From $A_1$ to $A_\infty$: New mixed inequalities for certain maximal operators

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in \cite{Berra}. Concretely, given $r\geq 1$, $u\in A_1$, $v^r\in A_\infty$ and a Young function $Φ$ with certain properties, we have that inequality $uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{M_Φv(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx$ holds for every positive $t$. The involved operator $\frac{M_Φ(fv)(x)}{M_Φv(x)}$ seems to be an adequate extension when $v^r\in A_\infty$, since when we assume $v^r\in A_1$ we can replace $M_Φv$ by $v$, yielding a mixed inequality for $M_Φ$ proved in \cite{Berra-Carena-Pradolini(MN)}. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator $M_{γ,Φ}$, where $0<γ<n$ and $Φ$ is a Young function of $L\log L$ type.

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