Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities

Cansu Keskin; Havva Nur Turkak

doi:10.53570/jnt.1696750

EN

Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities

Abstract

In this paper, we first establish the relation between $B$-maximal and sharp $B$-maximal functions generated by the generalized translation operator connected with the Laplace-Bessel differential operator. We then prove some sharp $B$-maximal function estimates and present an application using these sharp estimates to study singular integral operators. We finally obtain the boundedness of the Littlewood-Paley $g$-function related to the Laplace-Bessel differential operator on generalized $B$-Morrey spaces.

Keywords

$B$-Morrey space, Laplace-Bessel differential operator, Littlewood-Paley $g$-function, sharp $B$-maximal function

References

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Transactions of American Mathematical Society 43 (1938) 126-166.
F. Chiarenza, M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rendiconti del Seminario Matematico della Università di Padova 7 (1987) 273-279.
V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in $\mathbb{R}^n$, Doctoral Dissertation Steklov Mathematical Institute (1994) Moscow.
V. S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, Journal of Inequalities and Applications 2009 (2009) 1-20.
Y. Sawano, A thought on generalized Morrey spaces, Journal of The Indonesian Mathematical Society 25 (3) (2019) 210-281.
E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Mathematische Nachrichten 166 (1994) 95-10.
J. E. Littlewood, R. E. A. C. Paley, Theorems on Fourier series and power series, Journal of the London Mathematical Society 6 (1931) 230-233.
J. E. Littlewood, R. E. A. C. Paley, Theorems on Fourier series and power series (II), Proceedings of the London Mathematical Society 42 (1) (1936) 52-89.
J. E. Littlewood, R. E. A. C. Paley, Theorems on Fourier series and power series (III), Proceedings of the London Mathematical Society 43 (1937) 105-126.
A. Zygmund, Trigonometric series, 3rd Edition, Cambridge University Press, 2002.

E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley Theory, Princeton University Press, 1970.
E. M. Stein, S. Wainger, Problems in harmonic analysis related to curvature, Bulletin of the American Mathematical Society 84 (1978) 1239-1295.
E. M. Stein, The development of square functions in the work of A. Zygmund, Bulletin of the American Mathematical Society 7 (1982) 359-376.
M. H. Taibleson, Harmonic analysis on n-dimensional vector spaces over local fields, Mathematische Annalen 187 (1970) 259-271.
A. Uchiyama, Characterization of $H^p(\mathbb{R}^n)$ in terms of generalized Littlewood-Paley $g$-functions, Studia Mathematica 81 (1985) 135-158.
S. L. Wang, Boundedness of the Littlewood-Paley $g$-function on $\text{Lip}_\alpha(\mathbb{R}^n)$ $(0 < \alpha < 1)$, Illinois Journal of Mathematics 33 (1989) 531-541.
A. Akbulut, V. S. Guliyev, M. Dziri, Weighted norm inequalities for the $g$-Littlewood-Paley operators associated with Laplace-Bessel differential operators, Mathematical Inequalities and Applications 17 (1) (2014) 317-333.
D. Chen, H. Huang, Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators, Open Mathematics 19 (2021) 1554-1566.
A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Advances in Mathematics 226 (2011) 3912-3926.
A. Akbulut, M. Dziri, I. Ekincioglu, Maximal function and fractional integral associated with Laplace-Bessel differential operators on generalized Morrey spaces, Transactions Issue Mathematics. National Academy of Sciences of Azerbaijan. Series of Physical-Technical & Mathematical Sciences 42 (1) (2022) 8-25.
J. Garcıa-Cuerva, J. R. De Francia, Weighted norm inequalities and related topics, Elsevier B.V., 1985.
B. M. Levitan, Bessel function expansions in series and Fourier integrals, Uspekhi Matematicheskikh Nauk 6 (2) (1951) 102-143.
I. A. Kipriyanov, Fourier-Bessel transforms and imbedding theorems for weight classes, Trudy Matematicheskogo Instituta imeni V. A. Steklova 89 (1967) 130-213.
L. N. Lyakhov, Multipliers of the mixed Fourier-Bessel transform, Proceedings of the Steklov Institute of Mathematics 214 (1997) 234-249.
A. Serbetci, I. Ekincioglu, On boundedness of Riesz potential generated by generalized translate operator on Ba spaces, Czechoslovak Mathematical Journal 54 (3) (2004) 579-589.
E. L. Shishkina, S. M. Sitnik, Transmutations, singular and fractional differential equations with applications to mathematical physics, Elsevier, 2020.
I. Ekincioglu, J. J. Hasanov, C. Keskin, On the boundedness of $B$-Riesz potential and its commutators on generalized weighted $B$-Morrey spaces, Hacettepe Journal of Mathematics and Statistics 53 (2) (2023) 321-332.
V.S. Guliyev, Sobolev theorems for anisotropic Riesz-Bessel potentials on Morrey-Bessel spaces, Doklady Akademii Nauk 367 (2) (1999) 155-156.
C. Fefferman, E. M. Stein, $H^p$ spaces of several variables, Acta Mathematica 129 (1972) 137-193.
I. Ekincioglu, C. Keskin, A. B. Ozyurt Serim, New estimates for the $B$-Riesz transform, Filomat 39 (11) (2025) 3729-3740.
C. Keskin, I. Ekincioglu, V. S. Guliyev, Characterizations of Hardy spaces associated with Laplace-Bessel operators, Analysis and Mathematical Physics 9 (4) (2019) 2281-2310.
G, Sheng, H. Chuangxia, L. Lanzhe, Sharp maximal function estimates and boundedness for commutator related to generalized fractional integral operator, Analele Universitatii de Vest, Timişoara Seria Matematica-Informatica L 2 (2012) 97-115.
Q. Xue, X. Peng, K. Yabuta, On the theory of multilinear Littlewood-Paley $g$-function, Journal of the Mathematical Society of Japan 67 (2) (2015) 535-559.
R. R. Coifman, R. Rochberg, Another characterization of BMO, Proceedings of the American Mathematical Society 79 (2) (1980) 249-254.
I. Ekincioglu, E. Kaya, Bessel type Kolmogorov inequalities on weighted Lebesgue spaces, Applicable Analysis 100 (8) (2021) 1634-1643.

Details

Primary Language

English

Subjects

Lie Groups, Harmonic and Fourier Analysis

Journal Section

Research Article

Authors

Cansu Keskin ^*
0000-0002-0998-4419
Türkiye

Havva Nur Turkak This is me
0009-0006-9752-2850
Türkiye

Early Pub Date

June 30, 2025

Publication Date

June 30, 2025

Submission Date

May 10, 2025

Acceptance Date

June 26, 2025

Published in Issue

Year 2025 Number: 51

DOI

https://doi.org/10.53570/jnt.1696750

IZ

https://izlik.org/JA29AU76HS

APA

Keskin, C., & Turkak, H. N. (2025). Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. Journal of New Theory, 51, 65-75. https://doi.org/10.53570/jnt.1696750

AMA

1.Keskin C, Turkak HN. Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. JNT. 2025;(51):65-75. doi:10.53570/jnt.1696750

Chicago

Keskin, Cansu, and Havva Nur Turkak. 2025. “Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities”. Journal of New Theory, nos. 51: 65-75. https://doi.org/10.53570/jnt.1696750.

EndNote

Keskin C, Turkak HN (June 1, 2025) Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. Journal of New Theory 51 65–75.

IEEE

[1]C. Keskin and H. N. Turkak, “Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities”, JNT, no. 51, pp. 65–75, June 2025, doi: 10.53570/jnt.1696750.

ISNAD

Keskin, Cansu - Turkak, Havva Nur. “Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities”. Journal of New Theory. 51 (June 1, 2025): 65-75. https://doi.org/10.53570/jnt.1696750.

JAMA

1.Keskin C, Turkak HN. Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. JNT. 2025;:65–75.

MLA

Keskin, Cansu, and Havva Nur Turkak. “Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities”. Journal of New Theory, no. 51, June 2025, pp. 65-75, doi:10.53570/jnt.1696750.

Vancouver

1.Cansu Keskin, Havva Nur Turkak. Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. JNT. 2025 Jun. 1;(51):65-7. doi:10.53570/jnt.1696750