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Sharp Weak Type Inequalities for Fractional Integral Operators

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Abstract

Suppose that d≥1 is an integer, α∈(0,d) is a fixed parameter and let I α be the fractional integral operator associated with d-dimensional Walsh-Fourier series on (0,1]d. Let p, q be arbitrary numbers satisfying the conditions 1≤p<d/α and 1/q=1/p−α/d. We determine the optimal constant K, which depends on α, d and p, such that for any f∈L p((0,1]d) we have

$$ ||I_{\alpha } f||_{L^{q,\infty }((0,1]^{d})}\leq K||f||_{L^{p}((0,1]^{d})}. $$

In fact, we shall prove this inequality in the more general context of probability spaces equipped with a regular tree-like structures. This allows us to obtain this result also for non-integer dimension. The proof exploits a certain modification of the so-called Bellman function method and appropriate interpolation-type arguments. We also present a sharp weighted weak-type bound for I α , which can be regarded as a version of the Muckenhoupt-Wheeden conjecture for fractional integral operators.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Rodrigo Bañuelos

  2. Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Adam Osȩkowski

Authors
  1. Rodrigo Bañuelos
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  2. Adam Osȩkowski
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Corresponding author

Correspondence to Rodrigo Bañuelos.

Additional information

A. Osȩkowski is supported by Narodowe Centrum Nauki, Poland, grant DEC-2014/14/E/ST1/00532.

R. Bañuelos is supported in part by NSF Grant # 1403417-DMS

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Bañuelos, R., Osȩkowski, A. Sharp Weak Type Inequalities for Fractional Integral Operators. Potential Anal 47, 103–121 (2017). https://doi.org/10.1007/s11118-016-9610-x

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