New estimates for the maximal functions and applications

Domínguez, Óscar; Tikhonov, Sergey

doi:10.1090/tran/8632

New estimates for the maximal functions and applications
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by Óscar Domínguez and Sergey Tikhonov PDF

Trans. Amer. Math. Soc. 375 (2022), 3969-4018 Request permission

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, Calderón–Scott type inequalities, and extrapolation estimates. Our approach is based on the limiting interpolation techniques.

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