Home Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
Article
Licensed
Unlicensed Requires Authentication

Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator

  • Naqash Sarfraz and Ferit Gürbüz ORCID logo EMAIL logo
Published/Copyright: August 2, 2021

Abstract

In this paper, the boundedness of the Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of the p-adic fractional Hausdorff operator on weighted p-adic weak Lebesgue spaces. We also obtain the sufficient condition of commutators of the p-adic fractional Hausdorff operator by taking symbol function from Lipschitz spaces. Moreover, strong type estimates for fractional Hausdorff operator and its commutator on weighted p-adic Lorentz spaces are also acquired.

MSC 2010: 42B35; 26D15; 46B25; 47G10

Corresponding author: Ferit Gürbüz, Department of Mathematics, Kırklareli University, Kırklareli 39100, Turkey, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] I. V. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Singapore, World Scientific, 1994.10.1142/1581Search in Google Scholar

[2] A. V. Avestisov, A. H. Bikulov, and V. A. Osipov, “p-Adic description of Characterization relaxation in complex system,” J. Phys. A Math. Gen., vol. 36, pp. 4239–4246, 2003.10.1088/0305-4470/36/15/301Search in Google Scholar

[3] A. V. Avestisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “Application of p-adic analysis to models of spontaneous breaking of replica symmetry,” J. Phys. Math. Gen., vol. 32, pp. 8785–8791, 1999.10.1088/0305-4470/32/50/301Search in Google Scholar

[4] A. V. Avestisov, A. H. Bikulov, S. V. Kozyrev, and V. A. Osipov, “p-Adic models of ultrametric diffusion constrained by hierarchical energy landscape,” J. Phys. Math. Gen., vol. 35, pp. 177–189, 2002.10.1088/0305-4470/35/2/301Search in Google Scholar

[5] I. Y. Arefeva, B. Dragovich, P. Frampton, and I. V. Volovich, “The wave function of the universe and p-Adic gravity,” Mod. Phys. Lett. A, vol. 6, pp. 4341–4358, 1991.10.1142/S0217751X91002094Search in Google Scholar

[6] D. Dubischar, V. M. Gundlach, O. Steinkamp, and A. Khrennikov, “A p-adic model for the process of thinking disturbed by physiological and information noise,” J. Theor. Biol., vol. 197, no. 4, pp. 451–467, 1999. https://doi.org/10.1006/jtbi.1998.0887.Search in Google Scholar PubMed

[7] G. Parisi and N. Sourlas, “p-Adic numbers and replica symmetry break,” Eur. J. Phys. B., vol. 14, pp. 535–542, 2000. https://doi.org/10.1007/s100510051063.Search in Google Scholar

[8] V. S. Vladimirov and I. V. Volovich, “p-Adic quantum mechanics,” Commun. Math. Phys., vol. 123, pp. 659–676, 1989. https://doi.org/10.1007/bf01218590.Search in Google Scholar

[9] A. N. Kochubei, “Stochastic integrals and stochastic differential equations over the field of p-Adic numbers,” Potential Anal., vol. 6, pp. 105–125, 1997.Search in Google Scholar

[10] S. Haran, “Analytic potential theory over the p- adics,” Ann. Inst. Fourier, vol. 43, pp. 905–944, 1993. https://doi.org/10.5802/aif.1361.Search in Google Scholar

[11] S. Haran, “Riesz potential and explicit sums in arithmetic,” Invent Math., vol. 101, pp. 697–703, 1990. https://doi.org/10.1007/bf01231521.Search in Google Scholar

[12] A. Hussain, Sarfraz, I. Khan, and A. M. Alqahtani, “Estimates for commutators of bilinear fractional p-adic Hardy operator on Herz-type spaces,” J. Funct. Spaces Appl., vol. 2021, p. 7, 2021, Art no. 6615604. https://doi.org/10.1155/2021/6615604.Search in Google Scholar

[13] A. Hussain, N. Sarfraz, and F. Gürbüz, Sharp Weak Bounds for p-Adic Hardy Operators on p-Adic Linear Spaces, 2020, arXiv: 2002.08045.Search in Google Scholar

[14] S. V. Kozyrev, “Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics,” Proc. Steklov Inst. Math., vol. 274, pp. 1–84, 2011. https://doi.org/10.1134/s0081543811070017.Search in Google Scholar

[15] N. Sarfraz, N. Filali, A. Hussain, and F. Jarad, “Weighted estimates for commutator of rough p-adic fractional Hardy operator on weighted p-adic Herz- Morrey spaces,” J. Mat., vol. 2021, p. 14, 2021, Art no. 5559815. https://doi.org/10.1155/2021/5559815.Search in Google Scholar

[16] N. Sarfraz and A. Hussain, “Estimates for commutators of p-adic Hausdorff operator on Herz-Morrey spaces,” Mathematics, vol. 7, no. 2, p. 127, 2019. https://doi.org/10.3390/math7020127.Search in Google Scholar

[17] K. F. Andersen, “Boundedness of Hausdorff operator on Lp(Rn)${L}^{p}({\mathbb{R}}^{n})$, H1(Rn)${H}^{1}({\mathbb{R}}^{n})$ and BMO(Rn)$BMO({\mathbb{R}}^{n})$,” Acta Sci. Math., vol. 69, pp. 409–418, 2003.Search in Google Scholar

[18] A. K. Lerner and E. Liflyand, “Multidimensional Hausdorff operators on the real Hardy space,” Acta Sci. Math., vol. 83, pp. 79–86, 2007. https://doi.org/10.1017/s1446788700036399.Search in Google Scholar

[19] A. Ajaib and A. Hussain, “Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group,” Open Math., vol. 18, pp. 496–511, 2020. https://doi.org/10.1515/math-2020-0175.Search in Google Scholar

[20] V. I. Burenkov and E. Liflyand, “Hausdorff operators on Morrey-type spaces,” Kyoto J. Math., vol. 60, no. 1, pp. 93–106, 2020. https://doi.org/10.1215/21562261-2019-0035.Search in Google Scholar

[21] V. I. Burenkov and E. Liflyand, “On the boundedness of Hausdorff operators on Morrey-Herz spaces,” Eurasian Math. J., vol. 8, no. 2, pp. 97–104, 2017.Search in Google Scholar

[22] J. Chen, J. Dai, D. Fan, and X. Zhu, “Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces,” Sci. China Math., vol. 61, pp. 1647–1664, 2018. https://doi.org/10.1007/s11425-017-9246-7.Search in Google Scholar

[23] J. Chen, D. Fan, and J. Li, “Hausdorff operator on function spaces,” Chin. Ann. Math., vol. 33, pp. 537–556, 2012. https://doi.org/10.1007/s11401-012-0724-1.Search in Google Scholar

[24] J. Chen, S. Y. He, and X. Zhu, “Boundedness of Hausdorff operators on the power weighted Hardy spaces,” Appl. Math. J. Chin. Univ., vol. 32, no. 4, pp. 462–476, 2017. https://doi.org/10.1007/s11766-017-3523-3.Search in Google Scholar

[25] G. Gao, X. Wu, A. Hussain, and G. Zhao, “Some estimates of Hausdorff operators,” J. Math. Inequalities, vol. 9, no. 3, pp. 641–651, 2015. https://doi.org/10.7153/jmi-09-54.Search in Google Scholar

[26] K.-P. Ho, “Dilation operators and integral operators on amalgon spaces (Lp, lp),” Ricerche Matemat., vol. 68, pp. 661–677, 2019. https://doi.org/10.1007/s11587-019-00431-5.Search in Google Scholar

[27] A. Hussain and A. Ajaib, “Some weighted inequalities for Hausdorff operators and commutators,” J. Inequalities Appl., vol. 2018, p. 19, 2018. https://doi.org/10.1186/s13660-017-1588-4.Search in Google Scholar PubMed PubMed Central

[28] A. Hussain and A. Ajaib, “Some results for commutators of generalized Hausdorff operator,” J. Math. Inequalities, vol. 13, no. 4, pp. 1129–1146, 2019. https://doi.org/10.7153/jmi-2019-13-80.Search in Google Scholar

[29] A. Hussain and G. Gao, “Some new estimates for the commutators of n-dimensional Hausdorff operator,” Appl. Math., vol. 29, pp. 139–150, 2014. https://doi.org/10.1007/s11766-014-3169-3.Search in Google Scholar

[30] A. Hussain and G. Gao, “Multidimensional Hausdorff operators and commutators on Herz-type spaces,” J. Inequalities Appl., vol. 2013, p. 594, 2013. https://doi.org/10.1186/1029-242x-2013-594.Search in Google Scholar

[31] A. Hussain and N. Sarfraz, “The Hausdorff operator on weighted p-adic Morrey and Herz type spaces,” P-Adic Numbers Ultrametric Anal. Appl., vol. 11, pp. 151–162, 2019. https://doi.org/10.1134/s2070046619020055.Search in Google Scholar

[32] E. Liflyand and A. Miyachi, “Boundedness of multidimensional Hausdorff operator in Hp spaces, 0 < p < 1,” Trans. Am. Math. Soc., vol. 371, pp. 4793–4814, 2019.10.1090/tran/7572Search in Google Scholar

[33] J. Ruan, D. Fan, and Q. Wu, “Weighted Morrey estimates for Hausdorff operators and its commutator on the Heisenberg group,” Math. Inequalities Appl., vol. 22, no. 1, pp. 303–329, 2019.10.7153/mia-2019-22-24Search in Google Scholar

[34] X. Lin and L. Sun, “Some estimates on the Hausdorff operator,” Acta Sci. Math., vol. 78, pp. 669–681, 2012.10.1007/BF03651391Search in Google Scholar

[35] D. S. Fan and F. Y. Zhao, “Multilinear fractional Hausdorff operators,” Acta Math. Sin., vol. 30, pp. 1407–1421, 2014. https://doi.org/10.1007/s10114-014-3552-2.Search in Google Scholar

[36] G. Gao and F. Zhao, “Sharp weak bounds for Hausdorff operators,” Anal. Math., vol. 41, pp. 163–173, 2015. https://doi.org/10.1007/s10476-015-0204-4.Search in Google Scholar

[37] J. Chen, D. Fan, and J. Ruan, “The fractional Hausdorff operator on Hardy spaces Hp(Rn)${H}^{p}({\mathbb{R}}^{n})$,” Anal. Math., vol. 42, pp. 1–17, 2016. https://doi.org/10.1007/s10476-016-0101-5.Search in Google Scholar

[38] F. Gao, X. Hu, and C. Zhong, “Sharp weak estimates for Hardy-type operators,” Ann. Funct. Anal., vol. 7, no. 3, pp. 421–433, 2016. https://doi.org/10.1215/20088752-3605447.Search in Google Scholar

[39] Y. Haixia and L. Junferg, “Sharp weak estimates for n-dimensional fractional Hardy Operators,” Front. Math. China, vol. 13, no. 2, pp. 449–457, 2018.10.1007/s11464-018-0685-0Search in Google Scholar

[40] A. Hussain and N. Sarfraz, “Optimal weak type estimates for p-Adic Hardy operator,” P-Adic Numbers Ultrametric Anal. Appl., vol. 12, no. 1, pp. 12–21, 2020. https://doi.org/10.1134/s2070046620010033.Search in Google Scholar

[41] Q. Y. Wu and Z. W. Fu, “Weighted p-Adic Hardy operators and their commutators on p-adic central Morrey spaces,” Bull. Malaysian Math. Sci. Soc., vol. 40, pp. 635–654, 2015.10.1007/s40840-017-0444-5Search in Google Scholar

[42] Q. Y. Wu and Z. W. Fu, “Hardy-Littlewood-Sobolev inequalities on p-adic central Morrey spaces,” J. Funct. Spaces, vol. 2015, p. 7, 2015, Art no. 419532. https://doi.org/10.1155/2015/419532.Search in Google Scholar

Received: 2020-12-23
Revised: 2021-06-16
Accepted: 2021-07-20
Published Online: 2021-08-02

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
  4. Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
  5. Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
  6. Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
  7. Controllability of coupled fractional integrodifferential equations
  8. A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
  9. Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
  10. Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
  11. Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
  12. Two occurrences of fractional actions in nonlinear dynamics
  13. Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
  14. Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
  15. Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
  16. Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
  17. Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
  18. Pandemic management by a spatio–temporal mathematical model
  19. Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
  20. Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
  21. Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
  22. Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
  23. Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
  24. Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
Downloaded on 27.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2020-0290/pdf
Scroll to top button