Strictly hyperbolic Cauchy problems on ${\varvec{{\mathbb {R}}^n}}$ with unbounded and singular coefficients

Abstract

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times {\mathbb {R}}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and decay issues of the solution to an equation with coefficients polynomially bound in x with their x-derivatives and t-derivative of order $\text {O}(t^{-\delta }),\delta \in [0,1),$ and $\text {O}(t^{-1})$ respectively. This type of singular behavior allows coefficients to be either oscillatory or logarithmically bounded at $t=0$. We use the Planck function associated with the metric to subdivide the extended phase space and define an appropriate generalized parameter dependent symbol class. We report that the solution experiences a finite loss in the Sobolev space index in relation to the initial datum defined in the Sobolev space tailored to the metric. Our analysis suggests that an infinite loss is quite expected when the order of singularity of the first time derivative of the leading coefficients exceeds $O(t^{-1})$. We confirm this by providing a counterexample. Further, using the $L^1$ integrability of the logarithmic singularity in t and the global properties of the operator with respect to x, we derive an anisotropic cone condition in our setting.

Appendix A

We give a calculus for the parameter dependent global symbol classes defined in Sect. 4. The following two propositions give their relations to the symbol classes $G^{m_1,m_2}(\omega ,g_{\varPhi ,k}^{\rho ,r}).$ Let $N_1$ and $N_2$ be positive real numbers such that $N_1 \ge N_2.$

Proposition A.1

Let $a=a(t, x, \xi )$ be a symbol with

$$\begin{aligned} a \in G^{m_1',m_2'}\{0,0;0\}_{int,N_1}(\omega ,g_{\varPhi ,k})+G^{m_1,m_2}\{l_1,l_2,l_3,l_4;\delta \}_{ext,N_2}(\omega ,g_{\varPhi ,k}). \end{aligned}$$

Then, for $\tilde{m}_1=\max \{m_1',m_1\}$, $\tilde{m}_2=\max \{m_2',m_2\}$ and for any $\varepsilon >0$,

$$\begin{aligned} t^{l_1+\delta l_2}a \in C\left( [0, T] ; G^{{\tilde{m}}_1+\varepsilon ,1} (\omega ^{{\tilde{m}}_2}\varPhi ^{\varepsilon },g_{\varPhi ,k}^{(1,\delta ),(1-\delta ,0)})\right) . \end{aligned}$$

Proof

The hypothesis of the proposition implies that

$$\begin{aligned} \begin{aligned} \left| D_{x}^{\beta } \partial _{\xi }^{\alpha } a(t, x, \xi )\right|&\le C_{\alpha , \beta }\langle \xi \rangle _k^{m_1'-|\alpha |} \omega (x)^{m_2'}\varPhi (x)^{-|\beta |}+C_{\alpha , \beta }\langle \xi \rangle _k^{m_{1}-|\alpha |} \omega (x)^{m_2}\varPhi (x)^{-|\beta |} \\&\qquad \times t^{-(l_1+\delta (l_2+|\beta |))} (\log (1+1/t))^{l_3+l_4(|\alpha |+|\beta |)}. \end{aligned} \end{aligned}$$

From the definition of the regions, one can observe that for any $\varepsilon >0$

$$\begin{aligned} \begin{aligned} (\log (1+1/t))^{l_3+l_4(|\alpha |+|\beta |)}&\le (\varPhi (x)\langle \xi \rangle _k/N_2)^{\varepsilon }, \\ t^{-\delta |\beta |}&\le (\varPhi (x)\langle \xi \rangle _k/N_2)^{\delta |\beta |}. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} t^{l_1+\delta l_2}\left| D_{x}^{\beta } \partial _{\xi }^{\alpha } a(t, x, \xi )\right| \le C_{\alpha , \beta }\langle \xi \rangle _k^{{\tilde{m}}_{1}+\varepsilon -|\alpha |+\delta |\beta |} \omega (x)^{{\tilde{m}}_2}\varPhi (x)^{\varepsilon -(1-\delta )|\beta |}. \end{aligned}$$

$\square$

Remark A.1

Consider $\varepsilon ,\varepsilon '>0$ such that $\varepsilon<\varepsilon '<1-\delta$. Let

$$\begin{aligned} \begin{aligned} a&\in G^{0,0}\{0,0;0\}_{int,N}(\omega ,g_{\varPhi ,k})+G^{0,0}\{1,0,l_3,l_4;\delta \}_{ext,N}(\omega ,g_{\varPhi ,k}), \\ b&\in G^{0,0}\{0,0;0\}_{int,N}(\omega ,g_{\varPhi ,k})+G^{0,0}\{0,1,l_3,l_4;\delta \}_{ext,N}(\omega ,g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

Then, from (A), we have

$$\begin{aligned} \begin{aligned} a&\in C\left( [0, T] ; G^{\varepsilon ',1} (\varPhi ^{\varepsilon '},g_{\varPhi ,k}^{(1,\delta ),(1-\delta ,0)})\right) , \\ b&\in C\left( [0, T] ; G^{\varepsilon ,1} (\varPhi ^{\varepsilon }, g_{\varPhi ,k}^{(1,\delta ),(1-\delta ,0)}) \right) . \end{aligned} \end{aligned}$$

Let us consider an indicator function $\mathbf{{I}}_r: [0,\infty ) \rightarrow \{0,1\}$ defined as

$$\begin{aligned} \mathbf{{I}}_r = {\left\{ \begin{array}{ll} 0, &{} \text { if } r = 0\\ 1, &{} \text { otherwise } \end{array}\right. } \end{aligned}$$

and denote $1-\mathbf{{I}}_r$ as $\mathbf{{I}}_r^c.$

Proposition A.2

Let $a=a(t, x, \xi )$ be a symbol with

$$\begin{aligned} a \in G^{m_1',m_2'}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta \}_{int,N_1} (\omega ,g_{\varPhi ,k})+ G^{m_1,m_2}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k}), \end{aligned}$$

and let

$$\begin{aligned} {\tilde{l}} = {\left\{ \begin{array}{ll} l_2\delta &{} \text { if }{{\tilde{l}}_2>0}\\ \varepsilon &{} \text { if } {\tilde{l}}_2 \le 0 \text { and } {\tilde{l}}_1>0\\ 0 &{} \text { if } {\tilde{l}}_1 \le 0 \text { and } {\tilde{l}}_2 \le 0. \end{array}\right. } \end{aligned}$$

Then we have $t^{{\tilde{l}}}a \in C\left( [0,T] ; G^{{\tilde{m}}_1,{\tilde{m}}_2} (\omega ,g_{\varPhi ,k})\right)$ for $\tilde{m}_1=\max \{m_1',m_1\}$, $\tilde{m}_2=\max \{m_2',m_2\}$ and for any $\varepsilon >0$.

Proof

The proposition follows by observing the following estimate

$$\begin{aligned} \begin{aligned} \left| D_{x}^{\beta } \partial _{\xi }^{\alpha } a(t, x, \xi )\right|&\le C_{\alpha , \beta }\langle \xi \rangle _k^{m_{1}'-|\alpha |} \omega (x)^{m_2'}\varPhi (x)^{-|\beta |}(\log (1+1/t))^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c}t^{-{\tilde{l}}_2 \delta \mathbf{{I}}_{|\beta |}} \\&\qquad + C_{\alpha , \beta } \langle \xi \rangle _k^{m_{1}-|\alpha |} \omega (x)^{m_2} \varPhi (x)^{-|\beta |}. \end{aligned} \end{aligned}$$

$\square$

Remark A.2

Suppose $a \in G^{1,1}\{1,1;\delta \}_{int,N_1} + G^{0,0}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k})$. Then for any $\varepsilon >0$ satisfying $\varepsilon <1-\delta$, we have $t^{1-\varepsilon }a \in C\left( [0,T];G^{1,1}(\omega ,g_{\varPhi ,k})\right) .$

Remark A.3

Suppose $a \in G^{1,1}\{0,0;0\}_{int,6}(\omega ,{g}_{\varPhi ,k}) + G^{0,0}\{1,0,1,1;\delta \}_{ext,1}(\omega ,{g}_{\varPhi ,k}) +G^{0,0}\{0,1,2,3;\delta \}_{ext,3}(\omega ,{g}_{\varPhi ,k})$ ( as in (5.8)). Let $0<\varepsilon<\varepsilon '<1-\delta .$ In $Z_{int}(6),$

$$\begin{aligned} \omega (x)\langle \xi \rangle _k\le \varPhi (x)\langle \xi \rangle _k\le \big ( \varPhi (x)\langle \xi \rangle _k\big )^{\varepsilon '} \left( \frac{6}{t} \right) ^{1-\varepsilon '}. \end{aligned}$$

Note that $t^{1-\varepsilon } \left( \frac{1}{t} \left( \log \left( 1+1/t \right) \right) ^{1+|\alpha |+|\beta |} \right) \le \frac{1}{t^{\varepsilon '}} \le \big (\varPhi (x)\langle \xi \rangle _k\big )^{\varepsilon '}$ in $Z_{ext}(1)$ while in $Z_{ext}(3),$ $t^{1-\varepsilon } \left( \frac{1}{t^{\delta }} \left( \log \left( 1+1/t \right) \right) ^{2+3(|\alpha |+|\beta |)} \right) \le \frac{1}{t^{\varepsilon '}} \le \big (\varPhi (x)\langle \xi \rangle _k/3 \big )^{\varepsilon '}.$ Hence, by Proposition A.1, $t^{1-\varepsilon }a \in C\left( [0,T];G^{\varepsilon ',\varepsilon '}(\omega ,{g}_{\varPhi ,k}^{(1,\delta ),(1-\delta ,0)}) \right) .$

For $\mu >0$ and $r>1$, we set

$$\begin{aligned} Q_{r,\mu } = \{(x,\xi ) \in {\mathbb {R}}^{2n} : \varPhi (x)^{\mu }<r, \langle \xi \rangle ^{\mu } < r\}, \qquad Q_{r,\mu }^c = {\mathbb {R}}^{2n} \setminus Q_{r,\mu }. \end{aligned}$$

Proposition A.3

(Asymptotic expansion) Let $\{a_{j}\},j \ge 0$ be a sequence of symbols with

$$\begin{aligned} \begin{aligned} a_{j}&\in G^{\tilde{m}_1-j,1}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ^{\tilde{m}_2}\varPhi ^{-j},g_{\varPhi ,k}) \\&\qquad + G^{{m}_1-j,1}\{{l}_1+\delta _2j,{l}_2,l_3+2l_4j,l_4;\delta _2\}_{ext,N_2} (\omega ^{{m}_2}\varPhi ^{-j},g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

Then, there is a symbol

$$\begin{aligned} a \in G^{\tilde{m}_1,\tilde{m}_2}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})+ G^{{m}_1,{m}_2}\{{l}_1,{l}_2,l_3,l_4;\delta _2\}_{ext,N_2} (\omega ,g_{\varPhi ,k})\end{aligned}$$

such that

$$\begin{aligned} a(t, x, \xi ) \sim \sum _{j=0}^{\infty } a_{j}(t, x, \xi ) \end{aligned}$$

that is for all $j_{0} \ge 1$

$$\begin{aligned} \begin{aligned} a(t, x, \xi )-&\sum _{j=0}^{j_{0}-1} a_{j}(t, x, \xi ) \\&\in G^{\tilde{m}_1-j_0,1}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ^{\tilde{m}_2}\varPhi ^{-j_0},g_{\varPhi ,k}) \\&\;\,+ G^{{m}_1-(1-\delta _2)j_0+\varepsilon ,1}\{{l}_1,{l}_2,l_3,l_4;\delta _2\}_{ext,N_2} (\omega ^{{m}_2}\varPhi ^{-(1-\delta _2)j_0+\varepsilon },g_{\varPhi ,k}), \end{aligned} \end{aligned}$$

where $\varepsilon \ll 1-\delta _2$. The symbol is uniquely determined modulo $C\left( (0, T] ; \mathcal {S}({\mathbb {R}}^{2n})\right)$.

Proof

Let us fix $\varepsilon \ll 1-\delta _2$ and set $\mu =1-\delta _2-\varepsilon$. Consider a $C^{\infty }$ cut-off function, $\chi$ defined by

$$\begin{aligned} \chi (x,\xi )=\left\{ \begin{array}{ll} 1, &{} (x,\xi ) \in Q_{2,\mu } \\ 0, &{} (x,\xi ) \in Q_{4,\mu }^c \end{array}\right. \end{aligned}$$

and $0 \le \chi \le 1 .$ For a sequence of positive numbers $\varepsilon _{j} \rightarrow 0$, we define

$$\begin{aligned} \begin{aligned} \gamma _0(x,\xi )&\equiv 1, \\ \gamma _{{j}}(x,\xi )&=1-\chi \left( \varepsilon _{j} x,\varepsilon _{j} \xi \right) , \quad j \ge 1. \end{aligned} \end{aligned}$$

We note that $\gamma _{{j}}(x,\xi )=0$ in $Q_{1,\mu }$ for $j \ge 1.$ We choose $\varepsilon _{j}$ such that

$$\begin{aligned} \varepsilon _{j} \le 2^{-j} \end{aligned}$$

and set

$$\begin{aligned} a(t, x, \xi )=\sum _{j=0}^{\infty } \gamma _{{j}}(x,\xi ) a_{j}(t, x, \xi ). \end{aligned}$$

We note that $a(t, x, \xi )$ exists (i.e. the series converges point-wise), since for any fixed point $(t, x, \xi )$ only a finite number of summands contribute to $a(t, x, \xi ) .$ Indeed, for fixed $(t, x, \xi )$ we can always find a $j_{0}$ such that $\varPhi (x)^\mu < \frac{1}{\varepsilon _{j_0}}$, $\langle \xi \rangle ^\mu <\frac{1}{\varepsilon _{j_0}}$ and hence

$$\begin{aligned} a(t, x, \xi )=\sum _{j=0}^{j_{0}-1} \gamma _{j}(x,\xi ) a_{j}(t, x, \xi ). \end{aligned}$$

Observe that

$$\begin{aligned} \begin{aligned} |D_{x}^{\beta } \partial _{\xi }^{\alpha }&\left( \gamma _{{j}}(x,\xi ) a_{j}(t, x, \xi )\right) | \\&\le \sum \limits _{\begin{array}{c} \alpha ^{\prime }+\alpha ^{\prime \prime }=\alpha \\ \beta ^{\prime }+\beta ^{\prime \prime }=\beta \end{array}}\left( \begin{array}{c} \alpha \\ \alpha ^{\prime } \end{array}\right) \left( \begin{array}{c} \beta \\ \beta ^{\prime } \end{array}\right) |\partial _{\xi }^{\alpha ^{\prime }} D_x^{\beta ^\prime } \gamma _{j}(x,\xi ) D_{x}^{\beta ^{\prime \prime }} \partial _{\xi }^{\alpha ^{\prime \prime }} a_{j}(t, x, \xi )| \\&\le \mid \gamma _{{j}}(x,\xi ) D_{x}^{\beta } \partial _{\xi }^{\alpha } a_{j}(t, x, \xi ) \\&\qquad +\sum \limits _{\begin{array}{c} \alpha ^{\prime }+\alpha ^{\prime \prime }=\alpha ,|\alpha ^{\prime }|>0 \\ \beta ^{\prime }+\beta ^{\prime \prime }=\beta ,|\beta ^{\prime }|>0 \end{array} } C_{\alpha ^{\prime } \beta ^{\prime }} \frac{\tilde{\chi }_{{j}}(x,\xi )}{\varPhi (x)^{\mu |\beta ^{\prime }|} \langle \xi \rangle _k^{\mu |\alpha ^{\prime }|}}D_{x}^{\beta ^{\prime \prime }} \partial _{\xi }^{\alpha ^{\prime \prime }} a_{j}(t, x, \xi ) \mid , \end{aligned} \end{aligned}$$

where $\tilde{\chi }_{{j}}(x,\xi )$ is a smooth cut-off function supported in $Q_{1,\mu }^c \cap Q_{2,\mu }.$ This new cut-off function describes the support of the derivatives of $\gamma _{j}(x,\xi ) .$ In the last estimate, we also used that $\frac{1}{\varepsilon _{j}}\sim \langle \xi \rangle ^\mu$ and $\frac{1}{\varepsilon _{j}} \sim \varPhi (x)^\mu$ if $\tilde{\chi }_{j}(x,\xi ) \ne 0 .$ Noting the definition of the functions $\chi _{int}(N_1)$ and $\chi _{ext}(N_2)$ in Sect. 5.1, we conclude that

$$\begin{aligned} \begin{aligned}&|D_{x}^{\beta } \partial _{\xi }^{\alpha } \gamma _{j}(x,\xi ) a_{j}(t, x, \xi )| \\&\le \frac{1}{2^j} \Big [ \langle \xi \rangle _k^{{\tilde{m}}_1+\mu -j-|\alpha |} \omega (x)^{{\tilde{m}}_2} \varPhi (x)^{\mu -j-|\beta |} \left( \log (1+1/t)\right) ^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c} (1/t)^{\delta _1 {\tilde{l}}_2\mathbf{{I}}_{|\beta |}} \chi _{int}(N_1) \\&\quad + \langle \xi \rangle _k^{m_1+\mu -j-|\alpha |} \omega (x)^{m_2} \varPhi (x)^{\mu -j-|\beta |} (1/t)^{l_1+\delta _2(l_2+|\beta |+j)} \\&\qquad \quad \times (\log (1+1/t))^{l_3+l_4(|\alpha | + |\beta |+2j)} \chi _{ext}(N_2)) \Big ] \\&\le \frac{1}{2^j} \Big [ \langle \xi \rangle _k^{{\tilde{m}}_1+\mu -j-|\alpha |} \omega (x)^{{\tilde{m}}_2} \varPhi (x)^{\mu -j-|\beta |} \left( \log (1+1/t)\right) ^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c} (1/t)^{\delta _1 {\tilde{l}}_2\mathbf{{I}}_{|\beta |}} \chi _{int}(N_1) \\&\quad + \langle \xi \rangle _k^{m_1+\varepsilon \mathbf{{I}}_j+\mu -(1-\delta _2)j-|\alpha |} \omega (x)^{m_2} \varPhi (x)^{\mu -(1-\delta _2)j-|\beta |+\varepsilon \mathbf{{I}}_j} (1/t)^{l_1+\delta _2(l_2+|\beta |)} \\&\qquad \quad \times (\log (1+1/t))^{l_3+l_4(|\alpha | + |\beta |)}\chi _{ext}(N_2)\Big ], \end{aligned} \end{aligned}$$

where we have estimated $\frac{\varPhi (x)^\mu }{ 2^{j}} \ge 1$and $\frac{\langle \xi \rangle _k^\mu }{ 2^{j}} \ge 1$ (due to the support of cut-off functions) once in each summand and noted that in $Z_{ext}(N_2)$ for every $\varepsilon \ll 1$,

$$\begin{aligned} \begin{aligned} (\log (1+1/t))^{2l_4j}&\le (\varPhi (x)\langle \xi \rangle _k/N_2)^{\varepsilon \mathbf{{I}}_j}, \\ (1/t)^{\delta _2j}&\le (\varPhi (x)\langle \xi \rangle _k/N_2)^{\delta _2j}. \end{aligned} \end{aligned}$$

Using this relation, we obtain

$$\begin{aligned} |D_{x}^{\beta }&\partial _{\xi }^{\alpha } a(t,x, \xi ) | \\ \le&\left| D_{x}^{\beta } \partial _{\xi }^{\alpha }\left( \gamma _{0}(x,\xi ) a_{0}(t, x, \xi )\right) \right| + \sum _{j=1}^{j_{0}-1}\left| D_{x}^{\beta } \partial _{\xi }^{\alpha }\left( \gamma _{j}(x,\xi ) a_{j}(t, x, \xi )\right) \right| \\ \le&C_{\alpha \beta } \Big [ \langle \xi \rangle _k^{{\tilde{m}}_1-|\alpha |} \omega (x)^{{\tilde{m}}_2} \varPhi (x)^{-|\beta |} \left( \log (1+1/t)\right) ^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c} (1/t)^{\delta _1 {\tilde{l}}_2\mathbf{{I}}_{|\beta |}} \chi _{int}(N_1) \\&\quad + \langle \xi \rangle _k^{m_1-|\alpha |} \omega (x)^{m_2} \varPhi (x)^{-|\beta |} (1/t)^{l_1+\delta _2(l_2+|\beta |)} (\log (1+1/t))^{l_3+l_4(|\alpha | + |\beta |)} \\&\qquad \quad \times \chi _{ext}(N_2) \\&\quad +\sum _{j=1}^{j_{0}-1} \frac{1}{2^j} \Big [ \langle \xi \rangle _k^{{\tilde{m}}_1+\mu -j-|\alpha |} \omega (x)^{{\tilde{m}}_2} \varPhi (x)^{\mu -j-|\beta |} \left( \log (1+1/t)\right) ^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c} (1/t)^{\delta _1 {\tilde{l}}_2\mathbf{{I}}_{|\beta |}} \\&\qquad \quad \times \chi _{int}(N_1) + \langle \xi \rangle _k^{m_1+\mu -(1-\delta _2)j+\varepsilon -|\alpha |} \omega (x)^{m_2} \varPhi (x)^{\mu -(1-\delta _2)j+\varepsilon -|\beta |} \\&\qquad \quad \times (1/t)^{l_1+\delta _2(l_2+|\beta |)}(\log (1+1/t))^{l_3+l_4(|\alpha | + |\beta |)}\chi _{ext}(N_2)\Big ]\\ \le&C_{\alpha \beta } \Big [ \langle \xi \rangle _k^{{\tilde{m}}_1-|\alpha |} \omega (x)^{{\tilde{m}}_2} \varPhi (x)^{-|\beta |} \left( \log (1+1/t)\right) ^{{\tilde{l}}_1\mathbf{{I}}_{|\beta |}^c} (1/t)^{\delta _1 {\tilde{l}}_2\mathbf{{I}}_{|\beta |}} \chi _{int}(N_1) \\&\quad + \langle \xi \rangle _k^{m_1-|\alpha |} \omega (x)^{m_2} \varPhi (x)^{-|\beta |} (1/t)^{l_1+\delta _2(l_2+|\beta |)} (\log (1+1/t))^{l_3+l_4(|\alpha | + |\beta |)} \\&\qquad \quad \times \chi _{ext}(N_2) \Big ], \end{aligned}$$

where the last inequality holds by the choice $\mu .$ Thus, we have

$$\begin{aligned} a \in G^{\tilde{m}_1,\tilde{m}_2}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})+ G^{{m}_1,{m}_2}\{{l}_1,{l}_2,l_3,l_4;\delta _2\}_{ext,N_2} (\omega ,g_{\varPhi ,k}). \end{aligned}$$

Arguing as above, we have

$$\begin{aligned} \begin{aligned} \sum _{j=j_{0}}^{\infty } \gamma _{j} a_{j}&\in G^{\tilde{m}_1-j_0,1}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ^{\tilde{m}_2}\varPhi ^{-j_0},g_{\varPhi ,k}) \\&\qquad + G^{{m}_1-(1-\delta _2)j_0+\varepsilon ,1}\{{l}_1,{l}_2,l_3,l_4;\delta _2\}_{ext,N_2} (\omega ^{{m}_2}\varPhi ^{-(1-\delta _2)j_0+\varepsilon },g_{\varPhi ,k}), \end{aligned} \end{aligned}$$

and thus,

$$\begin{aligned} \begin{aligned} a(t, x, \xi )&-\sum _{j=0}^{j_{0}-1} a_{j}(t, x, \xi ) \\ {}&\in G^{\tilde{m}_1-j_0,1}\{\tilde{l}_1,\tilde{l}_2;\delta _1\}_{int,N_1} (\omega ^{\tilde{m}_2}\varPhi ^{-j_0},g_{\varPhi ,k}) \\&\quad + G^{{m}_1-(1-\delta _2)j_0+\varepsilon ,1}\{{l}_1,{l}_2,l_3,l_4;\delta _2\}_{ext,N_2} (\omega ^{{m}_2}\varPhi ^{-(1-\delta _2)j_0+\varepsilon },g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

Lastly, we use Proposition A.1 and A.2 to conclude that

$$\begin{aligned} t^{l}a_{j} \in C\left( [0, T] ; G^{ m_1^*-(1-\delta _2)j+\varepsilon \mathbf{{I}}_j,1}(\omega ^{ m_2^*}\varPhi ^{-(1-\delta _2)j+\varepsilon \mathbf{{I}}_j},g_{\varPhi ,k}^{(1,\delta ),(1-\delta ,0)}) \right) \end{aligned}$$

for $m_i^*=\max \{m_i,{\tilde{m}}_i\},i=1,2$, $\delta = \max \{\delta _1,\delta _2\}$ and $l=\max \{l_1+\delta l_2, {\tilde{l}}\}$. As j tends to $+\infty ,$ the intersection of all those spaces belongs to the space $C\left( (0, T] ; \mathcal {S}({\mathbb {R}}^{2n})\right) .$ This completes the proof. $\square$

Lemma A.4

Let $N_j$ and $N_j',j=1,2$ be positive real numbers such that $N_1 \ge N_2$ and $N_1' \ge N_2'.$ Suppose

$$\begin{aligned} \begin{aligned} a&\in G^{{\tilde{m}}_1,{\tilde{m}}_2}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})+ G^{m_1,m_2}\{l_1,l_2,l_3,l_4;\delta _2\}_{ext,N_2}(\omega ,g_{\varPhi ,k}),\\ b&\in G^{{\tilde{m}}_1^{\prime },{\tilde{m}}_2^{\prime }}\{{\tilde{l}}_1^{\prime },{\tilde{l}}_2^{\prime };\delta _1\}_{int,N_1'} (\omega ,g_{\varPhi ,k})+ G^{m_1^{\prime },m_2^{\prime }}\{l_1^{\prime },l_2^{\prime },l_3^{\prime },l_4^{\prime };\delta _2\}_{ext,N_2'}(\omega ,g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

Then, for ${\tilde{N}}_1=\max \{N_1,N_1'\}$ and ${\tilde{N}}_2=\min \{N_2,N_2'\},$

$$\begin{aligned} \begin{aligned} a b \in&\; G^{{\tilde{m}}_1 +{\tilde{m}}_1^{\prime },{\tilde{m}}_2+{\tilde{m}}_2^{\prime }}\{{\tilde{l}}_1+{\tilde{l}}_1^{\prime },{\tilde{l}}_2+{\tilde{l}}_2^{\prime };\delta _1\}_{int,{\tilde{N}}_1} (\omega ,g_{\varPhi ,k})\\&+G^{m_1+m_1^{\prime },m_2+m_2^{\prime }}\{l_1+l_1^{\prime },l_2+l_2^{\prime },l_3+l_3^{\prime },l_4+l_4^{\prime };\delta _2\}_{ext,{\tilde{N}}_2}(\omega ,g_{\varPhi ,k})\\&+G^{{\tilde{m}}_1,{\tilde{m}}_2}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})\cap G^{m_1^{\prime },m_2^{\prime }}\{l_1^{\prime },l_2^{\prime },l_3^{\prime },l_4^{\prime };\delta _2\}_{ext,N_2'}(\omega ,g_{\varPhi ,k})\\&+ G^{{\tilde{m}}_1^{\prime },{\tilde{m}}_2^{\prime }}\{{\tilde{l}}_1^{\prime },{\tilde{l}}_2^{\prime };\delta _1\}_{int,N_1'} (\omega ,g_{\varPhi ,k})\cap G^{m_1,m_2}\{l_1,l_2,l_3,l_4;\delta _2\}_{ext,N_2}(\omega ,g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

Notice that the symbols corresponding to the third summand of the above expression is non-zero only if $N_2'< t \varPhi (x)\langle \xi \rangle _k<N_1$ i.e., in $Z_{int}(N_1) \cap Z_{ext}(N_2')$. This requires $N_2'< N_1.$ Similarly, the fourth summand is nonvanishing in $Z_{int}(N_1') \cap Z_{ext}(N_2)$ which requires $N_2< N_1'.$ A straightforward computation proves the above lemma.

Lemma A.5

Let A and B be pseudodifferential operators with the respective symbols $a=\sigma (A)$ and $b=\sigma (B)$ as in Lemma A.4. Then, the pseudodifferential operator $C=A \circ B$ has a symbol $c=\sigma (C)$ such that

$$\begin{aligned} \begin{aligned} c \in&\; G^{{\tilde{m}}_1 +{\tilde{m}}_1^{\prime },{\tilde{m}}_2+{\tilde{m}}_2^{\prime }}\{{\tilde{l}}_1+{\tilde{l}}_1^{\prime },{\tilde{l}}_2+{\tilde{l}}_2^{\prime };\delta _1\}_{int,{\tilde{N}}_1} (\omega ,g_{\varPhi ,k})\\&+G^{m_1+m_1^{\prime },m_2+m_2^{\prime }}\{l_1+l_1^{\prime },l_2+l_2^{\prime },l_3+l_3^{\prime },l_4+l_4^{\prime };\delta _2\}_{ext,{\tilde{N}}_2}(\omega ,g_{\varPhi ,k})\\&+G^{{\tilde{m}}_1,{\tilde{m}}_2}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})\cap G^{m_1^{\prime },m_2^{\prime }}\{l_1^{\prime },l_2^{\prime },l_3^{\prime },l_4^{\prime };\delta _2\}_{ext,N_2'}(\omega ,g_{\varPhi ,k})\\&+ G^{{\tilde{m}}_1^{\prime },{\tilde{m}}_2^{\prime }}\{{\tilde{l}}_1^{\prime },{\tilde{l}}_2^{\prime };\delta _1\}_{int,N_1'} (\omega ,g_{\varPhi ,k})\cap G^{m_1,m_2}\{l_1,l_2,l_3,l_4;\delta _2\}_{ext,N_2}(\omega ,g_{\varPhi ,k})\end{aligned} \end{aligned}$$

for ${\tilde{N}}_1=\max \{N_1,N_1'\}$ and ${\tilde{N}}_2=\min \{N_2,N_2'\}$ and satisfies

$$\begin{aligned} c(t, x, \xi ) \sim \sum _{\alpha \in \mathbb {N}^{n}} \frac{1}{\alpha !} \partial _{\xi }^{\alpha } a(t, x, \xi ) D_{x}^{\alpha } b(t, x, \xi ). \end{aligned}$$

(A.1)

The operator C is uniquely determined modulo an operator with symbol from $C\left( (0, T] ; \mathcal {S}({\mathbb {R}}^{2n})\right) .$

Proof

In view of Proposition A.1, Proposition A.2, Proposition A.3 and Lemma A.4 it is clear that the operator C is a well-defined pseudodifferential operator. Relation (A.1) is a direct consequence of the standard composition rules (see [19, Section 1.2]). $\square$

Lemma A.6

Let A be a pseudodifferential operator with an invertible symbol

$$\begin{aligned} a=\sigma (A) \in G^{0,0}\{0,0;0\}_{int,N_1}(\omega ,g_{\varPhi ,k})+ G^{0,0}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k}). \end{aligned}$$

Then, there exists a parametrix $A^{\#}$ with symbol

$$\begin{aligned} a^{\#}=\sigma \left( A^{\#}\right) \in G^{0,0}\{0,0;0\}_{int,N_1}(\omega ,g_{\varPhi ,k})+ G^{0,0}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k}). \end{aligned}$$

Proof

We use the existence of the inverse of a and set

$$\begin{aligned} \begin{aligned} a_{0}^{\#}(t, x, \xi )=&a(t, x, \xi )^{-1} \\&\in G^{0,0}\{0,0;0\}_{int,N_1}(\omega ,g_{\varPhi ,k})+ G^{0,0}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k}). \end{aligned} \end{aligned}$$

In view of Propositions A.1 and A.2, one can define a sequence $a_{j}^{\#}(t, x, \xi )$ recursively by

$$\begin{aligned} \sum _{1 \le |\alpha | \le j} \frac{1}{\alpha !} \partial _{\xi }^{\alpha } a(t, x, \xi ) D_{x}^{\alpha } a_{j-|\alpha |}^{\#}(t, x, \xi )=-a(t, x, \xi ) a_{j}^{\#}(t, x, \xi ) \end{aligned}$$

with

$$\begin{aligned} a_{j}^{\#} \in G^{-j,1}\{0,0;0\}_{int,N_1}(\varPhi ^{-j},g_{\varPhi ,k}) + G^{-j,1}\{0,0,0,0;0\}_{ext,N_2} (\varPhi ^{-j},g_{\varPhi ,k}) . \end{aligned}$$

Proposition A.3 then yields the existence of a symbol

$$\begin{aligned} a_{R}^{\#} \in G^{0,0}\{0,0;0\}_{int,N_1}(\omega ,g_{\varPhi ,k})+ G^{0,0}\{0,0,0,0;0\}_{ext,N_2}(\omega ,g_{\varPhi ,k})\end{aligned}$$

and a right parametrix $A_{R}^{\#}(t, x, \xi )$ with symbol $\sigma \left( A_{R}^{\#}\right) =a_{R}^{\#} .$ We have

$$\begin{aligned} A A_{R}^{\#}-I \in C\left( [0, T] ; G^{-\infty ,-\infty }(\omega ,g_{\varPhi ,k})\right) . \end{aligned}$$

The existence of a left parametrix follows in similar lines. One can also prove the existence of a parametrix $A^{\#}$ by showing that right and left parametrix coincide up to a regularizing operator. $\square$

Now, we perform a conjugation of an operator A where $a=\sigma (A)$ is such that

$$\begin{aligned} a \in G^{{\tilde{m}}_1,{\tilde{m}}_2}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta _1\}_{int,N_1} (\omega ,g_{\varPhi ,k})+ G^{m_1,m_2}\{l_1,l_2,l_3,l_4;\delta _2\}_{ext,N_2}(\omega ,g_{\varPhi ,k})\nonumber \\ \end{aligned}$$

(A.2)

by the operator $\exp \{E(t,x,D_x) \}$ where the operator E is such that

$$\begin{aligned} E(t,x,\xi ) = \int _{0}^{t} \tilde{\psi }(r,x,\xi ) dr \end{aligned}$$

(A.3)

for ${\tilde{\psi }}$ as defined in (5.10) and (5.11). For the sake of generic presentation, one can replace the factors $\varphi (t\varPhi (x)\langle \xi \rangle _k/3)$ and $1-\varphi (t\varPhi (x)\langle \xi \rangle _k)$ in (5.10) by $\varphi (2t\varPhi (x)\langle \xi \rangle _k/N_1')$ and $1-\varphi (t\varPhi (x)\langle \xi \rangle _k/N_2')$, respectively, for $N_1' \ge N_2'.$ The conjugation operation is given by

$$\begin{aligned} A_E (t,x,D_x) = e^{-E(t,x,D_x) } A(t,x,D_x) e^{E(t,x,D_x) }. \end{aligned}$$

Notice that the operator $\exp \{\pm E(t,x,D_x) \}$ is a finite order pseudodifferential operator; in fact

$$\begin{aligned} e^{\pm E(t,x,\xi ) } \le (1+\varPhi (x)\langle \xi \rangle _k)^{\kappa _{00}}, \end{aligned}$$

(A.4)

where $\kappa _{00}>0$ is as in (5.11). When $|\alpha |+|\beta |>0$, we have

$$\begin{aligned} \partial _x^\beta \partial ^\alpha _\xi e^{\pm E(t,x,\xi ) } \le C^{\prime }_{\alpha \beta } e^{\pm E(t,x,\xi ) } \varPhi (x)^{-|\beta |} \langle \xi \rangle _k^{-|\alpha |} (\log (1+\varPhi (x)\langle \xi \rangle _k))^{|\alpha |+|\beta |} \chi _{int}(N_1')\nonumber \!\!\!\!\!\\ \end{aligned}$$

(A.5)

By successive composition of the operators while performing conjugation and using Proposition A.3 and Lemma A.4, one can prove the following lemma.

Lemma A.7

Let the operators A and E be as in (A.2) and (A.3). Then

$$\begin{aligned} A_E (t,x,D_x) = A(t,x,D)+ R(t,x,D_x), \end{aligned}$$

(A.6)

where

$$\begin{aligned} \begin{aligned} (\log (1+\varPhi (x)\langle \xi \rangle _k))^{-1}&\sigma (R)(t,x,\xi ) \\ {}&\in G^{{\tilde{m}}_1-1+{\tilde{\varepsilon }},1}\{{\tilde{l}}_1,{\tilde{l}}_2;\delta _1\}_{int,{\tilde{N}}_1} (\omega ^{{\tilde{m}}_2}\varPhi ^{-1+{\tilde{\varepsilon }}}, g_{\varPhi ,k}) \\&\qquad + G^{m_1-1,1}\{l_1,l_2,l_3,l_4;\delta _2\}_{ext,{\tilde{N}}_2} (\omega ^{ m_2}\varPhi ^{-1}, g_{\varPhi ,k}), \end{aligned} \end{aligned}$$

for every ${\tilde{\varepsilon }} \ll 1$ and ${\tilde{N}}_1=\max \{N_1,N_1'\}$ and ${\tilde{N}}_2=\min \{N_2,N_2'\}.$

Remark A.4

In Lemma A.7, one can ensure a compensation for the ${\tilde{\varepsilon }}$ increase in the order of remainder symbol in the interior region by an appropriate choice of the order of singularity in the interior region. For example, the conjugation of the operator $\mathcal {D}$ in (5.7) yields

$$\begin{aligned} \mathcal {D}_{E}(t,x,D_x) = \mathcal {D}(t,x,D_x) + R(t,x,D_x), \end{aligned}$$

where the operator $R(t,x,D_x)$ is such that its symbol satisfies

$$\begin{aligned} \begin{aligned} (\log (1+\varPhi (x)\langle \xi \rangle _k))^{-1} \sigma (R)(t,x,\xi )&\in G^{-1+{\tilde{\varepsilon }},1}\{0,0;0\}_{int,2} (\omega \varPhi ^{-1+{\tilde{\varepsilon }}}, g_{\varPhi ,k}) \\&\qquad + G^{0,1}\{0,0,1,1;\delta _1\}_{ext,1} (\omega \varPhi ^{-1}, g_{\varPhi ,k}), \end{aligned} \end{aligned}$$

for an arbitrary small ${\tilde{\varepsilon }}>0.$ By the definition of region $Z_{int}(2),$ we have the estimate

$$\begin{aligned} {(\varPhi (x)\langle \xi \rangle _k)^{{\tilde{\varepsilon }}}} \le \frac{2}{t^{{\tilde{\varepsilon }}}}. \end{aligned}$$

Hence, we have

$$\begin{aligned} \begin{aligned} t^{{\tilde{\varepsilon }}}(\log (1+\varPhi (x)\langle \xi \rangle _k))^{-1} \sigma (R)(t,x,\xi )&\in G^{0,1}\{0,0;0\}_{int,2} (\omega \varPhi ^{-1}, g_{\varPhi ,k}) \\&\qquad + G^{0,1}\{0,0,1,1;\delta _1\}_{ext,1} (\omega \varPhi ^{-1}, g_{\varPhi ,k}). \end{aligned} \end{aligned}$$