Gradient bounds for radial maximal functions
| Autor: | Emanuel Carneiro, Cristian González-Riquelme |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Physics
42B25 46E35 31B05 35J05 35K08 Maximal operators Articles Type (model theory) Convolution Combinatorics Sobolev space Mathematics - Analysis of PDEs Radial function Mathematics - Classical Analysis and ODEs Sobolev spaces Bounded variation Classical Analysis and ODEs (math.CA) FOS: Mathematics bounded variation convolution Maximal function sphere Nabla symbol Differentiable function Analysis of PDEs (math.AP) |
| Zdroj: | Annales Fennici Mathematici |
| ISSN: | 2737-114X 2737-0690 |
| Popis: | 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 $\mathbb{S}^d$. 28 pages. V2 with minor updates and typos corrected |
| Databáze: | OpenAIRE |
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