New estimates for the maximal functions and applications

TL;DR

This paper advances the understanding of maximal functions by establishing sharper inequalities, leading to improved embeddings and new inequalities in harmonic analysis using limiting interpolation techniques.

Contribution

It provides new sharp pointwise inequalities for maximal operators, strengthening classical results and deriving improved embeddings and inequalities in harmonic analysis.

Findings

Strengthened DeVore's inequality for moduli of smoothness.

Derived a logarithmic variant of Bennett--DeVore--Sharpley's inequality.

Improved the Stein--Zygmund embedding for Besov and BMO spaces.

Abstract

In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant of Bennett--DeVore--Sharpley's inequality for rearrangements. As a consequence, we improve the classical Stein--Zygmund embedding deriving $\dot{B}^{d/p}_\infty L_{p,\infty}(\mathbb{R}^d) \hookrightarrow \text{BMO}(\mathbb{R}^d)$ for $1 < p < \infty$. Moreover, these results are also applied to establish new Fefferman--Stein inequalities, Calder\'on--Scott type inequalities, and extrapolation estimates. Our approach is based on the limiting interpolation techniques.