Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCharacterizations of the Hardy space $mathcal{H}_{FIO}^{1}(mathbb{R}^{n})$ for Fourier Integral Operators
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 $pin(1,infty)$.
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 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 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.
We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $mathbb{R}^n$. Results for both solutions and subsolutions are given.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
