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Gradient bounds for radial maximal functions

By Emanuel Carneiro and Cristian González-Riquelme

Abstract

In this paper we study the regularity properties of certain maximal operators of convolution type at the endpoint $p=1$, when acting on radial data. In particular, for the heat flow maximal operator and the Poisson maximal operator, when the initial datum $u_0 \in W^{1,1}( \mathbb{R}^d)$ is a radial function, we show that the associated maximal function $u^*$ is weakly differentiable and $$\|\nabla u^*\|_{L^1(\mathbb{R}^d)} \lesssim_d \|\nabla u_0\|_{L^1(\mathbb{R}^d)}.$$ This establishes the analogue of a recent result of H. Luiro for the uncentered Hardy-Littlewood maximal operator, now in a centered setting with smooth kernels. In a second part of the paper, we establish similar gradient bounds for maximal operators on the sphere $\mathbb{S}^d$, when acting on polar functions. Our study includes the uncentered Hardy-Littlewood maximal operator, the heat flow maximal operator and the Poisson maximal operator on $\mathbb{S}^d$.Comment: 28 pages. V2 with minor updates and typos correcte

Topics: Mathematics - Classical Analysis and ODEs, Mathematics - Analysis of PDEs, 42B25, 46E35, 31B05, 35J05, 35K08
Year: 2020
OAI identifier: oai:arXiv.org:1906.01487
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