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March 2015 Weighted weak-type inequalities for some fractional integral operators
Adam Osȩkowski
Proc. Japan Acad. Ser. A Math. Sci. 91(3): 35-38 (March 2015). DOI: 10.3792/pjaa.91.35

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.

Citation

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Adam Osȩkowski. "Weighted weak-type inequalities for some fractional integral operators." Proc. Japan Acad. Ser. A Math. Sci. 91 (3) 35 - 38, March 2015. https://doi.org/10.3792/pjaa.91.35

Information

Published: March 2015
First available in Project Euclid: 3 March 2015

zbMATH: 1318.26015
MathSciNet: MR3317749
Digital Object Identifier: 10.3792/pjaa.91.35

Subjects:
Primary: 26A33
Secondary: 26D10

Keywords: best constant , fractional integral , Muckenhoupt-Wheeden conjecture , weight

Rights: Copyright © 2015 The Japan Academy

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Vol.91 • No. 3 • March 2015
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