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Weighted weak-type inequalities for some fractional integral operators. (English) Zbl 1318.26015

Summary: 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 \[ \|{W}_{\alpha} f\|_{L^{1/(1-\alpha),\infty}(w)}\leq \alpha^{-1}\|{f}\|_{L^{1}((M_{-}w)^{1-{\alpha}})} \] and \[ \|{R}_{\alpha} f\|_{L^{1/(1-{\alpha}),\infty}(w)}\leq \alpha^{-1}\|{f}\|_{L^{1}((M_{+}w)^{1-{\alpha}})}. \]
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.

MSC:

26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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References:

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