The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

2019/01/22 by David Cruz-Uribe, Estefanía Dalmasso, Cruz-Uribe, David +5
Mathematics · #42B25 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1901.07472

openalex publication_date 2019/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces L_w^p(⋅)(0,∞), assuming that the exponent function p(⋅) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals \ (0,b) : b>0\ on (0,∞). Our results extend those in \citeDMRO1 for the constant exponent L^p spaces with weights. We also give two applications: the first is a weighted version of Hilbert's inequality on variable Lebesgue spaces, and the second generalizes the results in \citeSW for integral operators to the variable exponent setting.

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