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