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Weighted Hardy operators and commutators on Morrey spaces. (English) Zbl 1206.42027

The authors discuss the Morrey space boundedness of the weighted Hardy operators \(U_\psi\) defined by \(U_\psi f(x)=\int_{0}^{1}f(tx)\psi(t)\,dt\) \((x\in \mathbb R^n)\), where \(\psi:[0,1)\to[0,\infty)\). When \(\psi\equiv 1\) and \(n=1\), this reduces to the classical Hardy operator \(U: Uf(x)=x^{-1}\int_{0}^{x}f(t)\,dt\). They show that when \(1<q<\infty\) and \(-1/q<\lambda<0\), \(U_\psi\) is bounded on the Morrey space \(L^{q,\lambda}(\mathbb R^n)\) if and only if \(\int_{0}^{1}t^{n\lambda }\psi(t)\,dt<\infty\), and \(\|U_\psi\|_{\text{op}}=\int_{0}^{1}t^{n\lambda }\psi(t)\,dt\). They also characterize those \(\psi\) for which the commutators of \(U_\psi\) and the function multipliers \(M_b\) are bounded on \(L^{q,\lambda}(\mathbb R^n)\) for all \(\text{BMO}(\mathbb R^n)\) functions \(b\). They give the same results for the Cesàro operators which are adjoint to \(U_\psi\), too. Their results generalize the corresponding ones in \(L^q(\mathbb R^n)\) spaces.

MSC:

42B99 Harmonic analysis in several variables
26D15 Inequalities for sums, series and integrals
42B25 Maximal functions, Littlewood-Paley theory
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[1] Anderson K F, Muckenhoupt B. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 1982, 72: 9–26 · Zbl 0501.47011
[2] Bastero J, Milman M, Ruiz F J. Commutators of the maximal and sharp functions. Proc Amer Math Soc, 2000, 128: 3329–3334 · Zbl 0957.42010 · doi:10.1090/S0002-9939-00-05763-4
[3] Carton-Lebrun C, Fosset M. Moyennes et quotients de Taylor dans BMO. Bull Soc Roy Sci Liège, 1984, 53: 85–87 · Zbl 0543.42013
[4] Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Mat, 1987, 7: 273–279 · Zbl 0717.42023
[5] Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103: 611–635 · Zbl 0326.32011 · doi:10.2307/1970954
[6] Ding Y, Lu S Z, Yabuta K. On commutators of Marcinkiewicz integrals with rough kernel. J Math Anal Appl, 2002, 275: 60–68 · Zbl 1019.42009 · doi:10.1016/S0022-247X(02)00230-5
[7] Duong X T, Yan L X. On commutators of fractional integrals. Proc Amer Math Soc, 2002, 132: 3549–3557 · Zbl 1046.42005 · doi:10.1090/S0002-9939-04-07437-4
[8] Fu Z W, Liu Z G, Lu S Z. Commutators of weighted Hardy operators. Proc Amer Math Soc, 2009, 137: 3319–3328 · Zbl 1174.42018 · doi:10.1090/S0002-9939-09-09824-4
[9] Fu Z W, Liu Z G, Lu S Z, Wang H B. Characterization for commutators of n-dimensional fractional Hardy operators. Sci China, Ser A, 2007, 50: 1418–1426 · Zbl 1131.42012 · doi:10.1007/s11425-007-0094-4
[10] Hardy G H, Littlewood J, Pólya G. Inequalities. 2nd ed. London/New York: Cambridge University Press, 1952
[11] Komori Y. Notes on commutators of Hardy operators. Int J Pure Appl Math, 2003, 7: 329–334 · Zbl 1054.46022
[12] Lacey M T. Commutators with Riesz potentials in one and several parameters. Hokkaido Math J, 2007, 36: 175–191 · Zbl 1138.42008
[13] Long S C, Wang J. Commutators of Hardy operators. J Math Anal Appl, 2002, 274: 626–644 · Zbl 1023.35077 · doi:10.1016/S0022-247X(02)00264-0
[14] Morrey C. On the solutions of quasi-linear elliptic partial differential equations. Trans Amer Math Soc, 1938, 43: 126–166 · Zbl 0018.40501 · doi:10.1090/S0002-9947-1938-1501936-8
[15] Muckenhoupt B. Hardy’s inequality with weights. Studia Math, 1972, 44: 31–38 · Zbl 0236.26015
[16] Perez C. Endpoints for commutators of Singular integrals operators. J Funct Anal, 1995, 128: 163–185 · Zbl 0831.42010 · doi:10.1006/jfan.1995.1027
[17] Xiao J. L p and BMO bounds of weighted Hardy-Littlewood averages. J Math Anal Appl, 2001, 262: 660–666 · Zbl 1009.42013 · doi:10.1006/jmaa.2001.7594
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