A local version of Hardy spaces associated with operators on metric spaces
type: Item
instance of: scholarly article
title: A local version of Hardy spaces associated with operators on metric spaces
zbMATH Open document ID: 1269.42014
author: Ji Li
author: Lixin Yan
author: Ruming Gong
published in: Science China. Mathematics
publication date: 2013-04-08T00:00:00Z
review text: Let \((X,d,\mu)\) a metric space, where \(d\) is a distance and \(\mu\) satisfies the doubling property. For \(L\) a non-negative and self-adjoint operator defined on \(L^2(X)\), let us consider the semigroup \(e^{-tL}\) generated by \(L\) on \(L^2(X)\) and whose kernel \(p_t(x,y)\) satisfies a Gaussian upper bound. The local Hardy space \(h^1_L(X)\) is defined as \(\{f\in L^2(X): \;\|s_{loc,L}f\|_{L^1(X)}<\infty\}\) where \(s_{loc,L}f\) is the following area function \[ s_{loc,L}f(x)=\left(\int_0^1 \int_{d(x,y)\leq t} |t^2 L e^{-t^2L} f(y)|^2 \frac{d\mu(y)}{V(x,t)}\frac{dt}{t}\right)^{1/2} \quad x\in X \] and \(V(x,t)=\mu(B(x,t))\). An atomic decomposition in terms of local \((1,p)\) atoms of this local Hardy space is proved. Also, a Moser type local boundedness condition for \(L\) is used to show that the local Hardy space \(h^1_L(X)\) can be characterized in terms of Littlewood-Paley functions. The work concludes with some applications to the study of second-order operators defined on Lipschitz domains and a class of Schrödinger operators which satisfy both, a Gaussian upper bound for the kernel, and a Moser type local condition.
reviewed by: Santiago Boza
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 6151455
zbMATH Keywords: local Hardy space
zbMATH Keywords: non-negative self-adjoint operator
zbMATH Keywords: semigroups
zbMATH Keywords: local (1, \(p\))-atom
zbMATH Keywords: Moser type local boundedness condition
zbMATH Keywords: Littlewood-Paley function
MaRDI profile type: Publication
full work available at URL: s11425-012-4428-5
OpenAlex ID: W1909743925
cites work: Q4352127
cites work: Maximal inequalities and Riesz transform estimates on \(L^p\) spaces for Schrödinger operators with nonnegative potentials
cites work: Hardy spaces of differential forms on Riemannian manifolds
cites work: Hardy spaces and divergence operators on strongly Lipschitz domains of \(\mathbb R^n\)
cites work: A T(b) theorem with remarks on analytic capacity and the Cauchy integral
cites work: Q2548398
cites work: Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem
cites work: Hardy spaces and regularity for the inhomogeneous Dirichlet and Neumann problems
cites work: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
cites work: Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality
cites work: A local version of real Hardy spaces
cites work: Littlewood-Paley and spectral multipliers on weighted \(L^p\) spaces
cites work: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates
cites work: Hardy and BMO spaces associated to divergence form elliptic operators
cites work: Localized Hardy spaces associated with operators
cites work: A decomposition into atoms of distributions on spaces of homogeneous type
cites work: Q4674789
cites work: Q3548322
cites work: Aspects of Sobolev-type inequalities
cites work: Riesz transform, Gaussian bounds and the method of wave equation
cites work: Hardy spaces and bmo on manifolds with bounded geometry
cites work: Q3607418
DOI: 10.1007/S11425-012-4428-5
description: scientific article; zbMATH DE number 6151455