Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups

Ushangi Goginava; Farrukh Mukhamedov

doi:10.33205/cma.1893230

Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups

Abstract

In the theory of orthonormal systems, the study of summability methods plays a central role. For the bounded Vilenkin system, a fundamental theorem asserts that the $L_{p}$-norm convergence of certain operator sequences follows from a mild $\ell_{\gamma}$-type condition on the coefficients, without requiring any positivity or monotonicity assumptions. In this paper, we extend this classical result (Theorem~V) to the setting of almost everywhere convergence. In particular, we show that if the coefficients $\{t_{k,n}\}$ satisfy \[ \left\Vert (t_{k,n})_{k=0}^{n}\right\Vert_{\ell_{\gamma}} = O\!\left(n^{-1/\gamma'}\right), \qquad 1 < \gamma \le 2, \] then the corresponding sequence converges almost everywhere with respect to the bounded Vilenkin system. This work continues the line of research initiated by Blahota and Gát (J. Geom. Anal., 2025). Moreover, we address an open problem posed by Blahota and Gát, providing both affirmative and negative results. Finally, we present several applications of the main theorem to classical and generalized summability methods, thereby illustrating its broad applicability within the context of Fourier analysis.

Keywords

Ethical Statement

The authors declare that this manuscript is original, has not been published previously, and is not under consideration elsewhere. All authors have contributed significantly to the work and approved the final version. The paper contains no plagiarism, fabricated, or falsified results. All sources are properly cited. The authors declare that there is no conflict of interest.

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Details

Primary Language

English

Subjects

Lie Groups, Harmonic and Fourier Analysis

Journal Section

Research Article

Authors

Ushangi Goginava ^*
0000-0002-8292-9715
United Arab Emirates

Farrukh Mukhamedov
0000-0001-5728-6394
United Arab Emirates

Early Pub Date

October 21, 2026

Publication Date

-

Submission Date

February 19, 2026

Acceptance Date

August 30, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.33205/cma.1893230

IZ

https://izlik.org/JA83RX67PB

Cite

RIS / Bibtex

APA

Goginava, U., & Mukhamedov, F. (2026). Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups. Constructive Mathematical Analysis, Advanced Online Publication, 199-216. https://doi.org/10.33205/cma.1893230

AMA

1.Goginava U, Mukhamedov F. Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups. CMA. 2026;(Advanced Online Publication):199-216. doi:10.33205/cma.1893230

Chicago

Goginava, Ushangi, and Farrukh Mukhamedov. 2026. “Weak-Type Inequalities for Maximal Operators Associated With Sequences on Bounded Vilenkin Groups”. Constructive Mathematical Analysis, no. Advanced Online Publication: 199-216. https://doi.org/10.33205/cma.1893230.

EndNote

Goginava U, Mukhamedov F (October 1, 2026) Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups. Constructive Mathematical Analysis Advanced Online Publication 199–216.

IEEE

[1]U. Goginava and F. Mukhamedov, “Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups”, CMA, no. Advanced Online Publication, pp. 199–216, Oct. 2026, doi: 10.33205/cma.1893230.

ISNAD

Goginava, Ushangi - Mukhamedov, Farrukh. “Weak-Type Inequalities for Maximal Operators Associated With Sequences on Bounded Vilenkin Groups”. Constructive Mathematical Analysis. Advanced Online Publication (October 1, 2026): 199-216. https://doi.org/10.33205/cma.1893230.

JAMA

1.Goginava U, Mukhamedov F. Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups. CMA. 2026;:199–216.

MLA

Goginava, Ushangi, and Farrukh Mukhamedov. “Weak-Type Inequalities for Maximal Operators Associated With Sequences on Bounded Vilenkin Groups”. Constructive Mathematical Analysis, no. Advanced Online Publication, Oct. 2026, pp. 199-16, doi:10.33205/cma.1893230.

Vancouver

1.Ushangi Goginava, Farrukh Mukhamedov. Weak-Type Inequalities for Maximal Operators Associated with Sequences on Bounded Vilenkin Groups. CMA. 2026 Oct. 1;(Advanced Online Publication):199-216. doi:10.33205/cma.1893230