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

Year 2025, Issue: 51, 65 - 75, 30.06.2025

Cansu Keskin , Havva Nur Turkak

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

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.

Year 2025, Issue: 51, 65 - 75, 30.06.2025

Cansu Keskin , Havva Nur Turkak

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

Abstract

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.
There are 35 citations in total.

Details

Primary Language English
Subjects Lie Groups, Harmonic and Fourier Analysis
Journal Section Research Article
Authors

Cansu Keskin 0000-0002-0998-4419

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

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 Issue: 51

Cite

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 Keskin C, Turkak HN. Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. JNT. June 2025;(51):65-75. doi:10.53570/jnt.1696750
Chicago 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): 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 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, June2025, 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 (June2025), 65-75. https://doi.org/10.53570/jnt.1696750.
JAMA 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, 2025, pp. 65-75, doi:10.53570/jnt.1696750.
Vancouver Keskin C, Turkak HN. Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities. JNT. 2025(51):65-7.
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