Weighted Lp Estimates for Multiple Generalized Marcinkiewicz Functions

Ali, Mohammed; Al-Qassem, Hussain

doi:10.3390/sym18020239

Open AccessArticle

Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions

by

Mohammed Ali

^1,*

and

Hussain Al-Qassem

²

¹

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

²

Department of Mathematics and Statistics, Qatar University, Doha 2713, Qatar

^*

Author to whom correspondence should be addressed.

Symmetry 2026, 18(2), 239; https://doi.org/10.3390/sym18020239

Submission received: 17 December 2025 / Revised: 24 January 2026 / Accepted: 27 January 2026 / Published: 29 January 2026

(This article belongs to the Section B: Mathematics)

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Abstract

In this paper we investigate the weighted

L^{p}

boundedness of generalized Marcinkiewicz integrals

M_{K}^{(ε)}

over multiple symmetric domains. Under the conditions

K \in L^{q} (S^{m - 1} \times S^{n - 1})

,

q > 1

, we establish suitable weighted

L^{p}

bounds for the integrals

M_{K}^{(ε)}

. These bounds are combined with Yano’s extrapolation argument to obtain the weighted

L^{p}

boundedness of

M_{K}^{(ε)}

from the Triebel–Lizorkin space

{\dot{F}}_{p}^{0, ε} (ω_{1}, ω_{2})

to the space

L^{p} (ω_{1}, ω_{2})

under the weaker conditions

K \in B_{q}^{(0, \frac{2}{ε} - 1)} (S^{m - 1} \times S^{n - 1}) \cup L {(\log L)}^{2 / ε} (S^{m - 1} \times S^{n - 1})

. Our findings are essential improvements and extension of several known findings in the literature.

Keywords: weighted L^p bounds; block space; generalized Marcinkiewicz operator; product domains; Triebel-Lizorkin space; extrapolation

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MDPI and ACS Style

Ali, M.; Al-Qassem, H. Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions. Symmetry 2026, 18, 239. https://doi.org/10.3390/sym18020239

AMA Style

Ali M, Al-Qassem H. Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions. Symmetry. 2026; 18(2):239. https://doi.org/10.3390/sym18020239

Chicago/Turabian Style

Ali, Mohammed, and Hussain Al-Qassem. 2026. "Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions" Symmetry 18, no. 2: 239. https://doi.org/10.3390/sym18020239

APA Style

Ali, M., & Al-Qassem, H. (2026). Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions. Symmetry, 18(2), 239. https://doi.org/10.3390/sym18020239

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Weighted L^p Estimates for Multiple Generalized Marcinkiewicz Functions

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