Optimal Bounds on the Modulus of Continuity of the Uncentered Hardy–Littlewood Maximal Function

Abstract

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and Hölder functions on proper subintervals ofIn higher dimensions, we determine the asymptotic behavior, as d → ∞, of the norm of the maximal operator associated with cross-polytopes, Euclidean balls, and cubes, that is, p balls for p = 1, 2, ∞. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and Hölder functions, the operator norm of the maximal operator is uniformly bounded by 2 −α/q , where q is the conjugate exponent of p = 1, 2, and as d → ∞ the norms approach this bound. When p = ∞, best constants are the same as when p = 1.

A stability version of Hölder's inequality

“…However, the methods used in [13] and [12] cannot tell us whether we actually have c p < 1, that is, whether the maximal operator M has a smoothing effect on f . For p = 1, Theorem 2.5 of [2] states that DMf 1 Df 1 , and c 1 = 1 is sharp, while for p = ∞, we have DMf ∞ ( √ 2 − 1) Df ∞ and c ∞ = ( √ 2 − 1) is best possible, by [1]. Thus, it is natural to conjecture "by interpolation" that whenever 1 < p < ∞, the optimal constant c p satisfies c p < 1, and furthermore, lim p→∞ c p = √ 2 − 1.…”

“…In fact, it is the standard argument: From a refined Young's inequality one obtains a refined Hölder inequality, which E-mail addresses: jesus.munarrizaldaz@dmc.unirioja.es, jesus.munarriz@uam.es. 1 Current address: Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco 28049, Madrid, Spain. 2 The author was partially supported by Grant MTM2006-13000-C03-03 of the DGI of Spain.…”

Sharp Inequalities for the Variation Of the discrete Maximal Function

“…, |m d |} ≤ r n }. We now introduce the local maxima and minima of a discrete function g : Z → R. 1 We say that an interval [n, m] is a string of local maxima of g if g(n − 1) < g(n) = . .…”

Endpoint Sobolev and BV continuity for maximal operators

“…Kinnunen's original result was later extended to a local setting in [14], to a fractional setting in [15] and to a multilinear setting in [9]. Other works on the regularity of maximal operators in Sobolev spaces and other function spaces include [1,16,18,23,25,28].The understanding of the action of M on the endpoint spacewas raised by Haj lasz and Onninen in [12] and remains unsolved in its full generality. A complete solution was reached only in dimension d = 1 in [2, 17, 29] and partial progress on the general case d > 1 was obtained by Haj lasz and Malý [11] and, more recently, by Saari [26] and Luiro [22].Let us elaborate on the achievements in dimension d = 1 since these will be quite important for our purposes here.…”