Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus
type: Item
instance of: scholarly article
title: Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus
zbMATH Open document ID: 1271.42033
author: Xuan Thinh Duong
author: Ji Li
published in: Journal of Functional Analysis
publication date: 2013-04-30T00:00:00Z
review text: Let \(X\) be a metric space, equipped with a distance function \(d\) and a non-negative, doubling Borel measure \(\mu\) on \(X\) with \(\mu(X)=\infty\). Let \(0\leq \omega<\nu<\pi\) and let \(S_\omega\) be the closed sector in the complex plane \({\mathbb C}\) defined by \[ S_\omega:=\{z\in{\mathbb C}:\,\, |\arg z|\leq \omega\}\cup\{0\} \] and \(S^0_\omega\) its interior. Let \(H(S^0_\nu)\) be the space of all holomorphic function on \(S^0_\nu\) and \[ H_\infty(S^0_\nu):=\{b\in H(S^0_\nu):\,\,\|b\|_\infty<\infty\}, \] where \(\|b\|_\infty:=\sup\{|b(z)|:\,\,z\in S^0_\nu\}\), and \[ \Psi(S^0_\nu):=\left\{\psi\in H_\infty(S^0_\nu):\,\, \exists s>0, |\psi(z)|\leq c\frac{|z|^s}{(1+|z|^{2s})}\right\}. \] Throughout the paper, \(L\) is assumed to be a linear operator of type \(\omega\) (that is, \(\sigma(L)\subset S_\omega\)) on \(L^2(X)\) with \(\omega<\pi/2\) and satisfy the following two conditions: {(H1)} The operator \(L\) has a bounded \(H_\infty\)-calculus on \(L^2(X)\). That is, there exists \(c_{\nu,\,2}>0\) such that for \(b\in H_\infty(S^0_\nu)\), \(b(L)\in {\mathcal L}(L^2, L^2)\) and for any \(g\in L^2(X)\), \[ \|b(L)g\|_{L^2(X)}\leq c_{\nu,\,2}\|b\|_\infty\|g\|_{L^2(X)}; \] {(H2)} The analytic semigroup \(\{e^{-tL}\}_{t>0}\) satisfies the Davies-Gaffney estimate. That is, there exist constants \(C\), \(c>0\) such that for any open subsets \(U_1\), \(U_2\subset X\), \(f_i\in L^2(X)\) with \({\text{\,supp\,}}(f_i)\subset U_i\), \(i=1,\,2\), and \(t>0\), \[ |\langle e^{-tL}f_1, f_2\rangle|\leq C\exp\left(-\frac{{\mathrm{\,dist\,}}(U_1, U_2)^2}{ct}\right)\|f_1\|_{L^2(X)}\|f_2\|_{L^2(X)}, \] where \({\text{\,dist\,}}(U_1, U_2):=\inf_{x\in U_1,\,y\in U_2}d(x, y)\). Fix \(\alpha\geq 0\), \(0<p\leq1\), and integers \(s\geq [\frac{n\alpha}{2}]\) and \(M>[\frac{n(2-p)}{4p}]\). The authors first introduce Lipschitz spaces \(\Lambda^{\alpha,\,s}_L(X)\), the molecular Hardy space \(H^p_{L,\,\text{mol},\,M}(X)\), the Hardy space \(H^p_{L,\,S_h}(X)\) defined in terms of the…
reviewed by: Dongyong Yang
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 6159679
zbMATH Keywords: Hardy space
zbMATH Keywords: molecular decomposition
zbMATH Keywords: atomic decomposition
zbMATH Keywords: Littlewood-Paley square function
zbMATH Keywords: Davies-Gaffney estimate
zbMATH Keywords: interpolation
MaRDI profile type: Publication
full work available at URL: j.jfa.2013.01.006
OpenAlex ID: W2075507562
DOI: 10.1016/J.JFA.2013.01.006
description: scientific article; zbMATH DE number 6159679