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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Mixed weak estimates of Sawyer type for generalized maximal operators
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by Fabio Berra PDF
Proc. Amer. Math. Soc. 147 (2019), 4259-4273 Request permission

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 \[ uw\left (\left \{x\in \mathbb {R}^n: \frac {M_\Phi (fv)(x)}{v(x)}>t\right \}\right )\leq C\int _{\mathbb {R}^n}\Phi \left (\frac {|f(x)|v(x)}{t}\right )u(x) dx\] 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
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4259-4273
  • MSC (2010): Primary 42B20, 42B25
  • DOI: https://doi.org/10.1090/proc/14495
  • MathSciNet review: 4002540