(2{\textsf{Im}}(\varPhi Au, [L, \varPhi ]L u)\). One can write
In fact since \(\partial _x^{\alpha }\partial _{\xi }^{\beta }\ell \in S(M^2\langle {\xi }\rangle _{\gamma }^{1-|\beta |}, g_{\epsilon })\) for \(|\alpha +\beta |=3\), Theorem 6.1 and Lemma 4.4 show \( (\ell \#\phi ^{-n}-\phi ^{-n}\#\ell )+i\{\ell , \phi ^{-n}\}\in S(\phi ^{-n}, g_{\epsilon })\). On the other hand one sees \( \{\ell , \phi ^{-n}\}=-in\,\omega ^{-1}\{\ell , \psi \}\phi ^{-n}+in\,\omega ^{-1}\{\ell , \langle {\xi }\rangle _{\gamma }^{-1}\}\phi ^{-n-1}\) in view of (4.5) and \(\omega ^{-1}\{\ell , \langle {\xi }\rangle _{\gamma }^{-1}\}\phi ^{-n-1}\in S(\phi ^{-n}, g_{\epsilon })\) by (4.4). Since \(\{\ell , \psi \}\in S(1, g_{\epsilon })\) Proposition 3.2 and Theorem 6.1 prove (4.22). Therefore from Lemma 6.11 we have
Since \( \Vert (\textrm{op}({\{\ell , \psi \}})v\Vert \le (|\kappa |+CM^{-1/2})\Vert v\Vert \) by Proposition 3.2 and Corollary 6.6 one obtains
Since the term \(|([\varPhi , L]Au, \varPhi Lu)|\) is estimated similarly one concludes
From \([A, L]=-i\,\textrm{op}({\partial _t\ell })\) and \(\partial _t\ell \in S(\langle {\xi }\rangle _{\gamma }, g_{\epsilon })\) it follows that \(\phi ^{-n}\#\phi ^{-n}\#(\partial _t\ell )=(\partial _t\ell )\phi ^{-2n}+r\) with \(r\in S(M^{-1}\langle {\xi }\rangle _{\gamma }\phi ^{-2n}, g_{\epsilon })\) then the estimate
follows from Lemma 6.11. Thus (4.20), (4.21), (4.23) and (4.24) give
Lemma 4.6
We have
Turn to \(-2{\textsf{Im}}(\varPhi A^2u, \varPhi Au)\). Choosing \(K=I\) and \(L=I\) in (4.19) and (4.21) respectively one has
Replacing \(\varPhi \) by \(\varPhi ^{2\sharp }\) a repetition of a similar argument shows
Since the left-hand side is bounded as
we conclude
Replacing u by Au in (4.25) one has
where we replace \(\nu \Vert \varPhi ^{\sharp } A u\Vert ^2\) (\(0<\nu <2\)) by the estimate (4.26) to obtain
Lemma 4.7
For any \(0<\nu <2\) the following estimate holds.
Finally we estimate \({\textsf{Im}}(\varPhi Q u, \varPhi A u)\). Study \({\textsf{Im}}([\varPhi , Q] u, \varPhi A u)\). From Proposition 4.1 and Theorem 6.1 it follows that \(\phi ^{-n}\#(\phi ^{-n}\#\langle {\xi }\rangle _{\gamma }-\langle {\xi }\rangle _{\gamma }\#\phi ^{-n})\in S(\omega ^{-1}\phi ^{-2n}, g_{\epsilon })\) hence Lemma 6.11 shows
To estimate \({\textsf{Im}}([\varPhi , \textrm{op}({q})] u, \varPhi A u)\) we shall examine that
Indeed since \(\partial _x^{\alpha }\partial _{\xi }^{\beta }q\in S(M\langle {\xi }\rangle _{\gamma }^{2-|\beta |}, g_{\epsilon })\) for \(|\alpha +\beta |=3\), Proposition 4.1 and Theorem 6.1 show \( \phi ^{-n}\#q-q\#\phi ^{-n}=-i\{\phi ^{-n}, q\}+r\) with \(r\in S(M\omega ^{-1}\phi ^{-n}, g_{\epsilon })\). Note \(\{\phi ^{-n}, q\}=n\omega ^{-1}\{\psi , q\}\phi ^{-n}-n\omega ^{-1}\{\langle {\xi }\rangle _{\gamma }^{-1}, q\}\phi ^{-n-1}/2\) by (4.5) where the second term on the right-hand is \(S(b\phi ^{-n}, {{\bar{g}}})\) because of Lemma 4.1 and (4.4), hence (4.28). Since \(\omega ^{-1}\{\phi ^{-n}, q\}\in S(M^{-1/2}\omega ^{-1}b\phi ^{-n}, {{\bar{g}}})\) by Lemma 4.1 it follows from Lemmas 6.12, 6.11 and (4.28) that
Lemma 4.8
The following estimate holds.
Proof
Note \(2{\textsf{Im}}(Q \varPhi u, [\varPhi , A]u)=2n{\textsf{Re}}\big (Q \textrm{op}({\phi ^{-n}})u, \textrm{op}({\omega ^{-1}\phi ^{-n}})u\big )\) and write \(\omega ^{-1}\phi ^{-n}=\omega ^{-1/2}\#(1+k)\#(\omega ^{-1/2}\phi ^{-n})\) and \(\omega ^{-1/2}\#\phi ^{-n}=(1+{{\tilde{k}}})\#(\omega ^{-1/2}\phi ^{-n})\) with \(k, {{\tilde{k}}}\in S(M^{-1}, g_{\epsilon })\) by use of Lemma 6.10, then one can write
where \([\textrm{op}({\omega ^{-1/2}}), Q]=\sum _{i=1}^3\textrm{op}({r_i})\) and
In fact \(\omega ^{-1/2}\#\langle {\xi }\rangle _{\gamma }-\langle {\xi }\rangle _{\gamma }\#\omega ^{-1/2}\in S(\omega ^{-3/2}, g_{\epsilon })\) is clear from Proposition 4.1 and Theorem 6.1. Similarly \(\omega ^{-1/2}\#q-q\#\omega ^{-1/2}+i\{\omega ^{-1/2}, q\}\in S(M\omega ^{-3/2}, g_{\epsilon })\), where
The first term on the right-hand is \(S(M^{-1/2}\omega ^{-3/2}b, {{\bar{g}}})\) because of Lemma 4.1 and \((t-\psi )\in S(\omega , g_{\epsilon })\) and the second term is \(S(\omega ^{-1/2}b, {{\bar{g}}})\) thanks to Lemma 4.1 and \(\omega ^{-2}\le \langle {\xi }\rangle _{\gamma }\), hence (4.30) is examined. Applying Lemma 6.12 we have
Turn to \(\big (\textrm{op}({1+{{\bar{k}}}})Q\textrm{op}({1+{{\tilde{k}}}})\varPhi ^{\sharp }u, \varPhi ^{\sharp } u)\). Since \({{\bar{k}}}\#(q+{\bar{\lambda }}\langle {\xi }\rangle _{\gamma })\in S(M^{-1}b^2, {{\bar{g}}})\) from Lemma 6.12 one has \( \big |(\textrm{op}({{{\bar{k}}}})Q\varPhi ^{\sharp } u, \varPhi ^{\sharp } u)\big | \le CM^{-1}\Vert \sqrt{Q}\,\varPhi ^{\sharp } u\Vert ^2\). Terms such as \(\big |(Q\textrm{op}({{{\tilde{k}}}})\varPhi ^{\sharp } u, \varPhi ^{\sharp } u)\big |\) are estimated similarly. To conclude the proof it suffices to apply Lemma 6.8 to \((Q\varPhi ^{\sharp } u, \varPhi ^{\sharp } u)\). \(\square \)
Lemma 4.9
There exists \(c_1>0\) such that
Proof
Write \(\phi ^{-n}\#\partial _tq\#\phi ^{-n}=(\omega ^{1/2}\langle {\xi }\rangle _{\gamma }\phi ^{-n})\#r\#(\omega ^{-1/2}\phi ^{-n})\) with \(r\in S(b, g_{\epsilon })\) by using Lemmas 4.1, 6.10 then \( |((\partial _tQ)\varPhi u, \varPhi u)|\le \Vert \textrm{op}({r})\varPhi ^{\sharp } u\Vert \Vert \varPsi ^{\sharp } u\Vert \). Write \((\omega ^{-1/2}\phi ^{-n})\#(1+k)\#(\omega ^{1/2}\phi ^{n})=1\), \((\omega ^{-1/2}\langle {\xi }\rangle _{\gamma }^{-1}\phi ^{n})\#(1+{\tilde{k}})\#(\omega ^{1/2}\langle {\xi }\rangle _{\gamma }\phi ^{-n})=1\) with \(k, {{\tilde{k}}}\in S(M^{-1}, g_{\epsilon })\) by using Lemma 6.10 it is clear
From Theorem 6.1 one sees \(\phi ^{-n}\#(1+k)\#(\omega ^{1/2}\phi ^{n})-\omega ^{1/2}=l\in S(M^{-1}\omega ^{1/2}, g_{\epsilon })\) and \((\omega ^{-1/2}\langle {\xi }\rangle _{\gamma }^{-1}\phi ^{n})\#(1+{\tilde{k}})\#\phi ^{-n}-\omega ^{-1/2}\langle {\xi }\rangle _{\gamma }^{-1}={{\tilde{l}}}\in S(M^{-1}\omega ^{-1/2}\langle {\xi }\rangle _{\gamma }^{-1}, g_{\epsilon })\) hence \(r=(\langle {\xi }\rangle _{\gamma }^{-1}\omega ^{-1/2}+{\tilde{l}})\#(\partial _tq)\#(\omega ^{1/2}+l)=(\langle {\xi }\rangle _{\gamma }^{-1}\omega ^{-1/2})\#(\partial _t q)\#\omega ^{1/2}+{{\tilde{r}}}\) where \({{\tilde{r}}}\in S(M^{-1}b, {{\bar{g}}})\) by Lemma 4.1. Noting \((\langle {\xi }\rangle _{\gamma }^{-1}\omega ^{-1/2})\#(\partial _t q)\#\omega ^{1/2}\in S(b, {{\bar{g}}})\) is independent of n we have \(\Vert \textrm{op}({r})v\Vert \le (c_1+CM^{-1})\Vert \sqrt{Q}\,v\Vert \) from Lemma 6.12 with some \(c_1>0\). Putting \(v=\varPhi ^{\sharp }u\) we conclude the proof. \(\square \)
Choosing \(K=Q\) in (4.19) it follows from (4.29) and Lemmas 4.8, 4.9 that
Writing \(\omega ^{1-k/2}\phi ^{-n}\langle {\xi }\rangle _{\gamma }=(\langle {\xi }\rangle _{\gamma }\omega )(\omega ^{-k/2}\phi ^{-n})\) we have from Lemma 6.11
Let \({{\tilde{b}}}\in S(b^{-1}, {{\bar{g}}})\) be given in Proposition 6.1 then \({{\tilde{b}}}\in S(\omega ^{-1}\langle {\xi }\rangle _{\gamma }^{-1}, {{\bar{g}}})\) by (4.3). Hence writing \(\langle {\xi }\rangle _{\gamma }\omega =(\langle {\xi }\rangle _{\gamma }\omega )\#{{\tilde{b}}}\#b\) with \( (\langle {\xi }\rangle _{\gamma }\omega )\#{{\tilde{b}}}\in S(1, {{\bar{g}}})\) there is \({{\hat{c}}}>0\) such that \( \Vert \textrm{op}({\langle {\xi }\rangle _{\gamma }\omega })v\Vert \le {{\hat{c}}}\,\Vert \sqrt{Q}\,v\Vert \) thanks to Theorem 6.2. Replacing v by \(\varPhi ^{k\sharp }u\) we have from (4.34) that
Lemma 4.10
There exist \({{\hat{c}}}>0, c>0\), \(C>0\) such that
for \(k=0, 1, 2\).
Proof
It remains to show the left side inequality. Write \(\phi ^{-n}\langle {\xi }\rangle _{\gamma }^{1/2+k/4}=(\omega ^{1/2}\langle {\xi }\rangle _{\gamma }^{1/4})^{-2+k}(\omega ^{1-k/2}\phi ^{-n}\langle {\xi }\rangle _{\gamma })\) then from Lemma 6.11 there is \(c>0\) such that \( c\Vert \langle {D}\rangle _{\gamma }^{1/2+k/4}\varPhi u\Vert \le (1+CM^{-1})\Vert \varPsi ^{k\sharp } u\Vert \) for \(k\le 2\). \(\square \)
In (4.33), replacing \(\Vert \varPsi ^{\sharp }u\Vert ^2\) by the estimate (4.35) one has
Lemma 4.11
We have
Finally we estimate the lower order term \(B_0A+B_1\). Since \({\tilde{a}_j}\in S(\langle {\xi }\rangle _{\gamma }^j, g_{\epsilon })\) Lemma 6.11 shows
Similarly \(2|(\varPhi B_0 Au, \varPhi Au)|\le C\Vert \varPhi Au\Vert ^2\). Then from Lemmas 4.6, 4.7, 4.11 and the estimates of lower order term one has
Proposition 4.3
We have
Lemma 4.12
For \(n\ge 1\) there is \(C>0\) such that
Proof
Since \(\phi ^{-n}\le C\omega ^{n}\langle {\xi }\rangle _{\gamma }^n\le C'\langle {\xi }\rangle _{\gamma }^n\) then \(\phi ^{-n}\in S(\langle {\xi }\rangle _{\gamma }^n, g_{\epsilon })\) by (4.4) hence the first inequality is clear from Lemma 6.11. Since \(\phi ^{-n}\ge (2\omega )^{-n}\ge C>0\) for \(\phi \le 2\omega \) hence \(1\in S(\phi ^{-n}, g_{\epsilon })\) which proves the second inequality. The third inequality follows from \(\omega \phi ^{-n}\langle {\xi }\rangle _{\gamma }\ge C\omega ^{1-n}\langle {\xi }\rangle _{\gamma }\ge C'\langle {\xi }\rangle _{\gamma }\) and Lemma 4.10. \(\square \)
In Proposition 4.3 we fix \(\nu >0\) such that \(1-|\kappa |-\nu /2>0\). Then choose n such that
and fix such a n. Note that (4.37) is always satisfied for any n greater than such a fixed n. Next, for such fixed n, choose M such that the arguments in this section should be justified, namely the assertions in Sect. 6.3 hold with
and the coefficients of the last four terms in Proposition 4.3 and that of Lemma 4.10 will be positive, and fix such a M then choose \(\gamma \) such that \(\gamma \ge M^{4}\) and \(\gamma \ge {\bar{\lambda }}M^{2}\) and fix such a \(\gamma \), while \(\theta \) is assumed to be free still. Once M and \(\gamma \) are fixed, denoting by \(g_0\) the metric \({\underline{g}}\) with \(\gamma =1\), there are \(C, C_s\) such that
then \(S(\langle {\xi }\rangle _{\gamma }^s, G)=S(\langle {\xi }\rangle ^s, g_0)=S^s\). In particular, \(\Vert \langle {D}\rangle _{\gamma }^s\cdot \Vert \) is equivalent to \(\Vert \langle {D}\rangle ^s\cdot \Vert \). The range of t is consequently fixed if M is fixed by (3.4). As long as \(\gamma \) is fixed, it is allowed to write \(\langle {\xi }\rangle _{\gamma }\) as \(\langle {\xi }\rangle \). After fixing n, M, \(\gamma \) and taking Lemma 4.10 into account we have
Proposition 4.4
There exist \(c>0\), \(c^*>0\), \(\delta _0>0\), \(\theta _0>0\) such that for \(|t|\le \delta _0\), \(\theta \ge \theta _0\) one has
Definition 4.2
Denote
Denote the substitution of A with \(D_t\) in the definition \({\mathcal {{\tilde{E}}}^2}(u)\) and \({\mathcal {{\tilde{E}}}^2_{\!\sharp }}(u)\) as \({{\mathcal {E}}^2}(u)\) and \({ {\mathcal {E}}_{\sharp }^2}(u)\) respectively.
To effectively utilize Proposition 4.4, noting that \(\ell ^2\in S(M^{-2}\langle {\xi }\rangle _{\gamma }^2, G)\) we introduce
where it can be assumed that \({\bar{\lambda }}\) is chosen so that both Proposition 6.1 and Lemma 6.8 hold.
Lemma 4.13
There is \(C>0\) such that
Proof
\(\Vert \sqrt{Q}\,\varPhi u\Vert /C\le \Vert \varPhi \sqrt{Q}\, u\Vert \le C\Vert \sqrt{Q}\, \varPhi u\Vert \) and \(C\Vert \sqrt{Q}\,\varPhi u\Vert ^2\ge (Q\varPhi u, \varPhi u)\) follow from Lemma 6.12 and \((Q\varPhi u, \varPhi u)\ge \Vert \sqrt{Q}\, \varPhi u\Vert ^2/2\ge \Vert \langle {D}\rangle ^{1/2}\varPhi u\Vert ^2/C\) by Lemmas 6.8 and 4.10. Similarly \(\Vert \varPhi L^{\dag } u\Vert \le C(\textrm{op}({b_1^2})\varPhi u, \varPhi u)\le C(\Vert \varPhi L u\Vert +\Vert \langle {D}\rangle ^{1/2}\varPhi u\Vert )\). Moreover one has \(\Vert L\varPhi u\Vert \le C(\Vert \varPhi L u\Vert +\Vert \langle {D}\rangle ^{1/2}\varPhi u\Vert )\) thanks to Proposition 4.1 and Theorem 6.1 hence to finish the proof it suffices to note \(\omega ^{-1}\phi ^{-n}\in S(\langle {\xi }\rangle _{\gamma }^{1/2}\phi ^{-n}, g_{\epsilon })\). The second assertion is proved similarly. \(\square \)
Therefore Proposition 4.4 can be stated as
Proposition 4.5
There exist \(c>0\), \(c^*>0\), \(\delta _0>0\), \(\theta _0>0\) such that for \(|t|\le \delta _0\), \(\theta \ge \theta _0\) one has \(2{\textsf{Im}}(\varPhi {{\hat{P}}_{\theta }}u, \varPhi Au)\ge \partial _t{\mathcal {{\tilde{E}}}^2}(u)+c\,\theta {\mathcal {\tilde{E}}^2}(u) +c\,{\mathcal {{\tilde{E}}}^2_{\!\sharp }}(u)\).
4.3 Estimates of higher order derivatives
Recall that n, M, \(\gamma \) are fixed such that the assertions in Sect. 6.3 hold with m or \(m_i\) in (4.38). For notational simplicity, we write
Lemma 4.14
There is \(C_s>0\) such that
Proof
The proof is clear from Lemma 4.13 and Corollaries 6.3, 6.5. \(\square \)
Estimate \(\langle {D}\rangle ^su\), \(s\in {{\mathbb {R}}}\). Noting \(\langle {D}\rangle ^s{{\hat{P}}_{\theta }}={{\hat{P}}_{\theta }}\langle {D}\rangle ^s+[\langle {D}\rangle ^s, {{\hat{P}}}]\) we consider \(|(\varPhi [\langle {D}\rangle ^s, {{\hat{P}}}]u, \varPhi A\langle {D}\rangle ^su)|\). Write \({{\hat{P}}_{\theta }}\) as
where \(B'_i=\textrm{op}({{{\tilde{a}}'_i}})\), \({{\tilde{a}}'_i}\in S^i\). From Theorem 6.1 and Lemma 6.7 we can write
then it is clear from Corollaries 6.5 and 6.3 that
Similarly it follows from Corollary 6.3 that
Proposition 4.6
For any \(s\in {{\mathbb {R}}}\) there exist \(c_s, \theta _s>0\) such that for \(|t|\le \delta _0\), \(\theta \ge \theta _s\) one has
Proof
Write \(2{\textsf{Im}}(\varPhi \langle {D}\rangle ^s{{\hat{P}}_{\theta }}u, \varPhi A\langle {D}\rangle ^s u)\) as a sum
and apply Proposition 4.4 to the first term. In view of (4.40) and (4.41), taking Lemma 4.14 into account, the term \(|(\varPhi [\langle {D}\rangle ^s, {{\hat{P}}}]u, \varPhi A\langle {D}\rangle ^s u)|\) is absorbed in \(\theta {\mathcal {{\tilde{E}}}_{\!s}^2}(u)\) choosing \(\theta \) large. \(\square \)
Proposition 4.7
Let \(|\tau |\le \delta _0\). For any \(s\in {{\mathbb {R}}}\) there are \(C_s, C'_s>0\) such that
holds for any \(u\in \cap _{j=0}^2C^j([\tau , \delta _0]; H^{s+n+2-j})\).
Proof
Replacing u by \(e^{-\theta t}u\) and noting \(Ae^{-\theta t}=e^{-\theta t}D_t\), \({{\hat{P}}_{\theta }}e^{-\theta t}=e^{-\theta t}{{\hat{P}}}\) it follows from Proposition 4.6 that
If we integrate from \(\tau \) to t (\(-\delta _0\le \tau <t\le \delta _0\)) and noting Lemma 4.12 we have
Denoting \(K=\sup _{\tau \le t'\le t}\big \{{{\mathcal {E}}_s}(u(t'))+\int _{\tau }^{t'}{{\mathcal {E}}_{\sharp s}}(u(t_1))dt_1\big \}\) we see that \(K^2\) is bounded by \(C_s'{{\mathcal {E}}_s^2}(u(\tau ))+C_s'K\int _{\tau }^t\Vert {{\hat{P}}}u(t')\Vert _{n+s}dt'\) hence we have
In virtue of Lemmas 4.12, 4.14 there exists \(C=C_s\) such that
from which the proof follows. \(\square \)
Here consider the adjoint operator \({{\hat{P}}^*}=\textrm{op}({\overline{{{\hat{P}}}(t, x, \tau ,\xi )}})\) of \({{\hat{P}}}\) where \(\overline{{{\hat{P}}}(t, x, \tau ,\xi )}\) is obtained from \({{\hat{P}}}(t, x, \tau ,\xi )\) replacing \(a_j(t, x, \xi )\) by \(\overline{ a_j(t, x, \xi )}\). Therefore replacing n by \(-n\) and \(\theta \) by \(-\theta \) the same argument can be repeated to obtain
where we have set
and \(\varPhi _{-n}^{k\sharp }=\textrm{op}({\omega ^{-k/2}\phi ^n})\), \(\varPhi ^{0\sharp }_{-n}=\varPhi _{-n}\) and \(L^{\dag }\) and \(\sqrt{Q}\) are as before. It is clear from \(\langle {\xi }\rangle _{\gamma }^{-1}\le C\phi \le C'\) that
Since the proof of Lemma 4.10 shows \(C\Vert \varPhi _{-n}\sqrt{Q}u\Vert \ge \Vert \textrm{op}({\omega \phi ^n\langle {\xi }\rangle })u\Vert \) noting \(\langle {\xi }\rangle ^{-n}\le (2\omega \phi )^n\le C\omega \phi ^n\) by (4.4) one has
Integrating (4.44) over \([t, \tau ]\) and repeating the proof of Proposition 4.7 we have
Proposition 4.8
Let \(|\tau |\le \delta _0\). For any \(s\in {{\mathbb {R}}}\) there exist \(C_s, C_s'>0\) such that
holds for any \(u\in \cap _{j=0}^2C^j([-\delta _0, \tau ]; H^{s+2-j})\).
5 Local existence and uniqueness theorem
In this section, we prove the existence of the solution operator of the localized operator with a finite speed of propagation. Making use of such solution operators we prove the local existence and uniqueness theorem for the original Cauchy problem.
5.1 Local existence theorem
We show the existence and uniqueness of the Cauchy problem for localized \({{\hat{P}}}\).
Theorem 5.1
Let \(|\tau |<\delta _0\), \(s\in {{\mathbb {R}}}\). For any \(f\in L^1((\tau , \delta _0);H^{s+n})\) and \(\phi _j\in H^{s+n+1-j}\) \((j=0, 1)\) there exists a unique solution \(u\in \cap _{j=0}^1C^{j}([\tau , \delta _0];H^{s+1-j})\) to the Cauchy problem
and (4.42) holds for this solution.
Proof
The uniqueness follows from (4.42). We show the existence of u. Consider the anti-linear from
on \(\{{{\hat{P}}^*}v; v\in C_0^{\infty }(\{(t,x); t<\delta _0\})\}\) where \(B_0=\textrm{op}({{{\tilde{a}}}_0})\) is given in (4.17). From (4.47) it is seen that \(\big |i(\phi _0, D_t v(\tau ))+i(\phi _1-B_0(\tau )\phi _0, v(\tau ))\big |\) is bounded by
and \(\big |\int _{\tau }^{\delta _0}(f,v)dt\big |\) is estimated by
Using the Hahn-Banach theorem to extend this form we conclude that there is some \(u\in L^{\infty }([\tau , \delta _0]; H^{s+1})\) such that
if \(v\in C_0^{\infty }(\{(t,x); t<\delta _0\})\). Thus \({{\hat{P}}}u=f\) in \((\tau , \delta _0)\times {{\mathbb {R}}}^d\) in the distribution sense. Then \(D_t^ju(t)\in L^2([\tau , \delta _0]; H^{s+1-j})\), \(j=0, 1, 2\) thanks to [4, Theorem B.2.9] hence \(u\in \cap _{j=0}^1C^{j}([\tau , \delta _0]; H^{s-j})\). Since \(v(\tau ), D_tv(\tau )\in C_0^{\infty }({{\mathbb {R}}}^d)\) are arbitrary we conclude \(D_t^ju(\tau )=\phi _j\), \(j=0, 1\). Choose \(\phi _{j\nu }\in {{\mathcal {S}}}({{\mathbb {R}}}^d)\), \(f_{\nu }\in {{\mathcal {S}}}({{\mathbb {R}}}^{1+d})\) so that
There is \(u_{\nu }(t)\in \cap _{j=0}^2C^{j}([\tau ,\delta _0];H^{s+n+2-j})\) satisfying \({{\hat{P}}}u_{\nu }=f_{\nu }\) and \(D_t^ju_{\nu }(\tau )=\phi _{j\nu }\) hence \(u_{\nu }\) is a Cauchy sequence in \(\cap _{j=0}^1C^j([\tau ,\delta _0];H^{s+1-j})\). The limit as \(\nu \rightarrow \infty \) is the desired solution. Clearly the limit u satisfies (4.42). \(\square \)
The Cauchy problem for the adjoint operator \({{\hat{P}}^*}\) can be treated similarly.
Theorem 5.2
Let \(|\tau |<\delta _0\), \(s\in {{\mathbb {R}}}\). For any \(f\in L^1((-\delta _0, \tau );H^{s+n})\) and \(\phi _j\in H^{s+n+1-j}\) \((j=0, 1)\) there is a unique solution \(u\in \cap _{j=0}^1C^{j}([-\delta _0, \tau ];H^{s+1-j})\) of
and (4.47) holds for this solution.
Study the solution operator of the Cauchy problem (5.1) with \(\phi _0=\phi _1=0\);
where \({{\hat{P}}}{{\hat{G}}} f=f\) in \((\tau , \delta _0)\times {{\mathbb {R}}}^d\) and the following estimate holds
Proposition 5.1
\({{\hat{G}}}\) has a finite speed of propagation, namely \({{\hat{G}}}\) satisfies the following Definition 5.1 with \(m=2\).
A conic set \(U\subset {{\mathbb {R}}}^d\times ({{\mathbb {R}}}^d{\setminus } 0)\) can be identified with \(\{(x, \xi /|\xi |); (x, \xi )\in U\}\), a subset of \({{\mathbb {R}}}^d\times S^{d-1}\). The topology for conic sets is induced through this identification. By \(\overset{\circ }{U}\) we denote the interior of U and by \(U^c\) the complement of U and \(U\Subset V\) means that U is relatively compact in \(\overset{\circ }{V}\).
Definition 5.1
We say that G has a finite speed of propagation if for any closed conic set \(U_1\) and compact conic set \(U_2\) with \(U_1\cap U_2=\emptyset \) there exists \(\delta >0\) such that for any \(l_i\in {{\mathbb {R}}}\) and \(h_i(x, \xi )\in S^0({{\mathbb {R}}}^{2d})\) with \(\textrm{supp}\,h_i\subset U_i\) one can find \(C>0\) such that the estimate
holds for any \(f\in L^1((\tau , T); H^{l_2})\) and \(\tau <t\le \min {(\tau +\delta , T)}\).
We postpone the proof of Proposition 5.1 to the next section.
Definition 5.2
Let \(P_i\) (\(i=1, 2\)) be two operators of the form
For \(\eta \in {{\mathbb {R}}}^d\), \(|\eta |\ne 0\) we say \(P_1\equiv P_2\) at \((0,\eta )\) if there are \(\delta >0\) and a conic neighborhood W of \((0, \eta )\) such that one can write
Before going on, we prepare a version of well-known relation on the wave front set under the pullback (e.g. [5, Theorem 8.2.1]). If \(\kappa \) is a diffeomorphism on \({{\mathbb {R}}}^d\) and \(U\subset {{\mathbb {R}}}^d\times ({{\mathbb {R}}}^d{\setminus } 0)\) is a conic set we denote
and \(\kappa ^*f=f(\kappa (x))\) is the pullback if f is a function on \({{\mathbb {R}}}^d\).
Proposition 5.2
Let \(\kappa \) be a diffeomorphism on \({{\mathbb {R}}}^d\) which is a linear transformation outside a compact set. Let U, V be two closed conic sets with \(V\cap \kappa ^*U=\emptyset \) and \(h, k\in S^0\) such that \(\textrm{supp}\, h\subset U\), \(\textrm{supp}\, k\subset V\). Then for any \(p, q\in {{\mathbb {R}}}\) there is C such that
We give the proof in the Appendix.
Lemma 5.1
If all critical points \((0, 0, \tau , \xi )\) of \(p=0\) are effectively hyperbolic then for any \(0\ne \eta \in {{\mathbb {R}}}^d\) there exists \(P_{\eta }\) of the form (5.6) such that \(P_{\eta }\equiv P\) at \((0, \eta )\) of whose solution operator has a finite speed of propagation.
Proof
If \(p(0, 0, \tau , \eta )=0\) has a double characteristic root, which is necessarily \(\tau =0\), and \((0, 0, 0, \eta )\) is effectively hyperbolic by assumption. Proposition 2.1 with \({\bar{\xi }}=\eta \) gives a diffeomorphism on \({{\mathbb {R}}}^d\): \(x\mapsto \kappa (x)\). Set \((Tu)(t, x)=u(t, \kappa (x))\) and let \({{\hat{P}}}\) be the localized operator defined in Sect. 3 and denote \(P_{\eta }=T {{\hat{P}}} T^{-1}\). Since \((y(x), \eta (\xi ))=(x, \xi )\) in some neighborhood of \((0, e_d)\) given by (3.3) it is clear that
The solution operator \({{\hat{G}}}\) of \({{\hat{P}}}\), given in Theorem 5.1, has a finite speed of propagation by Proposition 5.1. Set \(G_{\eta }=T {{\hat{G}}} T^{-1}\) then \(P_{\eta } G_{\eta }=I\) is obvious. We examine that \(G_{\eta }\) has a finite speed of propagation. Let \(U_1\), \(U_2\) be closed and compact conic set with \(U_1\cap U_2=\emptyset \). Choose open conic sets \(V_i\) and compact conic sets \(W_i\) such that \((\kappa ^{-1})^*U_2\Subset V_2\Subset W_2\Subset V_1\Subset W_1\) with \(W_1\cap (\kappa ^{-1})^*U_1=\emptyset \) and \(\phi _i\in S^0\) such that \(\phi _1=1\) on \(W_1^c\) with \(\textrm{supp}\,\phi _1\subset V_1^c\) and \(\phi _2=1\) on \(V_2\) with \(\textrm{supp}\, \phi _2\subset W_2\). Write \(\textrm{op}({h_2})G_{\eta }\textrm{op}({h_1})\) as a sum
Since \(\textrm{supp}\,\phi _1^c\subset W_1\), \(W_1\cap (\kappa ^{-1})^*U_1=\emptyset \) and \(\textrm{supp}\,\phi _2^c\subset V_2^c\), \(U_2\cap \kappa ^*V_2^c=\emptyset \) one can apply Proposition 5.2 to \(\textrm{op}({\phi _1^c})T^{-1}\textrm{op}({h_1})\) and \(\textrm{op}({h_2})T\textrm{op}({\phi _2^c})\) to obtain the desired estimates. On the other hand to estimate \( \textrm{op}({\phi _2}){{\hat{G}}}\textrm{op}({\phi _1})\) it suffices to use a finite speed of propagation of \({{\hat{G}}}\) for \( W_2\cap V_1^c=\emptyset \).
If \(p(0, 0, \tau , \eta )=0\) has a simple root one can find \(\delta >0\) and a conic neighborhood U of \((0, \eta )\) and real valued \(\lambda _i(t, x, \xi )\in C^{\infty }((-\delta , \delta )\times U)\), homogeneous of degree 1 in \(\xi \), such that \(\inf _{(-\delta , \delta )\times U}|\lambda _1(t,x,\xi )-\lambda _2(t, x, \xi )|/|\xi |>0\) which satisfy
Taking Theorem 6.1 into account one can find \(\lambda _{ij}\in C^{\infty }((-\delta ,\delta )\times U)\), \(j\in {{\mathbb {N}}}\), homogeneous of degree \(-j\), such that
is verified formally. Take a conic neighborhood \(V\Subset U\) of \((0, \eta )\) and \(\chi \in S^0\) such that \(\chi =1\) in \(V\cap \{|\xi |\ge 1\}\) and \(\textrm{supp}\,\chi \subset U\cap \{|\xi |\ge 1/2\}\). Then there is \({\tilde{\lambda }_i}\in S^1\) such that \({\tilde{\lambda }_i}\sim \chi \lambda _i+\sum _{j=0}^{\infty }\chi \lambda _{ij}\) (e.g. [4, Proposition 13.1.3]). If we set \(P_i=D_t+\textrm{op}({{\tilde{\lambda }_i}})\) it is clear that
Since \(P_i\) is a first order operator it is easily checked that there is a solution operator \(G_i\) with a finite speed of propagation (\(m=1\) in Definition 5.1) and consequently \(G_2G_1\) has a finite speed of propagation. Then \(P_{\eta }=P_1P_2\) is the desired one whose solution operator is \(G_{\eta }=G_2G_1\). \(\square \)
Theorem 5.3
If all critical points \((0, 0, \tau , \xi )\) of \(p=0\) are effectively hyperbolic then there are \(\delta >0\), \(n>0\) and a neighborhood \(\Omega \) of \(x=0\) such that for any \(|{\tau }|<\delta \) and \(f\in L^1(({\tau },\delta ); H^{s+n})\) there exists \(u\in \cap _{j=0}^1C^j([{\tau }, \delta ]; H^{s+1-j})\) satisfying \(Pu=f\) in \(({\tau },\delta )\times \Omega \) and
Proof
Thanks to Lemma 5.1, for any \(|\eta |=1\) there are \(\delta _{\eta }>0\), a conic neighborhood \(W_{\eta }\) of \((0, \eta )\), a second order operator \(P_{\eta }\) with solution operator \(G_{\eta }\) with a finite speed of propagation satisfying (5.4) with \(n=n_{\eta }\) and \(P_{\eta }\) satisfying
Since \(\{|\eta |=1\}\) is compact there are finite number of \(\eta _i\) and a neighborhood \(\Omega \) of \(x=0\) such that \(\cup _iW_{\eta _i}\supset \Omega \times ({{\mathbb {R}}}^d{\setminus } \{0\})\). Note that \(G_{\eta _i}\) satisfies (5.4) with \(n=\max _i n_{\eta _i}\). Take open conic coverings \(\{U_i\}\), \(\{V_i\}\) of \(\Omega \times ({{\mathbb {R}}}^d{\setminus }\{0\})\) such that \(U_i\Subset V_i\Subset W_{\eta _i}\) and a partition of unity \(\{\alpha _i(x,\xi )\}\), \(\alpha _i\in S^0\) associated to \(\{U_i\}\). If we set \( \sum _i\alpha _i(x,\xi )=\alpha (x)\) then \(\alpha (x)=1\) in a neighborhood of \(x=0\) and we may assume that \(\alpha (x)\) has a compact support. Define
then it is clear from (5.9) that
where \({{\tilde{R}}}=-\sum _i R_{\eta _i}G_{\eta _i}\textrm{op}({\alpha _i})\). Set \(R=\alpha {{\tilde{R}}}\) and we show that there are \(\delta _1, \delta '>0\) such that
for any \({{\tilde{s}}}\). Take \(\chi _i\in S^0\) be 1 on \(V_i\) with \(\textrm{supp}\,\chi _i\subset W_{\eta _i}\) and \({\tilde{\alpha }}\in C_0^{\infty }({{\mathbb {R}}}^d)\) be 1 in a neighborhood of \(x=0\) with \(\textrm{supp}\,\alpha \Subset \{{\tilde{\alpha }}=1\}\) and write
where \(\alpha (1-{\tilde{\alpha }})=0\). Since one can write \({\tilde{\alpha }}\#\chi _i=\kappa _i+r_i\), \(\textrm{supp}\, \kappa _i\subset W_{\eta _i}\), \(\kappa _i\in S^0\), \(r_i\in S^{-n}\) and \({\tilde{\alpha }}\#(1-\chi _i)={\tilde{\kappa }_i}+{{\tilde{r}}_i}\), \(\textrm{supp}\, {\tilde{\kappa }_i}\subset V^c_i\cap \textrm{supp}\,{\tilde{\alpha }}\), \({\tilde{\kappa }_i}\in S^0\), \({{\tilde{r}}_i}\in S^{-n}\) it is clear that \(\Vert \alpha R_{\eta _i}(1-{\tilde{\alpha }})G_{\eta _i}\textrm{op}({\alpha _i})f\Vert _{{{\tilde{s}}}}\), \(\Vert {\alpha }R_{\eta _i}\textrm{op}({\kappa _i+r_i})G_{\eta _i}\textrm{op}({\alpha _i})f\Vert _{{\tilde{s}}}\) and \(\Vert {\alpha }R_{\eta _i}\textrm{op}({{\tilde{r}_i}})G_{\eta _i}\textrm{op}({\alpha _i})f\Vert \) are bounded by
while \(\Vert \alpha R_{\eta _i}\textrm{op}({{\tilde{\kappa }_i}})G_{\eta _i}\textrm{op}({\alpha _i})f\Vert _{{{\tilde{s}}}}\le C\sum _{j=0}^1\Vert D_t^j\textrm{op}({{\tilde{\kappa }_i}})G_{\eta _i}\textrm{op}({\alpha _i})f\Vert _{{{\tilde{s}}}+2-j} \) to which we apply a finite speed of propagation of \(G_{\eta _i}\) for \((V^c_i\cap \textrm{supp}\,{\tilde{\alpha }})\cap U_i=\emptyset \). Thus (5.11) is proved.
Multiply (5.11) by \(e^{-\theta t}\) (\(\theta >0\)) and integrate from \(\tau \) (\(|\tau |\le \delta _1\)) to t one has
for any \(f\in L^1(({\tau }, { \tau }+\delta '); H^{{\tilde{s}}})\). Choose \(\theta =\theta _s\) such that \(C_{{{\tilde{s}}}}/\theta <1/2\) then \( Sf=\sum _{l=0}^{\infty }R^lf\) converges in the weighted \(L^1((\tau , \tau +\delta ');H^{{{\tilde{s}}}})\) with the weight \(e^{-\theta t}\) and it yields
Let \(\beta (x)\in C_0^{\infty }({{\mathbb {R}}}^d)\) be 1 in a neighborhood of \(x=0\) with \(\textrm{supp}\,\beta \Subset \{\alpha =1\}\). Noting \(\beta (\alpha -{{\tilde{R}}})=\beta (I-\alpha {{\tilde{R}}})\) it is clear \(\beta P G S f=\beta (I-R)Sf=\beta f\) hence \( P\big (GSf\big )=f\) on \(\{\beta (x)=1\}\). If \(f\in L^1((\tau , \tau +\delta '); H^{s+n})\) then \(u=G S f\in \cap _{j=0}^1C^j([\tau , \tau +\delta ']; H^{s+1-j})\) and choosing \({{\tilde{s}}}=s+n\) in (5.4), (5.12) one obtains
which proves (5.8). \(\square \)
5.2 Finite speed of propagation
Here we shall prove Proposition 5.1. Write \({{\hat{P}}_{\theta }}\) in the form (4.39).
Definition 5.3
\(f(t, x, \xi )\in C^{\infty }((-T, T); S^0)\) is called to be spacelike (for \({{\hat{P}}}\)) if there exist \(0<\delta _1\), \(0<\kappa <1\) such that
Following [8], for a spacelike f we denote
and set
It is clear that \({{\bar{f}}}\), \({{\bar{f}}}_1\), \(\partial _t{{\bar{f}}}\), \(m\in S^0\) and \({{\bar{f}}}-m\# {{\bar{f}}}_1\in S^{-1}\). Take a \(\ell \ge 0\) and with \(w_{\delta }=\langle {\delta \xi }\rangle ^{-\ell }\) \((0< \delta <1)\) we set
It is easy to see that \(|\partial _{\xi }^{\beta }w^{\pm 1}_{\delta }|\le C_{\beta }w_{\delta }^{\pm 1}\langle {\xi }\rangle ^{-|\beta |}\) with some \(C_{\beta }\) independent of \(\delta \). In the following, all arguments are uniform in \(0<\delta <1\) though we do not mention it.
Definition 5.4
Let \(S_i(t, \cdot )\) be two real functionals on \(C^2((-T, T); H^{s+n+1})\). We say \(S_1\overset{s}{\sim }S_2\) if for any \(\epsilon >0\) there is \(C_{\epsilon }>0\) independent of \(\delta \) such that
for any \(u(t)\in C^2((-T, T);H^{s+n+1})\). We write \(S_1\overset{s}{\lesssim }S_2\) or \(S_2\overset{s}{\gtrsim }S_1\) if \(S_1(t, u(t))-S_2(t, u(t))\) is bounded by the right-hand side.
In the following, all constants c, C may depend on s but not on \(\delta \) and may change from line to line. The main step to the proof of Proposition 5.1 is to estimate \((\varPhi \langle {D}\rangle ^s[F^{\delta }, {{\hat{P}}_{\theta }}]u, \varPhi \langle {D}\rangle ^s A F^{\delta } u)\).
Lemma 5.2
Let \(r_i\in S(\langle {\xi }\rangle ^{l_i}\phi ^{-n_i}, {{\bar{g}}})\) satisfy \(\partial _t r_i\in S(\langle {\xi }\rangle ^{l_i+1/2}\phi ^{-n_i}, {{\bar{g}}})\) \((i=1, 2, 3, 4)\). With \(R_j=\textrm{op}({r_j})\) one has
Proof
The proof is immediate from Corollary 6.3. \(\square \)
Lemma 5.3
We have
Proof
Since \((w_{\delta }{{\bar{f}}})\#h-h\#(w_{\delta }{{\bar{f}}})-i\{h, w_{\delta }{{\bar{f}}}\}\in S^0\) it follows from Lemma 5.2
Write \(\{h, w_{\delta }{{\bar{f}}}\}=\{h, {{\bar{f}}}\}w_{\delta }+\{h, w_{\delta }\}{{\bar{f}}}\). Since \(w_{\delta }{{\bar{f}}}-m\#(w_{\delta }{\bar{f}_1})\in S^{-1}\) and \(\{h, {{\bar{f}}\}}w_{\delta }\in S^1\) then \((\varPhi \langle {D}\rangle ^s\textrm{op}({i\{h, {{\bar{f}}}\}w_{\delta }}) u, \varPhi \langle {D}\rangle ^s A F^{\delta } u)\) is
Since \(r=\langle {\xi }\rangle ^s\#\phi ^{-n}\#\phi ^{-n}\#m-m\#\langle {\xi }\rangle ^s\#\phi ^{-n}\#\phi ^{-n}\in S(\langle {\xi }\rangle ^{s-1/2}\phi ^{-2n}, {{\bar{g}}})\) and \(\{h, {{\bar{f}}}\}w_{\delta }\in S^1\) it follows from Corollary 6.3 that for any \(\epsilon >0\) one has
hence (5.16) \(\overset{s}{\sim }(\varPhi \langle {D}\rangle ^s \textrm{op}({m}) \textrm{op}({i\{h, {{\bar{f}}}\}w_{\delta }}) u, \varPhi \langle {D}\rangle ^s A F^{\delta }_1u)\). Noting \(m\#(\{h, {{\bar{f}}}\}w_{\delta })+\big (\{h, f\}/\partial _tf\big )\#(w_{\delta }{{\bar{f}}_1})\in S^0\) we see that (5.16) is
this is still \( \overset{s}{\sim }\ -i(\textrm{op}({\{h, f\}/\partial _tf})\varPhi \langle {D}\rangle ^s F^{\delta }_1u, \varPhi \langle {D}\rangle ^s A F^{\delta }_1u)\) arguing as (5.17) for \(\phi ^{-n}\#\langle {\xi }\rangle ^s\#(\{h, f\}/\partial _tf)-(\{h, f\}/\partial _tf)\#\phi ^{-n}\#\langle {\xi }\rangle ^s\in S(\langle {\xi }\rangle ^{s+1/2}\phi ^{-n}, {{\bar{g}}})\).
For \(\{h, w_{\delta }\}{{\bar{f}}}\) setting \(k=\{h, w_{\delta }\}w_{\delta }^{-1}\in S^1\) one sees \(\{h, w_{\delta }\}{\bar{f}}-k\#(w_{\delta } {{\bar{f}}})\in S^0\) hence \((\varPhi \langle {D}\rangle ^s\textrm{op}({i\{h, w_{\delta }\}{{\bar{f}}}}) u, \varPhi \langle {D}\rangle ^s A F^{\delta } u) \overset{s}{\sim }(\varPhi \langle {D}\rangle ^s\textrm{op}({ik}) F^{\delta }u, \varPhi \langle {D}\rangle ^s A F^{\delta } u)\). Since \(\phi ^{-n}\#\langle {\xi }\rangle ^s\#k-k\#\phi ^{-n}\#\langle {\xi }\rangle ^s\in S(\langle {\xi }\rangle ^{s+1/2}\phi ^{-n}, {{\bar{g}}})\) this is still
Thanks to Lemma 6.7 we have \(\{h,w_{\delta }\}w_{\delta }^{-1}\in S(b, {{\bar{g}}})+S(b_1, {{\bar{g}}})\) then it follows that \((\textrm{op}({\{h, w_{\delta }\}w_{\delta }^{-1}})\varPhi \langle {D}\rangle ^s F^{\delta }u, \varPhi \langle {D}\rangle ^s A F^{\delta } u)\overset{s}{\sim }\ 0\) from Corollary 6.5. \(\square \)
Turn to \((\varPhi \langle {D}\rangle ^s[A^2, F^{\delta }]u, \varPhi \langle {D}\rangle ^s A F^{\delta }u)\) which is a sum
Noting \([A, F^{\delta }]=\textrm{op}({if^{-2}(\partial _tf)w_{\delta }{{\bar{f}}}})\in S^0\) and \(m\#(f^{-2}(\partial _tf)w_{\delta }{{\bar{f}}})-w_{\delta }{{\bar{f}}_1}\in S^{-1}\) it follows from a repetition of similar arguments that
Lemma 5.4
We have
Proof
The proof is easy if we note \(\partial _t\phi =\omega ^{-1}\phi \). \(\square \)
Noting \(A^2F^{\delta }=F^{\delta }A^2+A[A, F^{\delta }]+[A, F^{\delta }]A\) and \(\omega ^{-1}\phi ^{-n}\in S(\langle {\xi }\rangle ^{1/2}\phi ^{-n}, {{\bar{g}}})\) it follows from Lemma 5.4 with \(v=F^{\delta } u\) that
where replacing \(A^2\) by \(A^2=-{{\hat{P}}}_{\theta }+H+B'_0A+B'_1\) this is still
for \(B'_i=\textrm{op}({{{\tilde{a}}'_i}})\), \({{\tilde{a}}'_i}\in S^i\). We check the third term.
Lemma 5.5
\( (\varPhi \langle {D}\rangle ^s [A, F^{\delta }] u, \varPhi \langle {D}\rangle ^s F^{\delta } H u)\overset{s}{\sim }i(H \varPhi \langle {D}\rangle ^s F^{\delta }_1 u, \varPhi \langle {D}\rangle ^s F^{\delta }_1u)\).
Proof
Since \(h\in S^2\) this is \(\overset{s}{\sim }(\varPhi \langle {D}\rangle ^s [A, F^{\delta }] u, \varPhi H \langle {D}\rangle ^s F^{\delta } u)\). From Lemma (6.7) and Corollaries 6.5, 6.3 we see that \(|(\varPhi \langle {D}\rangle ^s[A, F^{\delta }]u, \textrm{op}({\{\phi ^{-n}, h\}})\langle {D}\rangle ^s F^{\delta } u)|\) is bounded by \( C\Vert \varPhi u\Vert _{s+1/2}{\mathcal {{\tilde{E}}}_{\!s}}(F^{\delta }u)\) hence this is still
If we move \(\textrm{op}({m})\) to in front of \([A, F^{\delta }]\), any term coming out in the process, is either \(S(b^2\langle {\xi }\rangle ^{2s-1/2}\phi ^{-2n}, {{\bar{g}}})\) or \(S(b_1^2\langle {\xi }\rangle ^{2s-1/2}\phi ^{-2n}, {{\bar{g}}})\) then thanks to Corollary 6.5 such a term is bounded by \({\mathcal {\tilde{E}}_{\!s-1/4}^2}(u)\). Finally, applying similar arguments to \((H \varPhi \langle {D}\rangle ^s \textrm{op}({m}) [A, F^{\delta }] u, \varPhi \langle {D}\rangle ^s F_1^{\delta } u)\) we conclude the proof. \(\square \)
Together with (5.19) we have
Lemma 5.6
We have
Let \(\kappa \) be the constant in Definition 5.3. It follows from Lemma 5.3 that
where, noting \((\{h, f\}/\partial _tf)\#(\{h, f\}/\partial _tf)-(\{h, f\}/\partial _tf)^2\in S^0\), the second term on the right-hand side is \(\overset{s}{\lesssim }4^{-1}\kappa ^{-1}(\textrm{op}({(\{h, f\}/\partial _tf)^2})\varPhi \langle {D}\rangle ^s F^{\delta }_1u, \varPhi \langle {D}\rangle ^s F^{\delta }_1u)\). In view of (5.13) Corollary 6.2 proves
Then using Lemma 5.6 we obtain
From Proposition 4.6 and (5.20) there are \(c>0, C>0\) such that
Choosing \(\epsilon >0\) and \(\theta \) such that \(\epsilon \le 2(1-\kappa )\), \(c\,\theta \ge C_{\epsilon }\) we have
Assume now \(\lim _{t\downarrow \tau }\sum _{j=0}^1\Vert D_t^ju(t)\Vert _{l+1-j}=0\) with some l. Since \({{\mathcal {E}}_s}(u(t))\le C\sum _{j=0}^1\Vert D_t^ju(t)\Vert _{n+s+1-j}\) choosing \(\ell \) so that \(\ell \ge n+s-l\) we have
for any \(\delta >0\). Since \(|(\varPhi \langle {D}\rangle ^s [A, F^{\delta }] u, \varPhi \langle {D}\rangle ^s A F^{\delta } u)| \le C{\mathcal {\tilde{E}}^2_{\!s-1/4}}(u)\) integrating (5.21) from \(\tau \) to t (\(|t|\le \delta _0\)) we have
One can replace \({\mathcal {{\tilde{E}}}_s}(\cdot )\), \({\mathcal {\tilde{E}}_{\sharp s}}(\cdot )\) and \({{\hat{P}}_{\theta }}\) by \({{\mathcal {E}}_s}(\cdot )\), \({{\mathcal {E}}_{\sharp s}}(\cdot )\) and \({{\hat{P}}}\) in (5.22) if we replace u by \(e^{-\theta t}u\). Denote
Since \(\Vert \varPhi \langle {D}\rangle ^s D_t F^{\delta }u\Vert +\Vert \varPhi \langle {D}\rangle ^s u\Vert \le C({{\mathcal {E}}_s^{1/2}}(F^{\delta }u)+{{\mathcal {E}}_{s-1/4}^{1/2}}(u))\) it follows that
from which we obtain
Letting \(\delta \downarrow 0\) we have
Proposition 5.3
Assume f is spacelike and \(u\in \cap _{j=0}^1C^j([\tau , \delta _0]; H^{l+1-j})\) verifies \(\lim _{t\downarrow \tau }\sum _{j=0}^1\Vert D_t^ju(t)\Vert _{l+1-j}=0\). Then if \({{\mathcal {N}}_{s-1/4}}(u; t)<+\infty \), \(\tau \le t\le \delta _0\) and \(F {{\hat{P}}} u\in L^1([\tau , \delta _0]; H^{n+s})\) with \(F=\textrm{op}({{\bar{f}}})\) one has \({{\mathcal {N}}_{s}}(Fu; t)<+\infty \) for \(\tau _1\le t\le \delta _0\) and
Let \(\chi (s)\in C^{\infty }({{\mathbb {R}}})\) be nondecreasing such that \(\chi (s)=s\) for \(|s|\le 1\) and \(|\chi (s)|=2\) for \(|s|\ge 2\) and let \(0\le {\tilde{\chi }}(\xi )\in C^{\infty }({{\mathbb {R}}}^d)\) be 0 in a neighborhood of the origin and \({\tilde{\chi }}=1\) for \(|\xi |\ge 1\). For \(w=(y, \eta )\in {{\mathbb {R}}}^{d}\times ({{\mathbb {R}}}^d{\setminus } 0)\) we set
Note that \(d_{\epsilon }^2(x, \xi ; w)\ge \min {\{1, |x-y|^2\}}+|\xi /|\xi |-\eta /|\eta ||^2+\epsilon ^2\) for \(|\xi |\ge 1\). We often write \(d_{\epsilon }(x,\xi )\) for \(d_{\epsilon }(x,\xi ; w)\) dropping w. It is clear that \(d_{\epsilon }\in S^0\) if \(\epsilon \ne 0\). For \(\nu >0\) we define
It is easy to see that \(|\partial _x^{\alpha }\partial _{\xi }^{\beta }d_{\epsilon }|\le C\langle {\xi }\rangle ^{-|\beta |}\) (\(|\alpha +\beta |=1\)) with \(C>0\) independent of \(\epsilon >0\). From \(0\le h\in G(M^{-2}\langle {\xi }\rangle ^2, G)\) one has \( \{h, \nu d_{\epsilon }\}^2\le C\nu ^2q\) in virtue of the Glaeser inequality then there is \(\nu _0>0\) such that \(f_{\epsilon }\) is spacelike for any \(0<\nu \le \nu _0\) and \(\epsilon >0\). We fix such a \(\nu >0\) and denote \(F_{\epsilon }=\textrm{op}({{{\bar{f}}_{\epsilon }}})\).
Lemma 5.7
Assume that \(u\in \cap _{j=0}^1 C^j([\tau , \delta _0]; H^{l+1-j})\), \({{\hat{P}}}u\in L^1([\tau , \delta _0]; H^{l'})\) and \(\lim _{t\downarrow \tau }\sum _{j=0}^1\Vert D_t^ju(t)\Vert _{l+1-j}= 0\) with some \(l, l'\in {{\mathbb {R}}}\). If \(F_{\epsilon _0} {{\hat{P}}} u\in L^1([\tau , \delta _0]; H^{s_0+n})\) with some \(\epsilon _0>0\), \(s_0\in {{\mathbb {R}}}\) then for any \(0<\epsilon <\epsilon _0\) and \(s\le s_0-1/4\) one has \( F_{\epsilon }u\in \cap _{j=0}^1C^j([\tau , \delta _0]; H^{s+1-j})\) and
where \(R_l(u; t)=\sup _{\tau \le t'\le t}\sum _{j=0}^1(\Vert D_t^ju(t')\Vert _{l+1-j}+\int _{\tau }^{t'}\Vert D_t^ju(t_1)\Vert _{l+1-j}dt_1)\).
Proof
Choose a strictly decreasing sequence \(\epsilon<\epsilon _j<\epsilon _0\) converging to \(\epsilon \) as \(j\rightarrow \infty \). Denoting \(F_j=F_{\epsilon _j}\), \(f_j=f_{\epsilon _j}\) we shall prove
for j with \(l+j/4\le s_0\) by induction on j.Take \(g_j\in S^0\) such that \(\textrm{supp}\,g_j\subset \{f_j<0\}\) and \(\{f_{j+1}<0\}\subset \{g_j=1\}\). Write \(F_{j+1} P \textrm{op}({g_j})u=F_{j+1}\textrm{op}({g_j}) P u+F_{j+1}[P, \textrm{op}({g_j})]u\) then it is seen that \(\Vert F_{j+1}{{\hat{P}}}\textrm{op}({g_j})u\Vert _{l+n+(j+1)/4}\) is bounded by
hence an application of Proposition 5.3 with \(s=l+(j+1)/4\le s_0\), \(F=F_{j+1}\), \(u=\textrm{op}({g_j})u\) gives
Repeating similar arguments one has \({{\mathcal {N}}_{l+j/4}}(\textrm{op}({g_j})u; t)\le C\big \{{{\mathcal {N}}_{l+j/4}}(F_ju; t) +R_l(u; t)\big \}\) and \({{\mathcal {N}}_{l+(j+1)/4}}(F_{j+1}u; t)\le C\big \{{{\mathcal {N}}_{l+(j+1)/4}}(F_{j+1}\textrm{op}({g_j})u; t)+R_l(u; t)\big \}\). Estimating \({{\mathcal {N}}_{l+j/4}}(\textrm{op}({g_j})u; t)\) by use of the inductive hypothesis we conclude that (5.26) holds for maximal \(j=j_0\) satisfying \(l+j/4\le s_0\). Since \(\epsilon <\epsilon _{j_0}\) one can write \({{\bar{f}}_{\epsilon }}-k\#{{\bar{f}}_{j_0}}\in S^{-\infty }\) with some \(k\in S^0\) the assertion follows. \(\square \)
Proof of Proposition 5.1
Take \(0<{ \epsilon }<1/4\) such that \(16\,\epsilon ^2\le |x-{\tilde{x}}|^2+|\xi /|\xi |-{\tilde{\xi }}/|{\tilde{\xi }}||^2\) holds for any \((x, \xi )\in U_1\) and \(({{\tilde{x}}}, {\tilde{\xi }})\in U_2\).Fix a \(0<\nu _1<\nu \) then there are finite many \(w_i=(y_i, \eta _i)\in { U_2}\), \(i=1, \ldots , n\) such that
Write \(f_{i,{ \epsilon }}=f_{ \epsilon }(\tau , x, \xi ; w_i)\), \(F_{i,{\epsilon }}=\textrm{op}({{{\bar{f}}}_{i,{\epsilon }}})\) then it is clear that \(\sum _if_{i, {\epsilon }}<0\) on \([\tau , \tau +\nu _1{\epsilon }]\times { U_2}\), while \(\{f_{i, 2\epsilon }(t, x, \xi )=t-\tau -4\nu \epsilon +\nu d_{2\epsilon }(x, \xi ; w_i)<0\}\) does not intersect with \([\tau , \tau +\nu _1 \epsilon ]\times (U_1\cap \{|\xi |\ge 1\})\). Therefore it follows that \(\int _{\tau }^{t}\Vert F_{i, 2{ \epsilon }}\textrm{op}({h_1}) f\Vert _pdt_1\le C\int _{\tau }^{t}\Vert f\Vert _{l_2}dt_1\) for any \(p\in {{\mathbb {R}}}\).Here we apply Lemma 5.7 with \(u={{\hat{G}}}\textrm{op}({h_1})f\in \cap _{j=0}^1 C^j([\tau , \delta _0]; H^{l_2-n+1-j})\), \(F_{\epsilon _0}=F_{i, 2{\epsilon }}\), \(F_{\epsilon }=F_{i, {\epsilon }}\), \(l=l_2-n\), \(l'=l_2\) to obtain
for any \(p, s\in {{\mathbb {R}}}\), \(s\le p-n-1/4\). From (5.4) one has \(R_{l_2-n}({{\hat{G}}}\textrm{op}({h_1})f; t)\le C\int _{\tau }^t\Vert \textrm{op}({h_1})f\Vert _{l_2}dt_1\) hence \( \sum _{j=0}^1\Vert D^j_tF_{i, {\epsilon }} {{\hat{G}}}\textrm{op}({h_1})f\Vert _{l_1-j}\le C\int _{\tau }^t\Vert f\Vert _{l_2}dt_1\) choosing \(s=l_1-1\) and \(l_1\le p-n+3/4\) in (5.28). Then Proposition 5.1 is proved if we remark
and take \(v={{\hat{G}}}\textrm{op}({h_1})f\) there. \(\square \)
For the solution operator of the Cauchy problem (5.3) with \(\phi _0=\phi _1=0\);
the same argument for \({{\hat{G}}}\) proves that \({{\hat{G}}^*}\) has a finite speed of propagation. Then repeating the proof of the local existence theorem for P one obtains
Theorem 5.4
If all critical points \((0, 0, \tau , \xi )\) of \(p=0\) are effectively hyperbolic then there are \(\delta >0\), \(n>0\) and a neighborhood \(\Omega \) of \(x=0\) such that for any \(|{\tau }|<\delta \) and \(f\in L^1((-\delta , {\tau }); H^{s+n})\) there exists \(u\in \cap _{j=0}^1C^j([-\delta , {\tau }]; H^{s+1-j})\) satisfying \(P^*u=f\) in \((-\delta , {\tau })\times \Omega \) and
5.3 Local uniqueness theorem
Consider the second order differential operator
with the principal symbol
where \(a_{j,\alpha }(t, x)\) are \(C^{\infty }\) functions defined in a neighborhood of \((t, x)=(0, 0)\in {{\mathbb {R}}}^{1+d}\). For notational convenience we write \(x_0\), \(\xi _0\) instead of t, \(\tau \) and denote \(x=(x_0,x_1,\ldots , x_d)=(x_0, x')\), \(\xi =(\xi _0,\xi _1,\ldots , \xi _d)=(\xi _0, \xi ')\). Let \(y=\kappa (x)\), \(\kappa (0)=0\) be a change of local coordinates x then, in y coordinates, the principal symbol \({{\tilde{p}}}(y, \eta )\) of P is \(p(\kappa ^{-1}(y), {^t\!}\kappa '(x)\eta )\).The following lemma is a special case of a well-known fact (e.g. [14]).
Lemma 5.8
If \((0, {\bar{\xi }})\) is effectively hyperbolic characteristic of p then \((0, {\bar{\eta }})\), \({\bar{\xi }}={^t\!}\kappa '(0){\bar{\eta }}\) is effectively hyperbolic characteristic of \({{\tilde{p}}}\) and vice versa.
Proof
Denote \(\kappa ^{-1}(y)=\lambda (y)\) and \(\kappa (x)=(\kappa _0(x), \kappa _1(x),\ldots , \kappa _d(x))\).If Q is the quadratic form associated with the Hessian of p then we have
where \(C=(c_{ij})\) is the \((d+1)\times (d+1)\) matrix
Therefore denoting by \({{\tilde{Q}}}\) the corresponding quadratic form of \({{\tilde{p}}}\) at \((0,{\bar{\eta }})\) one has
Checking that \(C \kappa '(0)\) is symmetric one concludes that \(F_{{{\tilde{p}}}}(0,{\bar{\eta }})=K^{-1}F_p(0, {\bar{\xi }})K\) hence the assertion. \(\square \)
Next, consider a new system of local coordinates y such that
which is so called Holmgren transform (e.g. [15]) where \(\epsilon >0\) is a small positive constant that will be fixed later. It is clear that
The following lemma is also well-known (e.g. [21]).
Lemma 5.9
If \(p(x, \xi _0, \xi ')=0\) has only real root in \(\xi _0\) for any x in a neighborhood of the origin of \({{\mathbb {R}}}^{1+d}\) and \(\xi '\in {{\mathbb {R}}}^d\) then there exist \(r>0\) and \(\epsilon _0>0\) such that for any \(|\epsilon |\le \epsilon _0\), \({{\tilde{p}}}(y, \eta _0, \eta ')=0\) has only real root in \(\eta _0\) for any \(|y|\le r\) and \(\eta '\in {{\mathbb {R}}}^d\).
Lemma 5.10
One can find a neighborhood \(\Omega \) of the origin of \({{\mathbb {R}}}^{1+d}\) and \({\bar{\epsilon }}>0\), \(\epsilon >0\) such that for any \(f(x)\in C_0^{\infty }(\Omega )\) with \(\textrm{supp}f\subset \{x; x_0\le {\bar{\epsilon }}-\epsilon |x'|^2\}\) there exists \(v(x)\in C^2(\Omega )\) with \(\textrm{supp}\, v\subset \{x; x_0\le {\bar{\epsilon }}-\epsilon |x'|^2\}\) satisfying \(P^*v=f\) in \( \Omega \).
Proof
Since \(P^*=\textrm{op}({p+{{\bar{P}}_1}+{{\bar{P}}_0}})\) then \(P^*\) in the local coordinates y is given by \(P^*=\textrm{op}({{\tilde{p}}+{P'_1}+{P''_0}})\).Thanks to Lemmas 5.8 and 5.9 one can apply Theorem 5.4 to conclude the assertion. \(\square \)
Now prove the local uniqueness theorem. Assume that \(u(x)\in C^2(\Omega )\) verifies \(Pu=0\) in \(\Omega \cap \{x_0>\tau \}\) and \(D_0^ju(\tau , x')=0\), \(j=0, 1\) on \(\Omega \cap \{x_0=\tau \}\) (\(|\tau |\le {\bar{\epsilon }}\)).For \(f\in C_0^{\infty }(\Omega )\) with \(\textrm{supp}f\subset \{x; x_0\le {\bar{\epsilon }}-\epsilon |x'|^2\}\) take v(x) in Lemma 5.10 then one has
Since f is arbitrary we conclude \(u=0\) in \(\{x; \tau< x_0< {\bar{\epsilon }}-\epsilon |x'|^2\}\). Returning to the original notation \(x_0=t\), \((x_0, x')=(t, x)\) the assertion can be stated as
Theorem 5.5
Assume that all critical points \((0, 0, \tau , \xi )\) of \(p=0\) are effectively hyperbolic. Then there are a neighborhood \(\omega \) of the origin and \(\epsilon >0\) such that if \(u\in C^2(\omega )\) satisfies \((|\tau |\le \epsilon )\)
then \(u=0\) in \(\omega \cap \{t>\tau \}\).
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Number JP20K03679.
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Appendix
Appendix
In this appendix, we summarize the properties of the pseudodifferential operators used in this paper and also give the proof of Proposition 5.2.
1.1 Pseudodifferential operators, composition, \(L^2\) continuity and inverse
In this paper, all metrics are supposed to be of the form
(Beals-Fefferman metric [1]) where \(\phi (z)\), \(\psi (z)\) are positive functions on \({{\mathbb {R}}}^{2d}\) depending on positive parameters \(\gamma \), M constrained by
For notational simplicity, we omit to write parameters in \(\phi \), \(\psi \), and all constants assumed to be independent of parameters \(\gamma \), M in what follows, although we do not mention this. Recall several notions related to Weyl-Hörmander calculus from Hörmander’s book [4, Chapter XVIII]. For a positive function m(z) we define S(m, g) the set of all \(a\in C^{\infty }({{\mathbb {R}}}^{2d})\) such that for every \(k\in {{\mathbb {N}}}\)
The left hand is denoted by \(|a|^{(k)}_{S(m, g)}\) which induces the topology in S(m, g) as a Fréchet space. Denote
where \(g_z^{\sigma }(w)=\sup _{0\ne v\in {{\mathbb {R}}}^{2n}}|\sigma (w, v)|^2/g_z(v)\). For a metric (6.1) it is easy to see
A metric (6.1) is \(\sigma \) temperate (see [4, Definition 18.5.1]) if there are positive constants c, C, N such that
Note that (6.6) implies
which is symmetric with respect to z, w. Let g be \(\sigma \) temperate metric. A positive function m(z) is called \(\sigma , g\) temperate weight (see [4, Definition 18.5.1]) if there are positive constants c, C, N such that
Note that (6.9) is equivalent to \( m(w)\le C'm(z)(1+g^{\sigma }_{z}(w-z))^{N'}\) because of (6.7). This paper uses more restricted weights than \(\sigma , g\) temperate weights.
Definition 6.1
Let g be \(\sigma \) temperate metric. A positive function m is called g admissible weight if there are positive constants C, N such that
It is clear from the definition that if m is g admissible weight then m is also \({{\tilde{g}}}\) admissible weight for any \(\sigma \) temperate metric \({{\tilde{g}}}\ge g\).
Lemma 6.1
Let g be \(\sigma \) temperate and satisfy (6.3). If m is g admissible weight then m is \(\sigma , g\) temperate weight.
Proof
If \(g_z(w-z)<c\) one has \(m(w)\le C'(1+c\,C)m(z)\) in view of (6.5) and (6.10). Since \(\max {\{g_w(w-z), g_z(w-z)\}}\) is symmetric for w, z one concludes (6.8). Noting \(\max {\{g_w(w-z), g_z(w-z)\}}\le \max {\{g^{\sigma }_w(w-z), g^{\sigma }_z(w-z)\}}\) by (6.3) we have (6.9) from (6.7). \(\square \)
Lemma 6.2
If m is g admissible weight so is \(m^s\) for any \(s\in {{\mathbb {R}}}\). If \(m_i\) \((i=1, 2)\) are g admissible weights so is \(m_1m_2\).
Proof
Since 1/m is g admissible weight by (6.10) then the first assertion is clear. The second assertion is also clear by (6.10). \(\square \)
In this paper we work with more restricted metrics (6.1) which satisfies with some \(0<\delta <1\) and \(c>0\) that
Lemma 6.3
A metric (6.1) satisfying (6.11), (6.12) is \(\sigma \) temperate and satisfies (6.3).
Proof
If \(g_z(z-w)<c_1^2\) then \(|\xi -\eta |<c_1\psi (z)\le c_1C\langle {\xi }\rangle _{\gamma }\) so (6.5) is immediate by (6.12) choosing \(c_1C\le c\). If \(\langle {\eta }\rangle _{\gamma }\le \langle {\xi }\rangle _{\gamma }/2\sqrt{2}\) then \(|\xi -\eta |\ge (\gamma +|\xi |)-(\gamma +|\eta |)\ge \langle {\xi }\rangle _{\gamma }-\sqrt{2}\langle {\eta }\rangle _{\gamma }\ge \langle {\xi }\rangle _{\gamma }/2\) which gives \(|\xi -\eta |\ge c \langle {\xi }\rangle _{\gamma }^{1-\delta }\langle {\eta }\rangle _{\gamma }^{\delta }\) with some \(c>0\). On the other hand if \(\langle {\eta }\rangle _{\gamma }\ge 2\sqrt{2}\langle {\xi }\rangle _{\gamma }\) then \(|\xi -\eta |\ge \langle {\eta }\rangle _{\gamma }/2\) hence \(|\xi -\eta |\ge c \langle {\eta }\rangle _{\gamma }^{1-\delta }\langle {\eta }\rangle _{\gamma }^{\delta }\). Therefore there is C such that
Note that \(g^{\sigma }_w(z-w)\ge \phi ^2(w)|\xi -\eta |^2\ge \langle {\eta }\rangle _{\gamma }^{-2\delta }|\xi -\eta |^2/C\) which proves (6.6) while (6.3) is obvious by (6.4) and (6.11). \(\square \)
Lemma 6.4
All metrics \({\underline{g}}\), \(g_{\epsilon }\), \({{\bar{g}}}\) in (4.1) satisfy (6.11), (6.12) and \(\langle {\xi }\rangle _{\gamma }^s\), \(s\in {{\mathbb {R}}}\) is \({\underline{g}}\), \(g_{\epsilon }\), \({{\bar{g}}}\) admissible weight.
Proof
Indeed (6.11) is verified with \(\delta =1/2\). If \(|\xi -\eta |<c\langle {\xi }\rangle _{\gamma }\). Here we remark that
which proves (6.12). Next since \(|\partial _{\xi }^{\alpha }\langle {\xi }\rangle _{\gamma }|\le C\) for \(|\alpha |=1\) we see that
hence \(\langle {\xi +\eta }\rangle _{\gamma }\le C\langle {\xi }\rangle _{\gamma }(1+{\underline{g}_z}(w))^{1/2}\) which shows \(\langle {\xi }\rangle _{\gamma }\) is \({\underline{g}}\) admissible weight hence the assertion because \({\underline{g}}\le g_{\epsilon }\le {{\bar{g}}}\). \(\square \)
We state the main theorem of the Weyl-Hörmander calculus [4, Theorem 18.5.4] for the present case.
Theorem 6.1
Let g satisfy (6.11), (6.12) and \(m_i\) be g admissible weights and \(a_i\in S(m_i, g)\). Then the oscillatory integral
defines \(c(z)\in S(m_1m_2,g)\). Denoting c(z) by \(a_1\#a_2\) one has
and for every \(l\in {{\mathbb {N}}}\) there are C, \(l'\) such that
Moreover if \(\partial _{x}^{\alpha }\partial _{\xi }^{\beta }a_i\in S(m^{\beta }_{i,\alpha }, g)\) for g admissible weights \(m^{\beta }_{i,\alpha }\) for \(|\alpha +\beta |=l\) we have
In particular with \(l=3\) one has
The theorem can be proved in a naive way (repeated use of integration by parts) taking into account the special features of such restricted metrics satisfying (6.11), (6.12) and weights given by Definition 6.1, or keeping that the “structural constants” of the metrics and weights are independent of parameters \(\gamma \), M in mind, it suffices to follow the general proof in [4, Theorem 18.5.4] or [13, Theorem 2.3.7].
Corollary 6.1
Set \(h^2(z)=\sup {g_{z}/g^{\sigma }_{z}}\) then for any \(N\in {{\mathbb {N}}}\) we have
In particular we have
Noting again that “structural constants” of the metric \({{\bar{g}}}\) is independent of \(\gamma \) it follows from the proof of [4, Theorem 18.6.3] or [13, Theorem 2.5.1] that
Theorem 6.2
The operator \(\textrm{op}({a})\) is \(L^2\) bounded for every \(a\in S(1, {{\bar{g}}})\). Namely there exist \(C>0\), \(\ell \in {{\mathbb {N}}}\) depending only on the dimension d such that
Similarly, following the proofs of [13, Lemma 2.6.26] and [13, Theorem 2.6.27] or [12, Theorem I.1] we have
Theorem 6.3
There exist \(C>0\), \(l^0\in {{\mathbb {N}}}\) such that if \(a(x, \xi )\in S(1, {{\bar{g}}})\) satisfies \(|a|^{(l^0)}_{S(1, {{\bar{g}}})}\le C^{-1}\) then
converges in \(S(1, {{\bar{g}}})\) and satisfies \((1-a)\#b=b\#(1-a)=1\). Moreover for any l there are \(C_{l}\), \(l'\) such that
1.2 Admissible weight related to nonnegative symbols
Let \(0\le a(x, \xi )\in S(M^{-2}\langle {\xi }\rangle _{\gamma }^2, G)\) where G is given in (3.5) and M, \(\gamma \) are constrained by (3.1). Since \(M^{|\alpha +\beta |}\langle {\xi }\rangle _{\gamma }^{-|\beta |}\le (M^2\langle {\xi }\rangle _{\gamma }^{-1})^{|\alpha +\beta |/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}\) it is clear that \(S(m, G)\subset S(m, {{\bar{g}}})\). Introducing a parameter \(\lambda \ge 1\) which is constrained by
we consider an approximate square root of a;
Lemma 6.5
We have \( \partial _x^{\alpha }\partial _{\xi }^{\beta }b^{\pm 1} \in S(\lambda ^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}\,b^{\pm 1}, {{\bar{g}}})\) for \(|\alpha +\beta |\ge 1\). In particular \(b^{\pm 1}\in S(b^{\pm 1}, {{\bar{g}}})\).
Proof
Set \( {{\bar{a}}}=a(x, \xi )\langle {\xi }\rangle _{\gamma }^{-2}\) and \({{\bar{b}}}=({{\bar{a}}}+\lambda \langle {\xi }\rangle _{\gamma }^{-1})^{1/2}\) so that \(b={{\bar{b}}}\langle {\xi }\rangle _{\gamma }\). In the proof we often use \(({{\bar{b}}}\lambda ^{-1/2})^k\ge \langle {\xi }\rangle _{\gamma }^{-k/2}\) \((k\ge 0)\) which follows from \(CM^{-1}\ge {{\bar{b}}}\ge \lambda ^{1/2}\langle {\xi }\rangle _{\gamma }^{-1/2}\). Since \({{\bar{a}}}\in S(M^{-2}, G)\) the Glaeser inequality shows
while it is clear that
Noting \(\sqrt{{{\bar{a}}}}\le {{\bar{b}}}\) it follows from (6.23) and (6.24) that
Assume (6.25) holds for \(1\le |\alpha +\beta |\le l\). Since \({{\bar{b}}}^2={{\bar{a}}}+\lambda \langle {\xi }\rangle _{\gamma }^{-1}\) then for \(|\alpha +\beta |\ge l+1\ge 2\) we see
where the second term on the right-hand side of (6.26) is estimated as
To estimate the third term it suffices to apply (6.24). Therefore we conclude from (6.26) that (6.25) holds for \(|\alpha +\beta |=l+1\) and hence for any \(\alpha , \beta \). The assertion for b follows immediately from (6.25). The estimate for \(b^{-1}\) can be obtained from those of b by differentiating \(bb^{-1}=1\). \(\square \)
Lemma 6.6
b is \({{\bar{g}}}\) admissible weight and \({ b}^{\pm 1}\in S({ b}^{\pm 1}, {{\bar{g}}})\).
Proof
Since \(\langle {\xi }\rangle _{\gamma }\) is \({{\bar{g}}}\) admissible weight it is enough to prove that \({{\bar{b}}}\) is \({{\bar{g}}}\) admissible weight. Note that \(|\partial _x^{\alpha }\partial _{\xi }^{\beta }{{\bar{b}}}|\lesssim \langle {\xi }\rangle _{\gamma }^{-|\beta |}\) for \(|\alpha +\beta |=1\) in view of (6.23) and (6.24). If \(|\eta |<c\langle {\xi }\rangle _{\gamma }\) then from (6.14) there is \(C>0\) such that
hence one has
which proves
when \(|\eta |\le c\langle {\xi }\rangle _{\gamma }\). If \(|\eta |\ge c\langle {\xi }\rangle _{\gamma }\) then \({{\bar{g}}}_z(w)\ge c^2\langle {\xi }\rangle _{\gamma }\) one has
so that (6.29) holds. Thus \({{\bar{b}}}\) is \({{\bar{g}}}\) admissible weight. \(\square \)
Proposition 6.1
One can find \(\lambda _1\ge 1\) independent of M and \(\gamma \) such that for \(\lambda \ge \lambda _1\) there exists \({{\tilde{b}}}\in S(b^{-1}, {{\bar{g}}})\) satisfying \(b\#{{\tilde{b}}}={{\tilde{b}}}\#b=1\).
Proof
Since \(b^{\pm 1}\in S(b^{\pm 1}, {{\bar{g}}})\) and \(\partial _x^{\alpha }\partial _{\xi }^{\beta }b^{\pm 1}\in S(\lambda ^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}b^{\pm 1}, {{\bar{g}}})\) for \(|\alpha +\beta |=1\) by Lemma 6.5 and \(b^{\pm 1}\) is \({{\bar{g}}}\) admissible weight then thanks to Theorem 6.1 one has \(b\#b^{-1}=1-r\) with \(r\in S(\lambda ^{-1/2}, {{\bar{g}}})\). Therefore there is \(\lambda _1\) such that for \(\lambda \ge \lambda _1\) one can apply Theorem 6.3 to obtain
and that \(b\#(b^{-1}\#{{\tilde{r}}})=1\). Similarly there exists \({{\hat{r}}}\in S(1, {{\bar{g}}})\) such that \({{\hat{r}}}\#b^{-1}\#b=1\) which proves \((b^{-1}\#{{\tilde{r}}})\#b=1\). Thus \({{\tilde{b}}}=b^{-1}\#{{\tilde{r}}}\in S(b^{-1}, {{\bar{g}}})\) is a desired one. \(\square \)
Lemma 6.7
We have \(a\in S(b^2, {{\bar{g}}})\) and \(\partial _x^{\alpha }\partial _{\xi }^{\beta }a\in S(b\langle {\xi }\rangle _{\gamma }^{1-|\beta |}, {{\bar{g}}})\) for \(|\alpha +\beta |=1\).
Proof
Since \(\langle {\xi }\rangle _{\gamma }\in S(b^2, {{\bar{g}}})\) is clear for \(\langle {\xi }\rangle _{\gamma }\le b^2\) the first assertion is obvious. Note that if \({{\tilde{a}}}\in S(M^{-1-k}\langle {\xi }\rangle _{\gamma }^{\ell }, G)\) noting \(b\ge \langle {\xi }\rangle _{\gamma }^{1/2}\) one has
For \(|\alpha +\beta |=1\) we have \(|\partial _x^{\alpha }\partial _{\xi }^{\beta }a|\le Cb\langle {\xi }\rangle _{\gamma }^{1-|\beta |}\) from (6.23). For \(|\alpha +\beta |\ge 2\) it is enough to apply (6.30) to \({\tilde{a}}=\partial _x^{\alpha '}\partial _{\xi }^{\beta '}a\in S(M^{-1}\langle {\xi }\rangle _{\gamma }^{2-|\beta '|}, G)\), \(|\alpha '+\beta '|=1\). \(\square \)
Lemma 6.8
There exists \(\lambda _2\ge \lambda _1\) independent of M and \(\gamma \) such that for \(\lambda \ge \lambda _2\) one has
Proof
Noting \(\partial _x^{\alpha }\partial _{\xi }^{\beta }b\in S(\lambda ^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}b, {{\bar{g}}})\) for \(|\alpha +\beta |=2\) in virtue of Lemma 6.5 it follows from Theorem 6.1 that \(b\#b=b^2+r\), \(r\in S(\lambda ^{-1}b^2, {{\bar{g}}})\). Taking \({{\tilde{b}}}\in S(b^{-1}, {{\bar{g}}})\) in Proposition 6.1 we set \(r=b\#({{\tilde{b}}}\#r\#{{\tilde{b}}})\#b\). Since \({{\tilde{b}}}\#r\#{{\tilde{b}}}\in S(\lambda ^{-1}, {{\bar{g}}})\), thanks to Theorem 6.2, there exists \(C>0\) independent of \(\lambda \) such that \(\Vert \textrm{op}({{{\tilde{b}}}\#r\#{{\tilde{b}}}}) v\Vert \le C\lambda ^{-1}\Vert v\Vert ^2\), hence \(|(\textrm{op}({r})v, v)|\le C\lambda ^{-1}\Vert \textrm{op}({b}) v\Vert ^2\). Then we see
It is enough to choose \(\lambda _2\ge \lambda _1\) so that \(1-C\lambda _2\le 1/2\). \(\square \)
Let \(\langle {\xi }\rangle \) stand for \(\langle {\xi }\rangle _{\gamma }\) with \(\gamma =1\). The following inequality is called sharp Gårding inequality ([2]).
Corollary 6.2
If \(0\le a(x, \xi )\in S^2_{1, 0}\) there is \(C>0\) such that
Proof
If we fix \(M=1\) and \(\gamma \ge \lambda _2\) then \(0\le a\in S(\langle {\xi }\rangle _{\gamma }^2, G)\) hence one can apply Lemma 6.8 with \(\lambda =\lambda _2\) to get
which is the assertion. \(\square \)
1.3 Pseudodifferential operators associated with metrics related to localization
In this subsection we study pseudodifferential operators associated with metrics g satisfying (6.11), (6.12) and
Lemma 6.9
Let m be g admissible weight such that \(m\in S(m, { g})\). Then there exist \(M_0>0\) and \(k\in S(M^{-1}, { g})\) \((M>M_0)\) such that
Proof
Since \(m^{-1}\) is g admissible weight and \(m^{-1}\in S(m^{-1}, { g})\) one has \(m\#m^{-1}=1-r\) with \(r\in S(M^{-1}, { g})\subset S(M^{-1}, {{\bar{g}}})\). Thanks to Theorem 6.3 there is \(M_0\) such that \(\sum _{l=1}^{\infty }r^{\#l}\) converges in \(S(1, {{\bar{g}}})\) to some \(k\in S(1, {{\bar{g}}})\) for \(M>M_0\) so that \((1-r)\#(1+k)=(1+k)\#(1-r)=1\) which shows the first two equalities. It remains to prove \(k\in S(M^{-1}, { g})\). It suffices to show
From (6.20) one sees that \(k\in S(M^{-1}, {{\bar{g}}})\) so that (6.32) holds when \(|\alpha |=0\). Suppose that (6.32) holds for \(|\alpha |\le l\) and consider the case \(|\alpha |=l+1\). Since k verifies \(k=r+r\#k\) one has
When \(|\alpha ''|=|\alpha |=l+1\) we have \(\partial _z^{\alpha }k\in S(M^{-1}{{\bar{g}}}_z^{1/2}(s)\prod g^{1/2}_z(t_j), {{\bar{g}}})\) where \(\partial _z^{\alpha }=\partial _z^s\prod \partial _z^{t_i}\), \(|s|=1\) and \(M^{-1}{{\bar{g}}}^{1/2}_z(s)\lesssim g_z^{1/2}(s)\) by assumption. Thus we have \(r\#(\partial _z^{\alpha }k)\in S(M^{-1}\prod g^{1/2}_z(t_i), {{\bar{g}}})\) with \(\partial _z^{\alpha }=\prod \partial _z^{t_i}\), \(|t_i|=1\). When \(|\alpha ''|\le l\) the assumption (6.32) shows that \((\partial _z^{\alpha '}r)\#(\partial _z^{\alpha ''}k)\in S(M^{-2}\prod g^{1/2}_z(t_i), {{\bar{g}}})\) with \(\partial _z^{\alpha }=\prod \partial _z^{t_i}\). Therefore (6.32) holds for \(|\alpha |=l+1\) and hence \(k\in S(M^{-1}, g)\) by induction on \(|\alpha |\). Similarly there is \({\tilde{k}}\in S(M^{-1}, g)\) such that \((1+{{\tilde{k}}})\#m^{-1}\#m=1\) (\(M>M_0\)) which proves the third equality. \(\square \)
Lemma 6.10
Let \(m_i\) \( (i=1, 2)\) be g admissible weights such that \(m_i\in S(m_i, g)\). If \({{\bar{m}}}\) is \({{\bar{g}}}\) admissible weight and \(a\in S({{\bar{m}}}m_1m_2, {{\bar{g}}})\) then there exist \(M_0>0\) and \({{\tilde{a}}}\in S({{\bar{m}}}, {{\bar{g}}})\) \((M> M_0)\) such that one has \( a=m_1\#{{\tilde{a}}}\#m_2\). Moreover if \({{\bar{m}}}\) is g admissible weight then \({{\tilde{a}}}\in S({{\bar{m}}}, g)\) is given by \((m_1 m_2)^{-1}a+r\) with \(r\in S(M^{-1}{{\bar{m}}}, g)\).
Proof
Since \(m_i^{-1}\) are g admissible weights such that \(m_i^{-1}\in S(m_i^{-1}, g)\), Lemma 6.9 gives \(k, {{\tilde{k}}}\in S(M^{-1}, g)\) (\(M>M_0\)) verifying \(m_1\#(1+k)\#m_1^{-1}=1\) and \(m_2^{-1}\#(1+{{\tilde{k}}})\#m_2=1\). Then \({{\tilde{a}}}=(1+k)\#m_1^{-1}\#a\#m_2^{-1}\#(1+{{\tilde{k}}})\in S({{\bar{m}}}, {{\bar{g}}})\) is a desired one. If \({{\bar{m}}}\) is g admissible weight it follows from Corollary 6.1 that \({{\tilde{a}}}-(m_1m_2)^{-1}a\in S(M^{-1}{{\bar{m}}}, g)\).
\(\square \)
Lemma 6.11
Let \(m_i\) \((i=1, 2)\) be g admissible weights such that \(m_i\in S(m_i, g)\). If \({{\bar{m}}}\) is \({{\bar{g}}}\) admissible weight and \(a\in S({{\bar{m}}}m_1m_2, {{\bar{g}}})\) or \(a\in S({{\bar{m}}}m_1, {{\bar{g}}})\) there are \(M_0\) and \({{\tilde{a}}}\in S({{\bar{m}}}, {{\bar{g}}})\) \((M>M_0)\) such that the followings hold for \(M>M_0\)
Moreover if \({{\bar{m}}}\) is g admissible weight such that \({\bar{m}}\in S({{\bar{m}}}, g)\) then with \({{\tilde{a}}}=(m_1 m_2)^{-1}a\in S({{\bar{m}}}, g)\) or \({{\tilde{a}}}=m_1^{-1}a\in S({{\bar{m}}}, g)\) the following estimates hold
for \(M>M_0\).
Proof
The first two assertions are direct consequences of Lemma 6.10. If \({{\bar{m}}}\) is g admissible weight with \({{\bar{m}}}\in S({{\bar{m}}}, g)\) one can write \(a=m_2\#({{\tilde{a}}}+r)\#m_1\) with \(r\in S(M^{-1}{{\bar{m}}}, g)\) by Lemma 6.10 from which it follows \( \big |(\textrm{op}({a})u, v)\big |\le \Vert \textrm{op}({{{\tilde{a}}}+r})\textrm{op}({m_1}) u\Vert \Vert \textrm{op}({m_2})v\Vert \). Writing \(r={{\tilde{r}}}\#{{\bar{m}}}\), \({{\tilde{r}}}\in S(M^{-1}, g)\) with use of Lemma 6.10 one has \( \Vert \textrm{op}({{{\tilde{a}}}+r})v\Vert \le \Vert \textrm{op}({{{\tilde{a}}}})v\Vert +CM^{-1}\Vert \textrm{op}({{\bar{m}}})v\Vert \) thanks to Theorem 6.2. Taking \(m_2=1\) in this proof one obtains the last assertion. \(\square \)
Corollary 6.3
If \(a\in S(\langle {\xi }\rangle _{\gamma }^sm_1m_2, {{\bar{g}}})\) and \(s_1+s_2=s\) then
Proof
Write \(a=\langle {\xi }\rangle _{\gamma }^{s_2}\#{{\tilde{a}}}\#\langle {\xi }\rangle _{\gamma }^{s_1}\) with \({{\tilde{a}}}=\langle {\xi }\rangle _{\gamma }^{-s_2}\#a\#\langle {\xi }\rangle _{\gamma }^{-s_1}\in S(m_1m_2, {{\bar{g}}})\) and apply Lemma 6.11 to \({{\tilde{a}}}\) to get \(|(\textrm{op}({a})u, v)|\le C\Vert \textrm{op}({m_1})\langle {D}\rangle _{\gamma }^{s_1}u\Vert \Vert \textrm{op}({m_2})\langle {D}\rangle _{\gamma }^{s_2}v\Vert \). The right hand-side is bounded by \(C\Vert \langle {D}\rangle _{\gamma }^{s_1}\textrm{op}({m_1})u\Vert \Vert \langle {D}\rangle _{\gamma }^{s_2}\textrm{op}({m_2})v\Vert \) with use of Lemma 6.11 again. \(\square \)
Corollary 6.4
Let m be g admissible weight with \(m\in S(m, g)\). Then there exists \(C>0\) such that \((\textrm{op}({m})u, u)\ge (1-CM^{-1})\Vert \textrm{op}({\sqrt{m}})u\Vert ^2\).
Proof
Since \(\sqrt{m}\) is g admissible weight such that \(\sqrt{m}\in S(\sqrt{m},g)\) one can write \(m=\sqrt{m}\#(1+r)\#\sqrt{m}\) with \(r\in S(M^{-1}, g)\) from Lemma 6.10 and the rest of the proof is clear. \(\square \)
Lemma 6.12
Let \(m_i\) be g admissible weights with \(m_i\in S(m_i, g)\) \((i=1, 2)\). Let w be \({{\bar{g}}}\) admissible weight with \(w\in S(w, {{\bar{g}}})\) for which there exists \({{\tilde{w}}}\in S(w^{-1}, {{\bar{g}}})\) such that \({{\tilde{w}}}\#w=w\#{{\tilde{w}}}=1\). If \({{\bar{m}}}\) is \({{\bar{g}}}\) admissible weight and \(a\in S({{\bar{m}}}m_1m_2w, {{\bar{g}}})\) there exist \(M_0\) and \({{\hat{a}}}\in S({{\bar{m}}}, {{\bar{g}}})\) \((M>M_0)\) such that the following estimates hold
for \(M>M_0\). If \(a\in S(m_1 m_2 w^2, {{\bar{g}}})\) one has
Proof
In virtue of Lemma 6.10 one can write \(a=m_2\#{{\tilde{a}}}\#m_1\) with \({{\tilde{a}}}\in S({{\bar{m}}} w, {{\bar{g}}})\). Write \({{\tilde{a}}}=({{\tilde{a}}}\#{{\tilde{w}}})\#w\) with use of \({{\tilde{w}}}\in S(w^{-1}, {{\bar{g}}})\) where \({{\tilde{a}}}\#{{\tilde{w}}}\in S({{\bar{m}}},{{\bar{g}}})\) hence the first assertion is proved. Noting \(m_2\#({{\tilde{a}}}\#{{\tilde{w}}})\in S({{\bar{m}}} m_2, {{\bar{g}}})\) this can be written as \({{\hat{a}}}\#m_2\) with \({{\hat{a}}}\in S({{\bar{m}}}, {{\bar{g}}})\) thanks to Lemma 6.10 which proves the second estimate. The last estimate can be obtained by taking \({{\bar{m}}}=w\) in the first estimate and applying the second estimate to it. \(\square \)
Corollary 6.5
Let \(a\in S(\langle {\xi }\rangle _{\gamma }^sm_1m_2w, {{\bar{g}}})\) or \(a\in S(\langle {\xi }\rangle _{\gamma }^sm_1m_2w^2, {{\bar{g}}})\) and \(s_1+s_2=s\) then one has
Proof
Writing \(a=\langle {\xi }\rangle _{\gamma }^{s_2}\#{{\tilde{a}}}\#\langle {\xi }\rangle _{\gamma }^{s_1}\) it suffices to apply Lemma 6.12 to \({{\tilde{a}}}\in S(m_1m_2w, {{\bar{g}}})\) or \({{\tilde{a}}}\in S(m_1m_2w^2, {{\bar{g}}})\) and repeat the proof of Corollary 6.3. \(\square \)
Next, consider pseudodifferential operators associated with the metric G.
Lemma 6.13
If \(a\in S(1, G)\) satisfies \(a\ge c\) with some c then there is \(C>0\) such that
Proof
Considering \(a-c\) one may assume \(c=0\). From \(0\le a\in S(1, G)\) it follows that \(|\partial _x^{\alpha }a|\le CM^2\) and \(|\partial _{\xi }^{\beta }a|\le CM^2\langle {\xi }\rangle _{\gamma }^{-2}\) for \(|\alpha |=|\beta |=2\) then we have \(|\partial _x^{\alpha }\partial _{\xi }^{\beta }a|\le CM\langle {\xi }\rangle _{\gamma }^{-1/2}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}\sqrt{a}\le CM^{-1}\langle {\xi }\rangle _{\gamma }^{(|\alpha |-|\beta |)/2}\sqrt{a}\) for \(|\alpha +\beta |=1\) by the Glaeser inequality. With \(b(x, \xi )=(a(x,\xi )+M^{-1})^{1/2}\ge M^{-1/2}\) we see
Differentiating \(b^2=a+M^{-1}\) one has
Noting \(M^{-1}\le b^2\) one can prove by the induction on \(|\alpha +\beta |\) that
We now show that b is \({{\bar{g}}}\) admissible weight. To do so it suffices to repeat the proof of Lemma 6.5, namely when \(|\eta |<\langle {\xi }\rangle _{\gamma }/2\) one has
and if \(|\eta |\ge \langle {\xi }\rangle _{\gamma }/2\) noting \({{\bar{g}}}_z(w)\ge \langle {\xi }\rangle _{\gamma }/4\ge M^4/4\) one has
Thus one can write \( a+M^{-1}=b\#b+r\) with \( r\in S(M^{-2}b^2, {{\bar{g}}})\subset S(M^{-2}, {{\bar{g}}})\) in virtue of (6.34) and Theorem 6.1. Applying Theorem 6.2 to \(\textrm{op}({r})\) to obtain
which proves the assertion. \(\square \)
Corollary 6.6
If \(a\in S(1, G)\) there is \(C>0\) such that
Proof
Note that \(\Vert \textrm{op}({a})u\Vert ^2=(\textrm{op}({{{\bar{a}}}\#a})u, u)\) and \({{\bar{a}}}\#a=|a|^2+r\) with \(r\in S(M^2\langle {\xi }\rangle _{\gamma }^{-1}, {{\bar{g}}})\) by Theorem 6.1. Since \(M^2\langle {\xi }\rangle _{\gamma }^{-1}\le M^{-2}\) it suffices to consider \((\textrm{op}({|a|^2})u, u)\). Applying Lemma 6.13 to \((\sup |a|)^2-|a|^2\ge 0\) to get
which ends the proof. \(\square \)
1.4 Proof of Proposition 5.2
For a conic set \(U\subset {{\mathbb {R}}}^d\times ({{\mathbb {R}}}^d{\setminus } 0)\) we denote \(\pi (U)=\{x \in {{\mathbb {R}}}^d; (x, \xi )\in U\}\) and \(U_{{{\bar{x}}}}=\{\xi \in {{\mathbb {R}}}^d{\setminus } 0; ({{\bar{x}}}, \xi )\in U\}\).It is clear that
We may assume that \(\kappa (x)=Ax\) for \(|x|\ge R>2\) with a nonsingular \(d\times d\) matrix A.Let \(\chi (x)\in C_0^{\infty }({{\mathbb {R}}}^d)\) be 1 for \(|x|\le R\) and 0 for \(|x|\ge R+1\).Write \(\textrm{op}({k})\kappa ^*\textrm{op}({h})=\textrm{op}({k})(\chi +(1-\chi ))\kappa ^*\textrm{op}({h})\) then \(\textrm{op}({k})(1-\chi )\kappa ^*\textrm{op}({h})=\textrm{op}({k})(1-\chi )\textrm{op}({h_A})\kappa ^*\) where \(h_A=h(Ax, {^t\!\!}A^{-1}\xi )\).Since \(k\#(1-\chi )\#h_A\in S^{-\infty }\) it suffices to consider \(\textrm{op}({k\#\chi })\kappa ^*\textrm{op}({h})\). Theorem 6.1 gives \(k_{N_1}\in S^0\) with compact support such that \(k\#\chi -k_{N_1}\in S^{-N_1}\) for any \(N_1\) hence it is enough to consider \(\textrm{op}({ k_{N_1}})\kappa ^*\textrm{op}({h})\).With \({\tilde{\chi }}(x)=\chi (\kappa ^{-1}(x))\) and writing \(\textrm{op}({k_{N_1}})\kappa ^*\textrm{op}({h})=\textrm{op}({k_{N_1}})\kappa ^*({\tilde{\chi }}+(1-{\tilde{\chi }}))\textrm{op}({h})\) it follows from the same argument that it suffices to consider \(\textrm{op}({k_{N_1}})\kappa ^*{\tilde{\chi }}\textrm{op}({h_{N_2}})\) with \(h_{N_2}\in S^0\) with compact support such that \({\tilde{\chi }}\#h-h_{N_2}\in S^{-N_2}\).Thus one can assume that U, V are compact conic sets.
Lemma 6.14
Let U be a closed conic set and \(\Gamma \subset {{\mathbb {R}}}^d{\setminus } 0\) be a closed cone with \(U_{{{\bar{x}}}}\cap \Gamma =\emptyset \).Then there exist a neighborhood \(\omega \) of \({\bar{x}}\) and a closed cone \({\tilde{\Gamma }}\Supset \Gamma \) such that for any \(\alpha (x)\in C_0^{\infty }(\omega )\) and \(h\in S^0\) with \(\textrm{supp}\,h\subset U\) and \(p, q\in {{\mathbb {R}}}\) there is C such that
Proof
One can take a compact neighborhood \(\omega \) of \({{\bar{x}}}\) and a closed cone \({\tilde{\Gamma }}\Supset \Gamma \) with \((\omega \times {\tilde{\Gamma }})\cap U=\emptyset \) such that the following holds with some \(\epsilon >0\)
Let \(\alpha (x)\in C_0^{\infty }(\omega )\). For any \(m\in {{\mathbb {N}}}\) there is \(h_m\in S^0\) with \(\textrm{supp}\,h_m\subset \pi ^{-1}(\omega )\cap U\) such that \(\alpha \#h-h_m=r_m\in S^{-m}\). Since
using \(e^{-2ix(\xi -\eta )}=\langle {\xi -\eta }\rangle ^{-2N}\langle {D_x/2}\rangle ^{2N}e^{-2ix(\xi -\eta )}\) integration by parts shows
where the right-hand side is bounded by that of (6.38) if \(\xi \in {\tilde{\Gamma }}\) because of (6.39).Replacing \(h_m\) by \(r_m\) in (6.40) and noting \(|\langle {D_x/2}\rangle ^{2N}r_m(x, \eta )|\le C(1+|x|)^{-d-1}(1+|\eta |)^{-m}\) it follows from integration by parts
Since N, m is arbitrary the right-hand side is bounded by that of (6.38) (for any \(\xi \)). Thus we conclude the assertion. For \({{\mathcal {F}}}( \textrm{op}({h})\alpha v) (\xi )\) the proof is similar. \(\square \)
Take compact conic sets W, Z such that \(U\Subset W\), \(V\Subset Z\) and \({ Z}\cap \kappa ^*{ W}=\emptyset \).Denote \({\tilde{\Gamma }_{\!y}}=\overset{\circ }{W}_{\!\!y}\), \(\Gamma _{\!x}=\overset{\circ }{Z}_{\!x}\) then by Lemma 6.14 there is a neighborhood \(\Omega _y\) of y such that for any \(\alpha \in C_0^{\infty }(\Omega _y)\) and \(p, q\in {{\mathbb {R}}}\) one has
Similarly there exist a neighborhood \(\omega _x\) of x and a closed cone \({\hat{\Gamma }_{\!x}}\Supset \Gamma _{\!x}^c\) such that for any \(\beta \in C_0^{\infty }(\omega _x)\) and \(p, q\in {{\mathbb {R}}}\) we have
Shrinking \(\omega _x\) if necessary one may assume \(\kappa (\omega _x)\Subset \Omega _y\) (\(y=\kappa (x)\)).Note that \(\pi (Z)\) can be covered by a finite number of \(\omega _{x_i}\). We denote \(\omega _{x_i}=\omega _i\) and \(\Omega _i=\Omega _{y_i}\), \({ \Gamma _{\!i}}={ \Gamma _{\!x_i}}\), \({\tilde{\Gamma }_{\!i}}\), \({\hat{\Gamma }_{\!i}}\) so on.Take \(\beta _i\in C_0^{\infty }(\omega _i)\) such that \(\sum _{i}\beta _i=1\) on \(\pi (Z)\).Since \(k\#(1-\sum \beta _i)\in S^{-\infty }\) it is enough to consider \(\sum _i\textrm{op}({k})\beta _i\).Similarly taking \(\alpha _i\in C_0^{\infty }(\Omega _i)\) which is 1 on \(\kappa (\omega _i)\) it suffices to consider \(\sum _i \textrm{op}({k})\beta _i\kappa ^* \alpha _i\).Denoting \(u=\alpha _i \textrm{op}({h}) v\) and using \(\kappa ^*u=(2\pi )^{-d}\int e^{i\langle {\kappa (x), \eta }\rangle }{{\hat{u}}}(\eta )d\eta \) one sees
where
Since \(d(\langle {\kappa (x), \eta }\rangle -\langle {x, \xi }\rangle )=\langle {dx, {^t\!}\kappa '(x)\eta -\xi }\rangle \) we have \(|{^t\!}\kappa '(x)\eta -\xi |\ge |\xi |/2\) for \(|\xi |\ge 2B|\eta |\) with some \(B>0\), while if \(|\xi |\le 2B|\eta |\) it is obvious \(|I(\xi , \eta )|\le C\le C(2B)^N(1+|\xi |)^{-N}(1+|\eta |)^N\). Thus for any \(N\in {{\mathbb {N}}}\) the following estimate holds
Next, one can assume that \(\omega _i\) is chosen such that
holds with some \(\epsilon >0\). Using (6.44) a repetition of integration by parts gives \(\big |I(\xi ,\eta )\big |\le C_N(1+|\xi |+|\eta |)^{-N}\) for \(\xi \in \Gamma _{\!i}\) and \(\eta \in {\tilde{\Gamma }_{\!i}}\). Summarizing we conclude
From (6.41) it follows that \(|{{\hat{u}}}(\eta )|\le C(1+|\eta |)^p\Vert v\Vert _q\) for any p if \(\eta \in {\tilde{\Gamma }^c_{\!i}}\) which together with (6.45) gives \(|{{\mathcal {F}}}(\beta _i \kappa ^*u)(\xi )|\le C\langle {\xi }\rangle ^{-N}\Vert v\Vert _q\) for any \(\xi \in \Gamma _{\!i}\) and \(N\in {{\mathbb {N}}}\).
Lemma 6.15
If the support of \(h\in S^0\) is contained in a compact conic set and \(u\in H^q\) satisfies \({{\hat{u}}}(\xi )=O(|\xi |^{-N})\) for any N in an open cone \(\Gamma \) then for any N and open cone \({\Gamma '}\Subset \Gamma \) there is C such that
Proof
Using (6.40) write \({{\mathcal {F}}}(\textrm{op}({h})u)(\xi )\) as
Choose \(0<c<1\) such that \(2\eta -\xi =\xi +2(\eta -\xi )\in \Gamma \) if \(|\xi -\eta |<c|\xi |\) and \(\xi \in {\Gamma '}\) then a repetition of integration by parts gives
for \(1+|\xi |+|\eta |\le 3(1+|\xi -\eta |)(1+|2\eta -\xi |)\). For \(I_2(\xi )\) noting \(|2\eta +\xi |\le (2+c^{-1})|\eta |\) if \(|\eta |\ge c|\xi |\) integration by parts proves
Since N is arbitrary the proof is completed. \(\square \)
Here apply Lemma 6.15 with \(\Gamma =\Gamma _{\!i}\), \({\Gamma '}={\hat{\Gamma }^c_{\!i}}\) to obtain
for \(\xi \in {\hat{\Gamma }^c_{\!i}}\). If \(\xi \in {\hat{\Gamma }_{\!i}}\) (6.42) shows that for any p one has
Combining these two estimates we complete the proof of the proposition. \(\square \)
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Nishitani, T. A more direct way to the Cauchy problem for effectively hyperbolic operators. J. Pseudo-Differ. Oper. Appl. 15, 20 (2024). https://doi.org/10.1007/s11868-024-00592-4
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DOI: https://doi.org/10.1007/s11868-024-00592-4
Keywords
- Effective hyperbolicity
- Coordinates changes
- Weyl-Hörmander calculus
- Geometric characterization
- Localized symbols