Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces

Abstract

We introduce the p-adic weighted multilinear Hardy-Cesàro operator. We also obtain the necessary and sufficient conditions on weight functions to ensure the boundedness of that operator on the product of Lebesgue spaces, Morrey spaces, and central bounded mean oscillation spaces. In each case, we obtain the corresponding operator norms. We also characterize the good weights for the boundedness of the commutator of weighted multilinear Hardy-Cesàro operator on the product of central Morrey spaces with symbols in central bounded mean oscillation spaces.

Keywords

Weighted multilinear Hardy-Cesàro operator / bounded mean oscillation (BMO) / commutator / p-adic analysis

References

[1] Albeverio S, Khrennikov A Yu, Shelkovich V M. Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems. J Fourier Anal Appl, 2006, 12: 393–425

[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: 1–47

[3] Chen Y Z, Lau K S, Some new classes of Hardy spaces. J Funct Anal, 1989, 84: 255–278

[4] Chuong N M, Egorov Yu V, Khrennikov A, Meyer Y, Mumford D. Harmonic, Wavelet and p-Adic Analysis.Singapore: World Scientific, 2007

[5] Chuong N M, Hong N T, Hung H D. Bounds of p-adic Hardy-Cesàro operators and their commutators on p-adic weighted spaces of Morrey types. P-Adic Numbers Ultrametric Anal Appl, 2016, 8: 31–44

[6] Chuong N M, Hong N T, Hung H D. Multilinear Hardy-Cesàro operator and commutator on the product of Morrey-Herz spaces. Anal Math (to appear)

[7] Chuong N M, Hung H D. Maximal functions and weighted norm inequalities on Local Fields. Appl Comput Harmon Anal, 2010, 29: 272–286

[8] Chuong N M, Hung H D. A Muckenhoupt’s weight problem and vector valued maximal inequalities over local fields. P-Adic Numbers Ultrametric Anal Appl, 2010, 2: 305–321

[9] 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

[10] Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math (2), 1976, 103: 611–635

[11] Fu Z W, Gong S L, Lu S Z, Yuan W. Weighted multilinear Hardy operators and commutators. Forum Math. 2014, DOI: https://doi.org/10.1515/forum-2013-0064

[12] 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

[13] Hung H D. p-Adic weighted Hardy-Cesàro operator and an application to discrete Hardy inequalities. J Math Anal Appl, 2014, 409: 868–879

[14] Hung H D, Ky L D. New weighted multilinear operators and commutators of Hardy-Cesàro type. Acta Math Sci Ser B Engl Ed, 2015, 35: 1411–1425

[15] Igusa J. Introduction to the Theory of Local Zeta Functions. Stud Adv Math, Vol 14. Boca Raton: Chapman & Hall/CRC, 2000

[16] Khrennikov A Yu. p-Adic Valued Distributions in Mathematical Physics.Dordrecht-Boston-London: Kluwer Academic Publishers, 1994

[17] Khrennikov A Yu. Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems, and Biological Models.Dordretch/Boston/London: Kluwer Academic Publishers, 1997

[18] Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282(2): 219–231

[19] Kozyrev S V. Wavelet analysis as a p-adic spectral analysis. Izv Ross Akad Nauk Ser Mat, 2002, 66(2): 149–158

[20] Kozyrev S V, Khrennikov A Yu. Pseudodifferential operators on ultrametric spaces and ultrametric wavelets. Izv Ross Akad Nauk Ser Mat, 2005, 69(5): 133–148

[21] Lu S Z, Yang D C. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl, 1995, 11: 72–94

[22] Rim K S, Lee J. Estimates of weighted Hardy-Littlewood averages on the p-adic vector space. J Math Anal Appl, 2006, 324: 1470–1477

[23] Taibleson M. Fourier Analysis on Local Fields.Princeton: Princeton Univ Press, 1975

[24] Vladimirov V S, Volovich I V, Zelenov E I. p-Adic Analysis and Mathematical Physics.Singapore: World Scientific, 1994

[25] Volosivets S S. Hausdorff operator of special kind on p-adic field and BMO-type spaces. P-Adic Numbers Ultrametric Anal Appl, 2011, 3: 149–156

[26] Volosivets S S. Hausdorff operator of special kind in Morrey and Herz p-adic spaces. P-Adic Numbers Ultrametric Anal Appl, 2012, 4: 222–230

[27] Wu Q Y, Fu Z W. Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces. Bull Malays Math Sci Soc, 2017, 40: 635–654

[28] Wu Q Y, Mi L, Fu Z W. Boundedness for commutators of fractional p-adic Hardy operators. J Inequal Appl, 2012, 2012: 293

[29] Wu Q Y, Mi L, Fu Z W. Boundedness of p-adic Hardy operators and their commutators on p-adic central Morrey and BMO spaces. J Funct Spaces Appl, 2013, 2013: 359193

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature