Weighted Inequalities for the Fractional Maximal Operator in Lorentz Spaces via Atomic Decomposition of Tent Spaces

Abstract

Consider the usual fractional maximal operator $M_{\alpha }$ with $0<\alpha <n$. A characterization of ${\mathbb{R}}^n$ weight functions $u(\cdot )$ and $\sigma (\cdot )$ for which $M_{\alpha }d\sigma$ sends the (generalized) Lorentz space ${\Lambda }_{\sigma }^s(w_1)$ into ${\Lambda }_u^r(w_2)$ with $1<s<r<\infty$ is obtained by using a suitable-atomic decomposition of tent spaces.