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Mixed weak estimates of Sawyer type for generalized maximal operators


Author: Fabio Berra
Journal: Proc. Amer. Math. Soc. 147 (2019), 4259-4273
MSC (2010): Primary 42B20, 42B25
DOI: https://doi.org/10.1090/proc/14495
Published electronically: June 27, 2019
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Abstract: We study mixed weak estimates of Sawyer type for maximal operators associated with the family of Young functions $ \Phi (t)=t^r(1+\log ^+t)^{\delta }$, where $ r\geq 1$ and $ \delta \geq 0$. More precisely, if $ u$ and $ v^r$ are $ A_1$ weights and $ w$ is defined as $ w=1/\Phi (v^{-1})$, then the estimate

$\displaystyle uw\left (\left \{x\in \mathbb{R}^n: \frac {M_\Phi (fv)(x)}{v(x)}>... ...\int _{\mathbb{R}^n}\Phi \left (\frac {\vert f(x)\vert v(x)}{t}\right )u(x)\,dx$

holds for every positive $ t$. This extends mixed estimates to a wider class of maximal operators, since when we put $ r=1$ and $ \delta =0$ we recover a previous result for the classical Hardy-Littlewood maximal operator.

This inequality generalizes the result proved by Sawyer in [Proc. Amer. Math. Soc. 93 (1985), no. 4, pp. 610-614]. Moreover, it includes estimates for some maximal operators related to commutators of Calderón-Zygmund operators.


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Additional Information

Fabio Berra
Affiliation: CONICET and Departamento de Matemática (FIQ-UNL), 3000 Santa Fe, Argentina
Email: fberra@santafe-conicet.gov.ar

DOI: https://doi.org/10.1090/proc/14495
Keywords: Young functions, maximal operators, Muckenhoupt weights
Received by editor(s): April 6, 2018
Received by editor(s) in revised form: October 2, 2018
Published electronically: June 27, 2019
Additional Notes: The author was supported by CONICET and UNL
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2019 American Mathematical Society