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

Abstract

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 $r\ge 1$, $u\in A_1$, $v^r\in A_{\infty }$ and a Young function $\Phi$ with certain properties, we have that inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{M_Φv(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. The involved operator $\frac{M_{\Phi }(fv)(x)}{M_{\Phi }v(x)}$ 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.

27 pages