The geometry of the dyadic maximal operator

Abstract

We prove a sharp integral inequality which connects the dyadic maximal operator with the Hardy operator. We also give some applications of this inequality.

Sharp Lorentz Estimates for Dyadic-like Maximal Operators and Related Bellman Functions

“…The construction in the next Lemma appears also in [11] and provides the other half of Theorem 1. We include a simpler proof for completeness.…”

Estimates for Bellman functions related to dyadic-like maximal operators on weighted spaces

“…A way to study estimates for the above maximal operator is through the symmetrization of φ as has been introduced in [5] and [13] and used in [8] to evaluate Bellman functions related to Lorentz norms. In order to apply this in the context of weights we introduce the following condition on a weight w on X.…”

“…Moreover the corresponding constants c, a can be easily seen to be a = 1 p−1−b , c = k p−1−b . Now we take into consideration the following theorem proved in [13] and [5]. Theorem 1.…”

A sharp integral rearrangement inequality for the dyadic maximal operator and applications

“…Following [10] we let a ∈ (0, 1). Using Lemma 2.1 we choose for every I ∈ T a family F(I) ⊆ T of disjoint subsets of I such that…”

“…By providing a generalization of the symmetrization principle given in [10] we give another proof of the computation for the Bellman function of three variables of the dyadic maximal operator,different from those given in [3] and [11].…”