Acta mathematica scientia, Series B >
NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESÁRO TYPE
Received date: 2015-04-13
Revised date: 2014-08-07
Online published: 2015-11-01
Supported by
The first author is supported by Vietnam National Foundation for Science and Technology Development (101.02-2014.51).
This paper deals with a general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesàro operators (with symbols in central BMO spaces) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.
Ha Duy HUNG , Luong Dang KY . NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESÁRO TYPE[J]. Acta mathematica scientia, Series B, 2015 , 35(6) : 1411 -1425 . DOI: 10.1016/S0252-9602(15)30063-1
[1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42(4): 765-778
[2] Alvarez J, Lakey J, Guzmán-Partida M. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 51(11): 147
[3] Andersen K F. Boundedness of Hausdorff operator on Lp(Rn), H1(Rn) and BMO(Rn). Acta Sci Math (Szeged), 2003, 69(1/2): 409-418
[4] Bradley J S. Hardy inequalities with mixed norms. Canad Math Bull, 1978, 21(4): 405-408
[5] Carton-Lebrun C, Fosset M. Moyennes et quotients de Taylor dans BMO. Bull Soc Roy Sci Li`ege, 1984, 53(2): 85-87
[6] Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function. Rend Math Appl, 1987, 7(3/4): 273-279
[7] Chuong N M, Hung H D. Bounds of weighted Hardy-Cesàro operators on weighted Lebesgue and BMO spaces. Integral Transforms Spec Funct, 2014, 25(9): 697-710
[8] Christ M, Grafakos L. Best constants for two nonconvolution inequalities. Proc Amer Math Soc, 1995, 123(6): 1687-1693
[9] Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103(3): 611-635
[10] Edmunds D E, Evans W D. Hardy Operators, Function Spaces and Embeddings. Springer Monographs in Mathematics. Berlin: Springer-Verlag, 2004
[11] Faris W G. Weak Lebesgue spaces and quantum mechanical binding. Duke Math J, 1976, 43(2): 365-373
[12] Folland G B. Real Analysis: Modern Techniques and Their Applications: Pure and Applied Mathematics. 2nd ed. New York: John Wiley & Sons, Inc, 1999
[13] Fu Z W, Liu Z G, Lu S Z. Commutators of weighted Hardy operators on Rn. Proc Amer Math Soc, 2009, 137(10): 3319-3328
[14] Fu Z W, Gong S L, Lu S Z, Yuan W. Weighted multilinear Hardy operators and commutators. Forum Math, 2014, DOI: 10.1515/forum-2013-0064
[15] Fu Z W, Grafakos L, Lu S Z, Zhao F. Sharp bounds for m-linear Hardy and Hilbert operators. Houston J Math, 2012, 38(1): 225-244
[16] Goldberg R R. Convolutions and general transforms on Lp. Duke Math J, 1960, 27: 251-259
[17] Gong S L, Fu Z W, Ma B. Weighted multilinear Hardy operators on Herz type spaces. The Scientific World Journal, 2014, Art ID 420408
[18] Grafakos L. Classical Fourier Analysis. Graduate Texts in Mathematics, 249. 2nd ed. New York: Springer, 2008
[19] Grafakos L, Martell J M. Extrapolation of weighted norm inequalities for multivariable operators and applications. J Geom Anal, 2004, 14(1): 19-46
[20] Hardy G H, Littlewood J E, Pólya G. Inequalities: 2nd ed. Cambridge: Cambridge University Press, 1952
[21] Hung H D. The P-adic weighted Hardy-Cesàro operator and an application to discrete Hardy inequalities. J Math Anal Appl, 2014, 409: 868-879
[22] John F, Nirenberg L. On functions of bounded mean oscillation. Comm Pure Appl Math, 1961, 14: 415-426
[23] Kenig C, Stein E M. Multilinear estimates and fractional integration. Math Res Lett, 1999, 6: 1-15
[24] Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282(2): 219-231
[25] Liflyand E, Móricz F. The Hausdorff operator is bounded on the real Hardy space H1(R). Proc Amer Math Soc, 2000, 128(5): 1391-1396
[26] Lerner A K, Liflyand E. Multidimensional Hausdorff operators on the real Hardy space. J Aust Math Soc, 2007, 83(1): 79-86
[27] Lu S Z, Yang D C. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl (NS), 1995, 11(3): 72-94
[28] Lu S Z, Yan D, Zhao F. Sharp bounds for Hardy type operators on higher-dimensional product spaces. J Inequal Appl, 2013, 2013: 148
[29] Muckenhoupt B. Hardy's inequalities with weights. Studia Math, 1972, 44: 31-38
[30] Sawyer E. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans Amer Math Soc, 1984, 281(1): 329-337
[31] Xiao J. Lp and BMO bounds of weighted Hardy-Littlewood averages. J Math Anal Appl, 2001, 262(2): 660-666
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