Real interpolation of generalized Besov-Hardy spaces and applications
type: Item
instance of: scholarly article
title: Real interpolation of generalized Besov-Hardy spaces and applications
zbMATH Open document ID: 1232.46029
author: Alexandre Almeida
author: António M. Caetano
published in: The Journal of Fourier Analysis and Applications
publication date: 2011-08-30T00:00:00Z
review text: The authors introduce the generalized homogeneous Hardy spaces \(H_q(\phi)\) and the generalized non-homogeneous Hardy spaces \(h_q(\phi),\) \(0<q<\infty,\) where \(\phi\in {\mathcal B},\) i.e., \(\phi: (0,\infty)\to (0,\infty)\) is continuous, \(\phi(1)=1\) and \(\sup_{s>0}\frac{\phi(st)}{\phi(s)}<\infty,\) \(0<t<\infty\). These generalizations correspond to the classical Hardy spaces \(H_p\) and \(h_p\), replacing the Lebesgue space \(L_p\) by the Lorentz space \(\Lambda_q(\phi).\) The Besov spaces \(B^s_{p,q}, s\in {\mathbb{R}}\), \(0<p,q\leq \infty\), are defined via the Fourier transform. More general Besov spaces \(B^\phi_{p,q}\) are introduced just replacing the power function \(t^s\) by \(\phi.\) Even more general Besov spaces \(B^\phi_{\psi,q,r}\) are defined, replacing \(L_p\) by the Hardy space \(h_q(\psi)\), \(\psi\in {\mathcal B}.\) The authors present results about the interpolation of generalized Hardy and Besov spaces using the \(K-\)method with a parameter \(L_p(1/\gamma),\) \(\gamma\in {\mathcal B}\). In particular, the following formula is proved: \[ (B^\phi_{p_0,q}, B^\phi_{p_1,q})_{\gamma,q}=B^\phi_{\psi,q,q},\; p_0\neq p_1,\; \phi, \gamma\in {\mathcal B}, \] \(1/\psi(t)=t^{-1/p_0} \gamma(t^{1/p_0-1/p_1})\) if the Boyd indices of \(\gamma\) are in the interval \((0,1)\). As an application, the boundedness of pseudodifferential operators in the generalized Besov and Hardy spaces is investigated, and a wavelet decomposition of \(h_q(\phi)\) is provided.
reviewed by: Georgi E. Karadzhov
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 46M35
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 5944315
zbMATH Keywords: generalized Hardy spaces
zbMATH Keywords: maximal functions
zbMATH Keywords: interpolation
zbMATH Keywords: generalized Besov spaces
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: wavelet decompositions
MaRDI profile type: Publication
full work available at URL: s00041-010-9145-2
OpenAlex ID: W2079501041
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DOI: 10.1007/S00041-010-9145-2
description: scientific article; zbMATH DE number 5944315