Mixed weak type inequalities for the fractional maximal operator

Abstract

Let $0\leq \alpha < n$, $q=\frac{n}{n-\alpha}$. We establish the mixed weak type inequality \begin{equation*} \sup_{\lambda > 0}\lambda^q \int_{\{x\in\mathbb{R}^n:M_{\alpha}f(x) > \lambda v(x) \}}u^qv^q\leq C\left(\int_{\mathbb{R}^n}|f|u \right)^q\, \end{equation*} for the fractional maximal operator \[ M_\alpha f(x)=\sup_{h > 0}|B(x,h)|^{\alpha/n-1}\int_{B(x,h)} |f| \] under the following assumptions: (a) $u^q$ belongs to the Muckenhoupt $A_1$ class, (b) $v$ is essentially constant over dyadic annuli, and (c) $(\lambda v)^q$ satisfies a certain condition $C_p(\gamma)$ for all $\lambda > 0$. The last condition is fulfilled by any Muckenhoupt weight but it is also satisfied by some non Muckenhoupt weights. Our approach is based on the study of the same kind of inequalities for the local fractional maximal operator, a Hardy type operator and its adjoint.