Improvements on Sawyer type estimates for generalized maximal functions

BERRA, FABIO; CARENA, MARILINA; PRADOLINI, GLADIS

MATHEMATISCHE NACHRICHTEN

WILEY-V C H VERLAG GMBH

Lugar: Weinheim; Año: 2020

0025-584X

In this paper we prove mixed inequalities for the maximal operator $M_Phi$, for general Young functions $Phi$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $rgeq 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $Phi$ is a Young function as above, then the inequality[uv^rleft(left{xin mathbb{R}^n: rac{M_Phi(fv)(x)}{v(x)}>tight}ight)leq Cint_{mathbb{R}^n}Phileft(rac{|f(x)|}{t}ight)u(x)v^r(x),dx]holds for every positive $t$.A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of $M_Phi$. Moreover, it is well-known that for the particular case $Phi(t)=t(1+log^+t)^m$ with $minmathbb{N}$ these maximal functions control, in some sense, certain operators in Harmonic Analysis.