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On the continuity of maximal operators of convolution type at the derivative level

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Abstract

In this paper we study a question related to the continuity of maximal operators of convolution type acting on W1,1(ℝ). We prove that the map u ↦ (u*)′ is continuous from W1,1(ℝ) to L1(ℝ), where u* is the maximal function associated to the Poisson kernel, the Heat kernel or a family of kernels related to the fractional Laplacian. This is the first result of this type for a centered maximal operator.

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Acknowlegdements

The author is grateful to E. Carneiro for encouragement and very helpful discussions. The author also thanks M. Sousa for helpful comments about earlier versions of this manuscript. The author was supported by CAPES-Brazil and from the STEP program of ICTP — Italy.

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  1. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, 22460-320, Brazil

    Cristian González-Riquelme

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  1. Cristian González-Riquelme
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Correspondence to Cristian González-Riquelme.

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González-Riquelme, C. On the continuity of maximal operators of convolution type at the derivative level. Isr. J. Math. 253, 745–759 (2023). https://doi.org/10.1007/s11856-022-2375-6

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  • DOI: https://doi.org/10.1007/s11856-022-2375-6

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