Abstract
Our aim in this paper is to define \(p\)-adic Herz spaces with variable exponents and prove the boundedeness of \(p\)-adic intrinsic square function in these spaces.
Similar content being viewed by others
References
V. A. Avetisov, A. Kh. Bikulov and V. A. Osipov, “\(p\)-Adic models of ultrametric diffusion in the conformational dynamics of macromolecules,” Proc. Steklov Inst. Math. 245, 48–57 (2004).
V. A. Avetisov, A. Kh. Bikulov and V. A. Osipov, “\(p\)-Adic description of characteristic relaxation in complex systems,” J. Phys. A 36 (15), 4239–4246 (2003).
B. Dragovich, A. Yu. Khrennikov, S.V. Kozyrev and I. V. Volovich, “On \(p\)-adic mathematical physics,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 1 (1), 1–17 (2009).
W. Orlicz, “Uber konjugierte Exponentenfolgen,” Studia Math. 3, 200–211 (1931).
O. Kováčik Rákosnǐk and J. Kovačik, “On spaces \(L^{q(x)}\) and\( W^{k,q(x)}\),” Czechosl. Math. J. 41 (4) (116), 592–618 (1991).
L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 2011).
R. Aboulaich, S. Boujena and E. E. Guarmah, “On a non-linear model for image denoising,” Math. Rep. 345 (8), 425–429 (2007).
R. Aboulaich, D. Meskine and A. Souissi, “New diffusion models in image processing,” Comp. Math. Appl. 56 (4), 874–882 (2008).
E. Acerbi and G. Mingione, “Regularity results for electrorheological fluids, the stationary case,” C. R. Math. Acad. Sci. Paris 334 (9), 817–822 (2002).
E. Acerbi and G. Mingione, “Regularity results for stationary electrorheological fluids,” Arch. Ration. Mech. Anal. 164 (3), 213–259 (2002).
S. N. Antontsev and J. F. Rodrigues, “On stationary thermo-rheological viscous flows,” Ann. Univ. Ferrara 52, 19–36 (2006). https://doi.org/10.1007/s11565-006-0002-9.
M. Ruzicka, “Electrorheological fluids: modeling and mathematical theory,” Lect. Not. Math. 1748, 16–38 (2000).
M. Ruzicka, “Modeling, mathematical and numerical analysis of electrorheological fluids,” Appl. Math. 49 (6), 565–609 (2004).
P. Harjulehto, P. Hästöo, Ú. V. L e and M. Nuortio, “Overview of differential equations with non-standard growth,” Nonl. Anal.: Theory Meth. Appl. 72 (12), 4551–4574 (2010).
G. Mingione, “Regularity of minima: an invitation to the dark side of the calculus of variations,” Appl. Math. 51 (4), 355–426 (2006).
L. F. Chacón-Cortés and H. Rafeiro, “Fractional operators in \(p\)-adic variable exponent Lebesgue spaces and application to \(p\)-adic derivative,” J. Func. Spac. 2021, Art. ID 3096701 (2021).
L. F. Chacón-Cortés and H. Rafeiro, “Variable exponent Lebesgue spaces and Hardy-Littlewood maximal function on \(p\)-adic numbers,” \(p\)-Adic Num. Ultrametr. Anal. Appl. 12, 90–111 (2020).
N. Sarfraz, M. Aslam, M. Zaman and F. Jarad, “Estimates for \(p\)-adic fractional integral operator and its commutators on \(p\)-adic Morrey-Herz spaces,” J. Inequal. Appl. 2022, Art. number 92 (2022).
A. Hussain, N. Sarfraz, I. Khan, A. Alsubie and N. N. Hamadneh, “The boundedness of commutators of rough \(p\)-adic fractional Hardy type operators on Herz-type spaces,” J. Inequal. Appl. 2021, Art. number 123 (2021).
S. Bashir, B. Sultan, A. Hussain, A. Khan and T. Abdeljawad, “A note on the boundedness of Hardy operators in grand Herz spaces with variable exponent,” AIMS Math. 8 (9), 22178–22191 (2023).
A. Ajaib and A. Hussain, “Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group,” Open Math. 18, 496–511 (2020).
B. Sultan, M. Sultan, M. Mehmood, F. Azmi, M. A. Alghafli and N. Mlaik, “Boundedness of fractional integrals on grand weighted Herz spaces with variable exponent,” AIMS Math. 8 (1), , 752-764 (2023).
M. Sultan, B. Sultan, A. Aloqaily and N. Mlaiki, “Boundedness of some operators on grand Herz spaces with variable exponent,” AIMS Math. 8 (6), 12964–12985 (2023).
B. Sultan, F. Azmi, M. Sultan, T. Mahmood, N. Mlaiki and N. Souayah, “Boundedness of fractional integrals on grand weighted Herz-Morrey spaces with variable exponent,” Fractal Fract. 2024 6 (11), 660 (2022).
B. Sultan, F. Azmi, M. Sultan, M. Mehmood and N. Mlaiki, “Boundedness of Riesz potential operator on grand Herz-Morrey spaces,” Axioms 11 (11), , 583 (2022).
M. Sultan, B. Sultan, A. Khan and T. Abdeljawad, “Boundedness of Marcinkiewicz integral operator of variable order in grand Herz-Morrey spaces,” AIMS Math. 8 (9), 22338–22353 (2023).
B. Sultan, M. Sultan, Q. Q. Zhang and N. Mlaiki, “Boundedness of Hardy operators on grand variable weighted Herz spaces,” AIMS Math. 8 (10), 24515–24527 (2023).
Funding
This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Publisher’s note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sultan, M., Sultan, B. Estimate for the Intrinsic Square Function on \(p\)-Adic Herz Spaces with Variable Exponent. P-Adic Num Ultrametr Anal Appl 16, 82–93 (2024). https://doi.org/10.1134/S2070046624010072
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046624010072