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For 0 < α < 1, let Wα and Rα 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,∞), we have ||Wαf ||L1/(1−α),∞(w) ≤ α−1||f ||L1((M−w)1−α) and ||Rαf ||L1/(1−α),∞(w) ≤ α−1||f ||L1((M+w)1−α). Here M−, M+ denote the one-sided Hardy-Littlewood maximal operators on [0,∞). In each of the estimates, the constant α−1 is the best possible.

Weighted weak-type inequalities for some fractional integral operators