Generalizations of Herz-Morrey spaces and boundedness of the Calderón-Zygmund operators

Ghada AlNemer; Ghada Ali Basendwah; Mehvish Sultan; Ioan-Lucian Popa; Ghada AlNemer; Ghada Ali Basendwah; Mehvish Sultan; Ioan-Lucian Popa

doi:10.3934/math.2025778

AIMS Mathematics

2025, Volume 10, Issue 8: 17403-17422. doi: 10.3934/math.2025778

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Generalizations of Herz-Morrey spaces and boundedness of the Calderón-Zygmund operators

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics King Abdulaziz University, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan
4.
Department of Computing, Mathematics and Electronics, "1 Decembrie 1918" University of Alba Iulia, 510009 Alba Iulia, Romania
5.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

Received: 01 April 2025 Revised: 17 July 2025 Accepted: 25 July 2025 Published: 01 August 2025
MSC : 46E30, 47B38

In this paper, we use the grand variable Herz-Morrey spaces, and our main objective is to prove the boundedness of multilinear Calderón-Zygmund operators on the product of grand variable Herz-Morrey spaces. We first define the Lebesgue spaces with variable exponents and some basic lemmas including Hölder's inequality for Lebesgue spaces. To prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz-Morrey spaces, we use the well known Hölder's inequality and Minkonwski's inequality. We split the summation of grand variable Herz-Morrey spaces to find the $ L^{p(\cdot)}(\mathbb{R}^n) $ estimate of the characteristic function under some conditions. After finding estimate of each term, we conclude that multilinear Calderón-Zygmund operators are bounded on grand variable Herz-Morrey spaces. Our results generalize some results on variable Herz spaces and grand variable Herz spaces.
- multilinear Calderón-Zygmund operators,
- grand Herz-Morrey spaces,
- grand Herz spaces
Citation: Ghada AlNemer, Ghada Ali Basendwah, Mehvish Sultan, Ioan-Lucian Popa. Generalizations of Herz-Morrey spaces and boundedness of the Calderón-Zygmund operators[J]. AIMS Mathematics, 2025, 10(8): 17403-17422. doi: 10.3934/math.2025778

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Abstract

In this paper, we use the grand variable Herz-Morrey spaces, and our main objective is to prove the boundedness of multilinear Calderón-Zygmund operators on the product of grand variable Herz-Morrey spaces. We first define the Lebesgue spaces with variable exponents and some basic lemmas including Hölder's inequality for Lebesgue spaces. To prove the boundedness of multilinear Calderón-Zygmund operators on grand variable Herz-Morrey spaces, we use the well known Hölder's inequality and Minkonwski's inequality. We split the summation of grand variable Herz-Morrey spaces to find the $ L^{p(\cdot)}(\mathbb{R}^n) $ estimate of the characteristic function under some conditions. After finding estimate of each term, we conclude that multilinear Calderón-Zygmund operators are bounded on grand variable Herz-Morrey spaces. Our results generalize some results on variable Herz spaces and grand variable Herz spaces.

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