On the variation of maximal operators of convolution type II

Abstract

In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space R d , on the torus T d and on the sphere S d . The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.where B 1 is the unit ball centered at the origin and m(B 1 ) is its ddimensional Lebesgue measure. In this case, due to the work of Kurka [13], the one-dimensional estimate (1.2) is known to hold with constant C = 240, 004, but the problem with C = 1 remains open. For the onedimensional right (or left) Hardy-Littlewood maximal operator, i.e. when ϕ(x, t) = 1 t χ [0,1] (x/t), estimate (1.2) holds with C = 1 due to the work of Tanaka [21]. The sharp bound (1.2) with constant C = 1 also Date: October= 4, 2018. 2010 Mathematics Subject Classification. 42B25, 46E35, 35B50, 31B05, 35J05, 35K08.holds for the one-dimensional uncentered version of this operator, as proved by Aldaz and Pérez Lázaro [1].Higher dimensional analogues of (1.2) for the Hardy-Littlewood maximal operator, centered or uncentered, are open problems (see, for instance, the work of Haj lasz and Onninen [9]). Other interesting works related to the regularity of the Hardy-Littlewood maximal operator and its variants, when applied to Sobolev and BV functions,are [2,3,4,5,8,10,11,12,14,15,20,22].In the precursor of this work [6, Theorems 1 and 2], Carneiro and Svaiter proved the variation-diminishing property, i.e. inequality (1.2) with C = 1, for the maximal operators associated to the Poisson kernel(1.3) and the Gauss kernelTheir proof is based on an interplay between the analysis of the maximal functions and the structure of the underlying partial differential equations (Laplace's equation and heat equation). The aforementioned examples are the only maximal operators of convolution type for which inequality (1.2) has been established (even allowing a constant C > 1). Maximal operators associated to elliptic equations. A question that derives from our precursor[6] is whether the variation-diminishing property is a peculiarity of the smooth kernels (1.3) and (1.4) or if these can be seen as particular cases of a general family. One could, for example, look at the semigroup structure via the Fourier transforms 1 (in space) of these kernels:A reasonable way to connect these kernels would be to consider the one-parameter family ϕ α (ξ, t) = e −t(2π|ξ|) α , for 1 ≤ α ≤ 2. However, in this case, the function u(x, t) = ϕ α (·, t) * u 0 (x) solves an evolution equation related to the fractional Laplacian u t + (−∆) α/2 u = 0 , for which we do not have a local maximum principle, essential to run the argument of Carneiro and Svaiter in [6]. The problem of proving that the corresponding maximal operator is variation-diminishing seems more delicate and it is currently open.

Poincaré inequalities for the maximal function

“…Similar results have later been established for maximal functions restricted to domains [24], for fractional maximal functions [20,19,25], and for certain convolution type maximal functions [10,8]. Continuity as well as action on some function spaces in the Triebel-Lizorkin scale have been studied in [26,28,29].…”

Sobolev regularity of polar fractional maximal functions

“…This work paved the way to several contributions of many researchers in this topic and its relations with other areas, see for instance [1,4,5,7,10,12,13,15,23,24,25] The most important open problem in this field is the W 1,1 -problem.…”

Sharp total variation results for maximal functions

“…This opened a new field of studies, and several other properties of this and other related maximal functions were studied. We mention, for example, [3,4,5,7,9].…”