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Register a new userRough pseudodifferential operators on Hardy spaces for Fourier integral operators
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Jan Rozendaal
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We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,eta)$ are elements of $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ and $mathcal{H}^{t,p}_{FIO}(mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$. Our main result implies that for $m=0$, $delta=1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for all $pin(1,infty)$.
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We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,eta)$ are elements of $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $mathcal{H}^{p,s}_{FIO}(mathbb{R}^{n})$ and $mathcal{H}^{p,t}_{FIO}(mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$. Our main result is that for all $r>0$, $m=0$ and $delta=1/2$, there exists an interval of $p$ around $2$ such that $a(x,D)$ acts boundedly on $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$.
We obtain new local smoothing estimates for the Euclidean wave equation on $mathbb{R}^{n}$, by replacing the space of initial data by a Hardy space for Fourier integral operators. This improves the bounds in the local smoothing conjecture for $pgeq 2(n+1)/(n-1)$, and complements them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators.
We define a scale of Hardy spaces $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$, $pin[1,infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.
The Hardy spaces for Fourier integral operators $mathcal{H}_{FIO}^{p}(mathbb{R}^{n})$, for $1leq pleq infty$, were introduced by Smith in cite{Smith98a} and Hassell et al. in cite{HaPoRo18}. In this article, we give several equivalent characterizations of $mathcal{H}_{FIO}^{1}(mathbb{R}^{n})$, for example in terms of Littlewood--Paley g functions and maximal functions. This answers a question from [Rozendaal,2019]. We also give several applications of the characterizations.
This is a introductory course focusing some basic notions in pseudodifferential operators ($Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and $Psi$DOs are introduced. In Chapter 3 we define the oscillatory integrals of different types. Chapter 4 is devoted to the stationary phase lemmas. One of the features of the lecture is that the stationary phase lemmas are proved for not only compactly supported functions but also for more general functions with certain order of smoothness and certain order of growth at infinity. We build the results on the stationary phase lemmas. Chapters 5, 6 and 7 covers main results in $Psi$DOs and the proofs are heavily built on the results in Chapter 4. Some aspects of the semi-classical analysis are similar to that of microlocal analysis. In Chapter 8 we finally introduce the notion of wavefront, and Chapter 9 focuses on the propagation of singularities of solution of partial differential equations. Important results are circulated by black boxes and some key steps are marked in red color. Exercises are provided at the end of each chapter.
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