Local Hardy Spaces with Variable Exponents Associated with Non-negative Self-Adjoint Operators Satisfying Gaussian Estimates

  • Víctor Almeida
  • Jorge J. Betancor
  • Estefanía Dalmasso
  • Lourdes Rodríguez-MesaEmail author


In this paper we introduce variable exponent local Hardy spaces \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) associated with a non-negative self-adjoint operator L. We assume that, for every \(t>0\), the operator \(e^{-tL}\) has an integral representation whose kernel satisfies a Gaussian upper bound. We define \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) by using an area square integral involving the semigroup \(\{e^{-tL}\}_{t>0}\). A molecular characterization of \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) is established. As an application of the molecular characterization, we prove that \(h_L^{p(\cdot )}({\mathbb {R}}^n)\) coincides with the (global) Hardy space \(H_L^{p(\cdot )}({\mathbb {R}}^n)\) provided that 0 does not belong to the spectrum of L. Also, we show that \(h_L^{p(\cdot )}({\mathbb {R}}^n)=H_{L+I}^{p(\cdot )}({\mathbb {R}}^{n})\).


Hardy spaces Molecules Local Variable exponent 

Mathematics Subject Classification

42B35 42B30 42B25 



The authors thank the referee for his/her corrections and insightful comments, which helped to improve the quality of the manuscript.


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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa Laguna (Sta. Cruz de Tenerife)Spain
  2. 2.Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FCE/FIQSanta FeArgentina

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