Gradient bounds for radial maximal functions

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 datumis a radial function, we show that the associated maximal function u * is weakly differentiable andThis 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 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 S d .

Sobolev regularity of polar fractional maximal functions

“…This recent result inspired the study the regularity theory of maximal functions in higher dimensions when restricted to radial data. For instance, in [6] the analogue of this result for some centered kernels is proved.…”

“…When working on the circle S 1 , an adaptation of the proof of Aldaz and Pérez Lázaro [2] yields Var( Mf ) ≤ Var(f ), where Var(f ) denotes the total variation of the function f . Recently, Carneiro and the author [6] settled affirmatively this question in the case of polar functions. The general case remains open.…”

“…Concerning to the polar case, the difficulties that Carneiro and the author faced in [6] also appear (in different ways) when dealing with this question. Our first theorem is to get the analogue of the main result of [21] in this context.…”

“…Our first theorem is to get the analogue of the main result of [21] in this context. We go further in the methods already developed in [6] in order to adapt the proof of [21]. We get the following.…”

Weak differentiability for fractional maximal functions of general L functions on domains

“…Extending such a statement to more general Sobolev functions of several variables is a difficult open problem, which has inspired many results in related topics. For instance, slightly stronger bounds have been proved for maximal operators with more special convolution kernels (see [7], [3], [4] and [20]), the continuity of the mapping has been studied in [17] and [6], and a part of the techniques used for continuity, also relevant for the current paper, have been extended to p = 1 in [11].…”