Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

Abstract

We establish continuity mapping properties of the non-centered fractional maximal operator M β in the endpoint input space W 1,1 (R d ) for d ≥ 2 in the cases for which its boundedness is known. More precisely, we provef is radial and for 1 ≤ β < d for general f . The results for 1 ≤ β < d extend to the centered counterpart M c β . Moreover, if d = 1, we show that the conjectured boundedness of that map for M c β implies its continuity. Af 1,q ≤ C f 1,p . At the endpoint p = 1, one cannot expect boundedness of M β from W 1,1 to W 1, d d−β to hold, as M β fails to be bounded at the level of Lebesgue spaces. However, one Date: June 4, 2019. 2010 Mathematics Subject Classification. 42B25, 46E35.

Sobolev regularity of polar fractional maximal functions

“…They also proved the analogous result for M β . Assuming the boundedness ∇M β f q β ∇f 1 for every f ∈ W 1,1 (R), the centered case has been settled affirmatively for d = 1 in [3]. In higher dimensions they also proved the continuity of the map…”

“…The proof of this fact use different ideas that the ones contained in [3]. Our approach seems to be quite general, and we discuss some other applications of it.…”

“…Moving our discussion to S d , we notice that for β ≥ 1, using inequality (1.2) the proof of [3] can be adapted to this case.…”

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

“…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…”

Gradient bounds for radial maximal functions

“…In the seminal paper [14], Kinnunen studied the action of the Hardy-Littlewood maximal operator on Sobolev functions, giving an elegant proof that M : W 1,p (R d ) → W 1,p (R d ) is bounded for 1 < p ≤ ∞. This work paved the way for several interesting contributions to the regularity theory of maximal operators over the past two decades, with interesting connections to potential theory and partial differential equations, see for instance [1,3,4,6,7,8,11,13,15,16,18,19,22,23,21,24,25,26]. One of the longstanding problems in this field is concerned with the regularity at the endpoint p = 1.…”