Improvements on Sawyer type estimates for generalized maximal functions

In this paper we prove mixed inequalities for the maximal operator $M_{\mathrm{\Phi }}$, for general Young functions $\mathrm{\Phi }$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $r\ge 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $\mathrm{\Phi }$ is a Young function as above, then the inequality $$u{v}^r\left(\{x\in {\mathbb{R}}^n:\frac{M_{\mathrm{\Phi }}(fv)(x)}{v(x)}>t\}\right)\le C{\int }_{{\mathbb{R}}^n}\mathrm{\Phi }\left(\frac{\mathrm{\mid }f(x)\mathrm{\mid }}{t}\right)u(x)v^r(x)dx$$ 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 $M_{\mathrm{\Phi }}$. Moreover, it is well-known that for the particular case $\mathrm{\Phi }(t)=t(1+{\log }^{+}t{)}^m$ with $m\in \mathbb{N}$ these maximal functions control, in some sense, certain operatos in Harmonic Analysis.