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The Riesz–Herz equivalence for capacitary maximal functions on metric measure spaces

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Published/Copyright: December 9, 2015
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Advances in Pure and Applied Mathematics
From the journal Advances in Pure and Applied Mathematics Volume 7 Issue 2

Abstract

We prove a Riesz–Herz estimate for the maximal function MCμf(x)=supx∈BC(B)-1∫B|f(y)|dμ(y) associated to a capacity C on a metric measure space (X,d,μ). This estimate extends the equivalence (Mf)*(t)≃f**(t) for the usual Hardy–Littlewood maximal function Mf and the Riesz–Herz estimate for the capacitary maximal function on ℝn. Essential tools are the extension of the Wiener–Stein estimate for the distribution function of MCμ and the existence of appropriate dyadic cubes in metric measure spaces. Finally, we obtain the Riesz–Herz estimate for a discrete version of the capacitary maximal function.

Keywords: Maximal functions; capacity; metric measure spaces; dyadic cubes; interpolation spaces; Morrey spaces
MSC: 42B25; 46B70; 28A12; 30L99

I am very grateful to Professor J. Kinnunen for his appropriate comments and Professor J. Cerdà for making me pay attention to these issues.

Received: 2015-6-29
Accepted: 2015-11-3
Published Online: 2015-12-9
Published in Print: 2016-4-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. The Riesz–Herz equivalence for capacitary maximal functions on metric measure spaces
  3. On graded classical primary submodules
  4. Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities
  5. Transversal hypersurfaces with (f,g,u,v,λ)-structure of a nearly trans-Sasakian manifold
  6. Generalized solution for a class of Hamilton–Jacobi equations
  7. Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones
https://doi.org/10.1515/apam-2015-0041
Keywords for this article
Maximal functions; capacity; metric measure spaces; dyadic cubes; interpolation spaces; Morrey spaces

Articles in the same Issue

  1. Frontmatter
  2. The Riesz–Herz equivalence for capacitary maximal functions on metric measure spaces
  3. On graded classical primary submodules
  4. Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities
  5. Transversal hypersurfaces with (f,g,u,v,λ)-structure of a nearly trans-Sasakian manifold
  6. Generalized solution for a class of Hamilton–Jacobi equations
  7. Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones
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