Weighted weak-type inequalities for some fractional integral operators

Abstract

For $0<\alpha<1$, let $W_{\alpha}$ and $R_{\alpha}$ denote Weyl fractional integral operator and Riemann-Liouville fractional integral operator, respectively. We establish sharp versions of Muckenhoupt-Wheeden conjecture for these operators. Specifically, we prove that for any weight $w$ on $[0,\infty)$, we have \begin{equation*} \|{W}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{-}w)^{1-α})} \end{equation*} and \begin{equation*} \|{R}_{α} f\|_{L^{1/(1-α),∞}(w)}≤ α^{-1}\|{f}\|_{L^{1}((M_{+}w)^{1-α})}. \end{equation*} Here $M_{-}$, $M_{+}$ denote the one-sided Hardy-Littlewood maximal operators on $[0,\infty)$. In each of the estimates, the constant $\alpha^{-1}$ is the best possible.