We establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, boundedness is also obtained for the commutators of p-adic multilinear Hausdorff operators on these spaces with symbols in the 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, No. 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: Math. Gen., 36, 4239–4246 (2003).
N. M. Chuong, Yu. V. Egorov, A. Yu. Khrennikov, Y. Meyer, and D. Mumford, Harmonic, Wavelet and p-Adic Analysis, World Scientific Publ., Hackensack, NJ (2007).
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, No. 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, No. 3, 204–216 (2016).
N. M. Chuong, D. V. Duong, and K. H. Dung, Multilinear Hausdorff Operators on Some Function Spaces with Variable Exponent (2017); arxiv.org/abs/1709.08185.
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, No. 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, No. 1, 1–17 (2009).
M. Dyachenko, E. Nursultanov, and S. Tikhonov, “Hardy–Littlewood and Pitt’s inequalities for Hausdorff operators,” Bull. Sci. Math., 147, 40–57 (2018).
L. Grafakos, “Modern Fourier analysis, 2nd edn.,” Graduate Texts in Mathematics, 250, Springer, New York (2009).
F. Hausdorff, “Summationsmethoden und Momentfolgen. I,” Math. Z., 9, No. 1-2, 74–109 (1921).
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).
W. A. Hurwitz and L. L. Silverman, “The consistency and equivalence of certain definitions of summabilities,” Trans. Amer. Math. Soc., 18, 1–20 (1917).
T. Hytönen, C. Pérez, and E. Rela, “Sharp reverse Hölder property for A1 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, No. 4, 905–944 (1993).
A. Yu. Khrennikov, “p-Adic valued distributions in mathematical physics,” Mathematics and Its Applications, 309, Kluwer AP, Dordrecht (1994).
A. N. Kochubei, “Pseudo-differential equations and stochastics over non-Archimedean fields,” Monographs and Textbooks in Pure and Applied Mathematics, 244, Marcel Dekker, New York (2001).
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).
S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics, World Scientific, Singapore (2007).
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).
K. S. Rim and J. Lee, “Estimates of weighted Hardy–Littlewood averages on the p-adic vector space,” J. Math. Anal. Appl., 324, No. 2, 1470–1477 (2006).
E. M. Stein, “Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals,” Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton Univ. Press, Princeton, NJ (1993).
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, 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 Physics, World Scientific, Singapore (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 operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces,” Math. Notes, 3, 382–391 (2013).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 7, pp. 979–1004, July, 2021. Ukrainian DOI: 10.37863/umzh.v73i7.441.
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Chuong, N.M., Duong, D.V. & Dung, K.H. Weighted Lebesgue and Central Morrey Estimates for p-Adic Multilinear Hausdorff Operators and Their Commutators. Ukr Math J 73, 1138–1168 (2021). https://doi.org/10.1007/s11253-021-01983-2
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DOI: https://doi.org/10.1007/s11253-021-01983-2