Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Positivity Preserving, Translation Invariant, Operators in $L^p(mathbb{R}^n)^m$
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.
rate research
Read More
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.
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 $H_V=-Delta +V$ 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.
Fix integers $m geq 2$, $n geq 1$. Let $C^{m-1,1}(mathbb{R}^n)$ be the space of $(m-1)$-times differentiable functions $F : mathbb{R}^n rightarrow mathbb{R}$ whose $(m-1)$st order partial derivatives are Lipschitz continuous, equipped with a standard seminorm. Given $E subseteq mathbb{R}^n$, let $C^{m-1,1}(E)$ be the trace space of all restrictions $F|_E$ of functions $F$ in $C^{m-1,1}(mathbb{R}^n)$, equipped with the standard quotient (trace) seminorm. We prove that there exists a bounded linear operator $T : C^{m-1,1}(E) rightarrow C^{m-1,1}(mathbb{R}^n)$ satisfying $Tf|_E = f$ for all $f in C^{m-1,1}(E)$, with operator norm at most $exp( gamma D^k)$, where $D := binom{m+n-1}{n}$ is the number of multiindices of length $n$ and order at most $m-1$, and $gamma,k > 0$ are absolute constants (independent of $m,n,E$). Our results improve on the previous construction of linear extension operators with norm at most $ exp(gamma D^k 2^D)$.
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~$S$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
