Better bounds on mixed inequalities involving radial functions and applications

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.