New estimates for the maximal functions and applications

Oscar Domínguez, Sergey Tikhonov

Published 2021-02-09Version 1

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.