Good lambda inequalities for non-doubling measures in ${\mathbb{R}}^n$

Abstract

We establish a good lambda inequality relating to the distribution function of Riesz potential and fractional maximal function on $\left({\mathbb{R}}^n,d\mu \right)$ where $\mu$ is a positive Radon measure which doesn't necessarily satisfy a doubling condition. This is extended to weights $w$ in $A_{\infty }(\mu )$ associated to the measure $\mu$. We also derive potential inequalities as an application.