Regularity of fractional maximal functions through Fourier multipliers

Abstract

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n ≥ 2. We also show that the spherical fractional maximal function maps L p into a first order Sobolev space in dimensions n ≥ 5. t α |ϕ t * f (x)|.

Sobolev regularity of polar fractional maximal functions

“…Moreover, we prove that under the assumption of a local boundedness conjecture we can conclude the centered version of Theorem 2 for both the S d and the R d contexts (see Subsection 4.2 and Theorem 20). Applications for some of the maximal functions discussed in [4] are expected, but not proved here.…”

“…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.…”

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

“…Finally, we remark that the techniques used to get results for smooth kernels as in [2] are insensitive to the ambient domain, because one does not use precise information about the maximizing radius. The arguments there only rely on sublinearity of maximal functions.…”

“…The fractional maximal function is given by M α f (x) = sup r>0 r α |B(x, r)| B(x,r) |f (y)| dy, and it defines a bounded operator L p (R n ) → L q (R n ) when q = np/(n − p) and p > 1. This boundedness fails at the endpoint p = 1, but the question about boundedness of ∇M α from W 1,1 (R n ) to L n/(n−α) (R n ) has not been answered so far for α < 1 (see [19], [1] and [2] for related research and partial results). The case α ≥ 1 turned out to be very simple, and the reason can be traced back to the inequality…”

Regularity of Maximal Operators: Recent Progress and Some Open Problems

“…We can hence ask ourselves the follow up question, which is a particular case of Question 7. in Question 7. They consider the slightly different setting of maximal operators either associated to smooth convolution kernels or to a lacunary set of radii in dimensions d ≥ 2 (see [3,Theorem 1]). In this work, they also show that the spherical maximal operator maps L p into a first order Sobolev spaces in dimensions d ≥ 5.…”