Pseudodifferential operators and weighted normed symbol spaces

TL;DR

This paper investigates classes of pseudodifferential operators with symbols characterized by phase space estimates, aiming to understand their properties within weighted normed symbol spaces.

Contribution

It introduces a framework for analyzing pseudodifferential operators using phase space estimates in weighted normed symbol spaces, expanding existing theoretical understanding.

Findings

Characterization of pseudodifferential operators in weighted normed spaces

Development of phase space estimate techniques for symbols

Enhanced understanding of operator boundedness and regularity

Abstract

In this work we study some general classes of pseudodifferential operators whose symbols are defined in terms of phase space estimates.

Equations

$$|\widehat{{\chi }_{\gamma }a}(x^{\ast })|\le F(x^{\ast }),\forall \gamma \in \Gamma$$

$$a^{w}u(x)=\frac{1}{(2\pi {)}^n}\int \int e^{i(x-y)\cdot \theta }a(\frac{x+y}{2},\theta )u(y)dyd\theta .$$

$$m(x^{\ast })\le C_0⟨x^{\ast }-y^{\ast }{⟩}^{N_0}m(y^{\ast }),x^{\ast },y^{\ast }\in E^{\ast }.$$

$$m(\rho )\le C_0⟨\rho -\mu {⟩}^{N_0}m(\mu ),\forall \rho ,\mu \in E.$$

$$\sum _{\gamma \in \Gamma }{\tau }_{\gamma }\chi =1,{\tau }_{\gamma }\chi (\rho )=\chi (\rho -\gamma ).$$

$$\Vert {\chi }_{\gamma }^{w}a\Vert \le Cm(\gamma ),\gamma \in \Gamma ,$$

$$\Vert {\tilde{\chi }}_{\tilde{\gamma }}^{w}a{\Vert }_{L^p}\le \tilde{C}m(\tilde{\gamma }),\tilde{\gamma }\in \tilde{\Gamma }.$$

$$\sum _{\gamma \in \Gamma }({\tilde{\chi }}_{\gamma }^{ϵ}{)}^w{\chi }_{\gamma }^w=1-\sum _{\gamma \in \Gamma }(1-{\tilde{\chi }}_{\gamma }^{ϵ}{)}^w{\chi }_{\gamma }^w$$

$${\tilde{\chi }}_{\tilde{\gamma }}^{w}a=\sum _{\gamma \in \Gamma }{\tilde{\chi }}_{\tilde{\gamma }}^w{\psi }_{\gamma }^w{\chi }_{\gamma }^{w}a.$$

$$\Vert {\tilde{\chi }}_{\tilde{\gamma }}{\psi }_{\gamma }^w{\Vert }_{\mathcal{L}(L^2,L^p)}\le C_{p,N}⟨\tilde{\gamma }-\gamma {⟩}^{-N},1\le p\le \infty ,N\ge 0.$$

$$\Vert {\tilde{\chi }}_{\tilde{\gamma }}^{w}a{\Vert }_{L^p}$$

$$\tilde{S}(m)=\{a\in {\mathcal{S}}^{\prime }(E);\exists C>0,|\widehat{{\chi }_{j}u}(x^{\ast })|\le Cm(j,x^{\ast })\}.$$

$$\Vert {\chi }_k^{\ast }(D)({\chi }_j(x)u(x)){\Vert }_{L^2}\le \tilde{C}m(j,k).$$

$$\Vert {\chi }_k^{\ast }(x^{\ast })\widehat{{\chi }_{j}u}(x^{\ast }){\Vert }_{L^{\infty }}\le \hat{C}m(j,k),$$

$$a\#b=(e^{\frac{i}{2}\sigma (D_x,D_y)}a(x)b(y){)}_{y=x}$$

$$S(m)=\{a\in C^{\infty }(E);|D_x^{\alpha }a(x)|\le C_{\alpha }m(x)\},$$

$$e^{i\ell }\#a$$

$$a\#e^{i\ell }=e^{i\ell (x)}(e^{-\frac{1}{2}{H}_{\ell }}a).$$

$$e^{i\ell }\#a\#e^{-i\ell }=e^{H_{\ell }}a,$$

$$e^{i\frac{m}{2}}\#a\#e^{i\frac{m}{2}}=e^{im}a,$$

$$b=e^{-H_{y_0^{\ast }}}{e}^{i{x}_0^{\ast }}a=e^{-i{y}_0^{\ast }}\#e^{i\frac{x_0}{2}}\#a\#e^{i\frac{x_0}{2}}\#e^{i{y}_0^{\ast }},$$

$$Tu=C\int e^{i\varphi (x,y)}u(y)dy,C>0,$$

$$\Phi (x)=\underset{y\in {\mathbf{R}}^n}{\sup }-\mathrm{I}m\varphi (x,y).$$

$${\Lambda }_{\Phi }={\kappa }_T(E),$$

$${\kappa }_T:{\mathbf{C}}^{2n}\simeq E^{\mathbf{C}}\ni (y,-{\varphi }_y^{\prime }(x,y))\to (x,{\varphi }_x^{\prime }(x,y))\in {\mathbf{C}}^{2n}$$

$$T{a}^w{T}^{-1}={\tilde{a}}^w,$$

$${\tilde{a}}^{w}u(x)=\int e^{\Phi (x)}{K}_{\tilde{a}}^{\mathrm{eff}}(x,y)u(y)e^{-\Phi (y)}L(dy),u\in H_{\Phi }({\mathbf{C}}^n),$$

$$K_{\tilde{a}}^{\mathrm{eff}}(x,y)={\mathcal{O}}_N(1)⟨x-y{⟩}^{-N},$$

$$|K_{T{a}^w{T}^{-1}}^{\mathrm{eff}}(x,y)|\le C_N(a)⟨x{⟩}^{-N}⟨y{⟩}^{-N},x,y\in {\mathbf{C}}^n,$$

$$|K_{a^w}^{\mathrm{eff}}(x,y)|\le C_N(a)⟨x{⟩}^{-N}⟨y{⟩}^{-N},x,y\in E.$$

Full text

Résumé

On étudie des classes générales d’opérateurs pseudodifférentiels dont les classes de symboles sont définis en termes d’éstimations dans l’espace de phase.

Keywords and Phrases: Pseudodifferential operator, symbol, modulation space. Mathematics Subject Classification 2000: 35S05

1. 1 Introduction
2. 2 Symbol spaces
3. 3 Effective kernels and $L^{2}$-boundedness
4. 4 Composition
5. 5 More direct approach using Bargmann transforms
6. 6 $C_{p}$ classes
7. 7 Further generalizations

1 Introduction

This paper is devoted to pseudodifferential operators with symbols of limited regularity. The author [28] introduced the space of symbols $a(x)$ on the phase space $E={\bf R}^{n}\times({\bf R}^{n})^{*}$ with the property that

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for some $L^{1}$ function $F$ on $E^{*}$. Here the hat indicates that we take the Fourier transform, $\Gamma\subset E$