Mixed weak estimates of Sawyer type for generalized maximal operators

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.