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Weak type (1,1) estimates for some integral operators related to rough maximal functions

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Abstract

We study the problem of determining for which integrable functionsG:R → (0, ∞) the operatorf → 1/yG(y.) *f(x), which maps functions on the real line into functions defined on the upper half-planeR 2+ , is of weak type (1,1). Here,R 2+ is endowed with the measurey dx dy. The conditions we will impose are related to the distribution of the mass ofG.

One of the motivations for this study comes from the problem of deciding whether there is a weak type (1,1) inequality for the “rough” modification of the standard maximal function, obtained by inserting in the mean values a factor Ω which depends only on the angle. Here, Ω≥0 is any integrable function on the sphere. Our estimates for the first-mentioned problem allow us to answer in the affirmative, the second one in dimension two, when we restrict the operator to radial functions. Some extensions to higher dimensions in the context of both problems are also discussed.

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Authors and Affiliations

  1. Department of Mathematics, Chalmers Institute of Technology and University of Göteborg, S-41296, Göteborg, Sweden

    Peter Sjögren

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, E-28049, Madrid, Spain

    Fernando Soria

Authors
  1. Peter Sjögren
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  2. Fernando Soria
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Correspondence to Peter Sjögren.

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Both authors were partially supported by DGICYT PB90/187.

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Sjögren, P., Soria, F. Weak type (1,1) estimates for some integral operators related to rough maximal functions. Israel J. Math. 95, 211–229 (1996). https://doi.org/10.1007/BF02761040

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  • DOI: https://doi.org/10.1007/BF02761040

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