Abstract
In this paper, we give some sufficient conditions for the boundedness of \(p\)-adic Hausdorff operators on weighted Triebel–Lizorkin spaces and two-weighted central Morrey spaces, Morrey–Herz spaces. In particular, the norms of \(p\)-adic Hausdorff type operators are obtained on Morrey spaces and Morrey–Herz spaces with power weights. As a consequence, we establish some Hardy–Hilbert type inequalities on \(p\)-adic fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 (2006), 393–425.
R. A. Bandaliyev and S. S. Volosivets, “Hausdorff Operator on Weighted Lebesgue and Grand Lebesgue p-Adic Spaces”, p-Adic Numbers, Ultrametric Anal. Appl., 11 (2019), 114–122.
G. Brown and F. Móricz, “Multivariate Hausdorff Operators on the Spaces \(L^p(\mathbb{R}^n)\)”, J. Math. Anal. Appl., 271 (2002), 443–454.
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, “Weighted Hardy- Littlewood Operators and Commutators on \(p\)-Adic Functional Spaces”, \(p-\)Adic numbers, Ultrametric Anal. Appl., 5:1 (2013), 65–82.
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 (2016), 204–216.
N. M. Chuong, D. V. Duong and K. H. Dung, “Some Estimates for \(p\)-Adic Rough Multilinear Hausdorff Operators and Commutators on Weighted Morrey–Herz Type Spaces”, Russ. J. Math. Phys., 26 (2019), 9–31.
N. M. Chuong, D. V. Duong, and K. H. Dung, “Weighted Lebesgue and Central Morrey Estimates for \(p\)-Adic Multilinear Hausdorff Operators and Its Commutators”, Ukrain Mat. Zh., 73:7 (2021), 979–1004.
N. M. Chuong, D. V. Duong and K. H. Dung, “Two-Weighted Inequalities for Hausdorff Operators in Herz-Type Hardy Spaces”, Math. Notes, 106 (2019), 20–37.
N. M. Chuong, D. V. Duong and N. D. Duyet, “Weighted Morrey–Herz Space Estimates for Rough Hausdorff Operator and Its Commutators”, J. Pseudo-Differ. Oper. Appl., 11 (2020), 753–787.
N. M. Chuong, D. V. Duong and H. D. Hung, “Bounds for the Weighted Hardy–Cesàro Operator and Its Commutator on Morrey–Herz Type Spaces”, Z. Anal. Anwend, 35 (2016), 489–504.
D. V. Duong, “Hausdorff Operator and Commutator on Weighted Morrey–Herz Spaces on \(p\)-Adic Fields”, Anal. Math. Physics, 11:31 (2021), 1–26.
B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich, “On \(p\)-Adic Mathematical Physics”, \(p\)-Adic Numbers, Ultrametric Anal. Appl., 1 (2009), 1–17.
R. R. Goldberg, “Convolutions and General Transforms on \(L^p\)”, Duke Math. J., 27 (1960), 251–259.
C. Georgakis, “The Hausdorff Mean of a Fourier–Stieltjes Transform”, Proc. Amer. Math. Soc, 116 (1992), 465–471.
G. H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.
F. Hausdorff, “Summation Methoden und Momentfolgen, I”, Math. Z, 9 (1921), 74–109.
W. A. Hurwitz and L. L. Silverman, “The Consistency and Equivalence of Certain Definitions of Summabilities”, Trans. Amer. Math. Soc., 18 (1917), 1–20.
A. Yu. Khrennikov, \(p\)-Adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers, Dordrecht-Boston-London, 1994.
S. V. Kozyrev and A. Yu. Khrennikov, “Pseudodifferential Operators on Ultrametric Spaces and Ultrametric Wavelets”, Izv. Ross. Akad. Nauk Ser. Mat., 69 (2005), 133–148.
A. Kochubei, “Radial Solutions of Non-Archimedean Pseudodifferential Equations”, Pacific J. Math., 269 (2014), 355–369.
H. Li and J. Jin, “On a \(p\)-Adic Hilbert-Type Integral Operator and Its Applications”, J. Appl. Ana. Comput., 10 (2020), 1326–1334.
A. Lerner and E. Liflyand, “Multidimensional Hausdorff Operators on Real Hardy Spaces”, J. Austr. Math. Soc., 83 (2007), 79–86.
S. Z. Lu, D. C. Yang and G. E. Hu, Herz Type Spaces and Their Applications, Beijing Sci. Press, Beijing, 2008.
B. Muckenhoupt, “Weighted Norm Inequalities for the Hardy Maximal Function”, Trans. Amer. Math. Soc., 165 (1972), 207–226.
K. S. Rim and J. Lee, “Estimates of Weighted Hardy–Littlewood Averages on the \(p\)-Adic Vector Space”, J. Math. Anal. Appl., 324 (2006), 1470–1477.
E. M. Stein, Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993.
H. Triebel, Theory of Function Spaces, Birkhäuser, Basel–Boston–Stuttgart, 1983.
V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, \(p\)-Adic Analysis and Mathematical Physics, World Scientific, 1994.
S. S. Volosivets, “Multidimensional Hausdorff Operator on \(p\)-Adic Field”, \(p\)-Adic Numbers, Ultrametric Anal. Appl., 2 (2010), 252–259.
S. S. Volosivets, “Hausdorff Operators on p-Adic Linear Spaces and Their Properties in Hardy, BMO, and Hölder Spaces”, Math. Notes, 3 (2013), 382–391.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Dung, K.H., Duong, D.V. The \(p\)-Adic Hausdorff Operator and Some Applications to Hardy–Hilbert Type Inequalities. Russ. J. Math. Phys. 28, 303–316 (2021). https://doi.org/10.1134/S1061920821030043
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920821030043