On the Hardy--Littlewood maximal functions in high dimensions: Continuous and discrete perspective

Abstract

This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy-Littlewood averaging operators associated with convex symmetric bodies in R d and Z d .

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

“…The next lemma is variant of a comparison principle that was recently used in [17, Theorem 1, p. 859], see also [18]. Lemma 4.9 will transfer the problem from the set of integers Z 3 to the continuous setting R 3 , where the properties of the maximal functions corresponding to the kernels K λ will be investigated.…”

“…Once (4.28), (4.29), (4.30) and (4.31) are established we may proceed much the same way as in [63] to deduce (4.26) and (4.27), we refer also to [18, (4.7) and (4.8), pp. [16][17][18][19] or [17]. More precisely, to prove (4.27) it suffices to use (4.28), (4.29), (4.30) with the standard Littlewood-Paley theory and appeal to [63, Theorem 2.14, p. 537].…”