Abstract
The aim of this paper is to introduce and study the boundedness of a new class of p-adic rough multilinear Hausdorff operators on the product of Herz, central Morrey and Morrey–Herz spaces with both power weights and Muckenhoupt weights. We also establish the boundedness for the commutators of p-adic rough multilinear Hausdorff operators on the weighted spaces with symbols in central BMO space.
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S. Albeverio, A.Yu. Khrennikov, and V. M. Shelkovich, “Harmonic Analysis in the p–Adic Lizorkin Spaces: Fractional Operators, Pseudo–Differential Equations, p–Wavelets, Tauberian Theorems,” J. Fourier Anal. Appl. 12 (4), 393–425 (2006).
A. V. Avetisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “p–Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes,” J. Phys. A 35, 177–189 (2002).
A. V. Avetisov, A. H. Bikulov, and V. A. Osipov, “p–Adic Description of Characteristic Relaxation in Complex Systems,” J. Phys. A 36, 4239–4246 (2003).
K. F. Andersen and B. Muckenhoupt, “Weighted Weak Type Hardy Inequalities with Applications to Hilbert Transforms and Maximal Functions,” Studia Math. 72 (1), 9–26 (1982).
O. V. Beloshapka, “Feynman Formulas for the Schrödinger Equations with the Vladimirov Operator,” Russ. J. Math. Phys. 17 (3), 267–271 (2010).
C. Carton–Lebrun and M. Fosset, “Moyennes et quotients de Taylor dans BMO,” Bull. Soc. Roy. Sci. Liége 53 (2), 85–87 (1994).
R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization Theorems for Hardy Spaces in Several Variables,” Ann. Math. 103, 611–635 (1976).
J. Chen, D. Fan and J. Li, “Hausdorff Operators on Function Spaces,” Chinese Ann. Math. Ser. B. 33, 537–556 (2012).
N. M. Chuong, Pseudodifferential Operators and Wavelets over Real and p–Adic Fields (Springer–Basel, 2018).
N. M. Chuong, Yu. V. Egorov, A. Yu. Khrennikov, Y. Meyer, and D. Mumford, Harmonic, Wavelet and p–Adic Analysis (World Scientific, 2007).
N. M. Chuong, and D. V. Duong, “Wavelet Bases in the Lebesgue Spaces on the Field of p–Adic Numbers,” p−Adic numbers, Ultrametric Anal. Appl. 5 (2), 106–121 (2013).
N. M. Chuong and D. V. Duong, “Weighted Hardy–Littlewood Operators and Commutators on p−Adic Functional Spaces,” p−Adic numbers, Ultrametric Anal. Appl. 5 (1), 65–82 (2013).
N. M. Chuong and D. V. Duong, “The p–Adic Weighted Hardy–Cesàro Operators on Weighted Morrey–Herz Space,” p−Adic numbers, Ultrametric Anal. Appl. 8 (3), 204–216 (2016).
N. M. Chuong and N. V. Co, “The Cauchy Problem for a Class of Pseudo–Differential Equations over p–Adic Field,” J. Math. Anal. Appl. 340 (1), 629–643 (2008).
N. M. Chuong and H. D. Hung, “Maximal Functions and Weighted Norm Inequalities on Local Fields,” Appl. Comput. Harmon. Anal. 29, 272–286 (2010).
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, “On p−Adic Mathematical Physics,” p−Adic numbers, Ultrametric Anal. Appl. 1 (1), 1–17 (2009).
L. Grafakos, Modern Fourier Analysis (Second Edition, Springer, 2008).
H. D. Hung, “The p–Adic Weighted Hardy–Cesàro Operator and an Application to Discrete Hardy Inequalities,” J. Math. Anal. Appl. 409, 868–879 (2014).
T. Hytönen, C. Pérez, and E. Rela, “Sharp Reverse Hölder Property for A∞ Weights on Spaces of Homogeneous Type,” J. Funct. Anal. 263, 3883–3899 (2012).
S. Indratno, D. Maldonado, and S. Silwal, “A Visual Formalism for Weights Satisfying Reverse Inequalities,” Expo. Math. 33, 1–29 (2015).
S. Haran, “Riesz Potentials and Explicit Sums in Arithmetic,” Invent. Math. 101, 697–703 (1990).
S. Haran, “Analytic Potential Theory over the p–Adics,” Ann. Inst. Fourier (Grenoble) 43 (4), 905–944 (1993).
A.Yu. Khrennikov, p–Adic Valued Distributions in Mathematical Physics (Kluwer Academic Publishers, Dordrecht–Boston–London, 1994).
A.Yu. Khrennikov, V. M. Shelkovich, and M. Skopina, “p–Adic Refinable Functions and MRA–Based Wavelets,” J. Approx. Theory 161, 226–238 (2009).
S. V. Kozyrev, “Methods and Applications of Ultrametric and p–Adic Analysis: From Wavelet Theory to Biophysics,” Proc. Steklov Inst. Math. 274, 1–84 (2011).
Z. W. Fu, Z. G. Liu, and S. Z. Lu, “Commutators of Weighted Hardy Operators,” Proc. Amer. Math. Soc. 137, 3319–3328 (2009).
Z. W. Fu, Q. Y. Wu, and S. Z. Lu, “Sharp Estimates of p–Adic Hardy and Hardy–Littlewood–P´olya Operators,” Acta Math. Sin. 29, 137–150 (2013).
S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics (World Scientific Publishing Company, Singapore, 2007).
C. Morrey, “On the Solutions of Quasi–Linear Elliptic Partial Differential Equations,” Trans. Amer. Math. Soc. 43, 126–166 (1938).
B. Muckenhoupt, “Weighted Norm Inequalities for the Hardy Maximal Function,” Trans. Amer. Math. Soc. 165, 207–226 (1972).
J. Ruan, D. Fan, and Q. Wu, “Weighted Herz Space Estimates for Hausdorff Operators on the Heisenberg Group,” Banach J. Math. Anal. 11, 513–535 (2017).
J. Ruan, D. Fan, and Q. Wu, “Weighted Morrey Estimates for Hausdorf Op–erator and Its Commutator on the Heisenberg Group,” Math. Inequal. Appl. 22 (1), 303–329 (2019).
K. S. Rim and J. Lee, “Estimates ofWeighted Hardy–Littlewood Averages on the p–Adic Vector Space,” J. Math. Anal. Appl. 324 (2), 1470–1477 (2006).
Elias M. Stein, Harmonic Analysis, Real–Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, 1993).
C. Tang, F. Xue, and Y. Zhou, “Commutators of Weighted Hardy Operators on Herz–Type Spaces,” Ann. Polon. Math. 101, 267–273 (2011).
V. S. Varadarajan, “Path Integrals for a Class of p–Adic Schrödinger Equations,” Lett. Math. Phys. 39, 97–106 (1997).
V. S. Vladimirov, “Tables of Integrals of Complex–Valued Functions of p–Adic Arguments,” Proc. Steklov Inst. Math. 284 (2), 1–59 (2014).
V. S. Vladimirov and I. V. Volovich, “p–Adic Quantum Mechanics,” Comm. Math. Phys. 123, 659–676 (1989).
V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p–Adic Analysis and Mathematical Physis (World Scientific, 1994).
S. S. Volosivets, “Multidimensional Hausdorff Operator on p–Adic Field,” p–Adic numbers, Ultrametric Anal. Appl. 2, 252–259 (2010).
S. S. Volosivets, “Hausdorff Operator of Special Kind in Morrey and Herz p–Adic Spaces,” p–Adic numbers, Ultrametric Anal. Appl. 4, 222–230 (2012).
S. S. Volosivets, “Hausdorff Operators on p–Adic Linear Spaces and Their Properties in Hardy, BMO, and Hölder Spaces,” Math. Notes 3, 382–391 (2013).
S. S. Volosivets, “Weak and Strong Estimates for Rough Hausdorff Type Operator Defined on p–Adic Linear Space,” p–Adic numbers, Ultrametric Anal. Appl. 9 (3), 236–241 (2017).
Q. Y. Wu, L. Mi, and Z. W. Fu, “Boundedness of p–Adic Hardy Operators and Their Commutators on p–Adic Central Morrey and BMO Spaces,” J. Funct. Spaces Appl. (2013), Article ID 359193, 10 pages.
J. Xiao, “L p and BMO Bounds of Weighted Hardy–Littlewood Averages,” J. Math. Anal. Appl. 262, 660–666 (2001).
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This paper is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.51.
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Chuong, N.M., Duong, D.V. & Dung, K.H. Some Estimates for p-Adic Rough Multilinear Hausdorff Operators and Commutators on Weighted Morrey–Herz Type Spaces. Russ. J. Math. Phys. 26, 9–31 (2019). https://doi.org/10.1134/S1061920819010023
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DOI: https://doi.org/10.1134/S1061920819010023