Mixed weak estimates of Sawyer type for generalized maximal operators

BERRA, FABIO

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2019

0002-9939

We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions $Phi(t)=t^r(1+log^+t)^{delta}$, where $rgeq 1$ and $deltageq 0$. More precisely, if $u$ and $v^r$ are $A_1$ weights, and $w$ is defined as $w=1/Phi(v^{-1})$ then the following estimate[uwleft(left{xin mathbb{R}^n: rac{M_Phi(fv)(x)}{v(x)}>tight}ight)leq Cint_{mathbb{R}^n}Phileft(rac{|f(x)|v(x)}{t}ight)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. extbf{93} (1985), no.~4, 610--614. Moreover, it includes estimates for some maximal operators related with commutators of Calder´on-Zygmund operators.