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Register a new user$L^p$-mapping properties for Schrodinger operators in open sets of $mathbb R ^d$
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Let $H_V=-Delta +V$ be a Schrodinger operator on an arbitrary open set $Omega$ of $mathbb R^d$, where $d geq 3$, and $Delta$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Omega$. The purpose of this paper is to show $L^p$-boundedness of an operator $varphi(H_V)$ for any rapidly decreasing function $varphi$ on $mathbb R$. $varphi(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $varphi(H_V)$ are also obtained.
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We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.
Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Hans proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $hat{mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.
In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the $s$-nuclearity, $0<s leq 1,$ of periodic and discrete pseudo-differential operators. To accomplish this, we classify those $s$-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $sigma$-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.
Let $S subset mathbb{R}^{n}$ be a~closed set such that for some $d in [0,n]$ and $varepsilon > 0$ the~$d$-Hausdorff content $mathcal{H}^{d}_{infty}(S cap Q(x,r)) geq varepsilon r^{d}$ for all cubes~$Q(x,r)$ centered in~$x in S$ with side length $2r in (0,2]$. For every $p in (1,infty)$, denote by $W_{p}^{1}(mathbb{R}^{n})$ the classical Sobolev space on $mathbb{R}^{n}$. We give an~intrinsic characterization of the restriction $W_{p}^{1}(mathbb{R}^{n})|_{S}$ of the space $W_{p}^{1}(mathbb{R}^{n})$ to~the set $S$ provided that $p > max{1,n-d}$. Furthermore, we prove the existence of a bounded linear operator $operatorname{Ext}:W_{p}^{1}(mathbb{R}^{n})|_{S} to W_{p}^{1}(mathbb{R}^{n})$ such that $operatorname{Ext}$ is right inverse for the usual trace operator. In particular, for $p > n-1$ we characterize the trace space of the Sobolev space $W_{p}^{1}(mathbb{R}^{n})$ to the closure $overline{Omega}$ of an arbitrary open path-connected set~$Omega$. Our results extend those available for $p in (1,n]$ with much more stringent restrictions on~$S$.
Let $D^alpha, alpha>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^alpha D^{-alpha}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
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