Better bounds on mixed inequalities involving radial functions and applications

TL;DR

This paper establishes improved mixed inequalities for generalized maximal operators with radial power functions, extending to fractional and commutator operators, with applications to fractional integrals.

Contribution

It provides new bounds for mixed inequalities involving radial functions and generalized maximal operators, improving previous results and applying to fractional and commutator operators.

Findings

Proved mixed inequalities for $M_ ext{Phi}$ with radial power functions.

Extended results to fractional maximal operators and fractional integrals.

Established bounds for commutators with Lipschitz symbols.

Abstract

We prove mixed inequalities for the generalized maximal operator $M_\Phi$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^\beta$ with $\beta<-n$ and $r\geq 1$. If $\Phi$ is a Young function with certain properties, then the inequality \[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_\Phi (fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\] holds for every $t>0$ and every bounded function. This improves a similar mixed estimate proved in \cite{BCP-M}. As an application, we give mixed estimates for the generalized fractional maximal operator $M_{\gamma,\Phi}$, where $0<\gamma<n$ and $\Phi$ is of $L\log L$ type. A special case involving the fractional maximal operator $M_\gamma$ allows to obtain a similar estimate for the fractional integral operator $I_\gamma$ through an extrapolation result. Furthermore, we also give mixed estimates for commutators of singular integral Calder\'on-Zygmund operators and of $I_\gamma$, both with Lipschitz symbol.