Hardy spaces related to Schrödinger operators with potentials which are sums of \(L^p\)-functions
type: Item
instance of: scholarly article
title: Hardy spaces related to Schrödinger operators with potentials which are sums of \(L^p\)-functions
zbMATH Open document ID: 1253.42012
author: Jacek Dziubański
author: Marcin Preisner
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.5626
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 6086919
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: Hardy space
zbMATH Keywords: maximal function
zbMATH Keywords: atom
zbMATH Keywords: Riesz transform
reviewed by: Dachun Yang
MaRDI profile type: Publication
OpenAlex ID: W2964032747
arXiv ID: 1109.5626
cites work: Q3142876
cites work: New abstract Hardy spaces
cites work: Atomic characterization of the Hardy space 𝐻¹_{𝐿}(ℝ) of one-dimensional Schrödinger operators with nonnegative potentials
cites work: Hardy spaces associated with some Schrödinger operators
cites work: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality
cites work: Hpspaces associated with Schrödinger operators with potentials from reverse Hölder classes
cites work: Hardy spaces H1for Schrödinger operators with certain potentials
cites work: Remarks on spectral multiplier theorems on Hardy spaces associated with semigroups of operators
cites work: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates
cites work: Hardy spaces \(H^1\) for Schrödinger operators with compactly supported potentials
cites work: Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials
cites work: A local version of real Hardy spaces
cites work: Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates
DOI: 10.1016/J.JMAA.2012.06.012
review text: The authors consider a Schrödinger operator on \(\mathbb{R}^n\) given by \(Lf(x):=-\Delta f(x)+V(x)f(x)\), where \(x\in\mathbb{R}^n\) and \(\Delta\) denotes the Laplace operator. Let \(d\in\mathbb{N}\). The authors assume that the potential \(V\) satisfies:NEWLINENEWLINE(A1) for any \(j\in\{1,\dots,d\}\), there exists \(V_j\geq0\), \(V_j\not\equiv 0\), such that, for all \(x\in\mathbb{R}^n\), \(V(x)=\sum_{j=1}^d V_j(x)\),NEWLINENEWLINE(A2) for every \(j\in\{1,\dots,d\}\), there exists a linear subspace \(\mathbb{V}_j\) of \(\mathbb{R}^n\) of dimension \(n_j\geq 3\) such that, if \(\Pi_{\mathbb{V}_j}\) denotes the orthogonal projection on \(\mathbb{V}_j\), then, for all \(x\in\mathbb{R}^n\), \(V_j(x)=V_j(\Pi_{\mathbb{V}_j}x)\),NEWLINENEWLINE(A3) there exists \(\kappa\in(0,\infty)\) such that, for any \(j\in\{1,\dots,d\}\) and all \(r\) satisfying \(|r-n_j/2|\leq \kappa\), \(V_j\in L^r(\mathbb{V}_j)\).NEWLINENEWLINELet \(K_t:=\exp(-tL)\) and \(P_t:=\exp(t\Delta)\), with \(t\in(0,\infty)\), be the semigroups of linear operators associated with \(L\) and \(\Delta\), respectively. Let \(K_t(x,y)\) and \(P_t(x-y)\) be the integral kernels of these semigroups. Let \(M_L\) and \(M_{\Delta}\) be the associated maximal operators, i.\,e., for all \(x\in\mathbb{R}^n\), \(M_Lf(x):=\sup_{r\in(0,\infty)}|K_tf(x)|\) and \(M_{\Delta}f(x):=\sup_{t\in(0,\infty)}|P_tf(x)|\). The Hardy spaces \(H^1_L(\mathbb{R}^n)\) and \(H^1_{\Delta}(\mathbb{R}^n)\) are the subspaces of \(L^1(\mathbb{R}^n)\) defined by NEWLINE\[NEWLINEf\in H^1_L(\mathbb{R}^n)\Longleftrightarrow M_Lf\in L^1(\mathbb{R}^n),\;\;f\in H^1_{\Delta}(\mathbb{R}^n)\Longleftrightarrow M_{\Delta}f\in L^1(\mathbb{R}^n)NEWLINE\]NEWLINE with the norms \(\|f\|_{H^1_L(\mathbb{R}^n)}:=\|M_L f\|_{L^1(\mathbb{R}^n)}\) and \(\|f\|_{H^1_{\Delta}(\mathbb{R}^n)}:=\|M_{\Delta}f\|_{L^1(\mathbb{R}^n)}\). Denote by \(L^{-1}\) the operator with kernel \(\Gamma(x,y)=\int_0^{\infty}K_t(x,y)\,dt\).NEWLINENEWLINEAssume that \(f\in L^1(\mathbb{R}^n)\).…
description: scientific article; zbMATH DE number 6086919