A class of rough generalized Marcinkiewicz integrals along twisted surfaces

Hussain Al-Qassem; Mohammed Ali; Hussain Al-Qassem; Mohammed Ali

doi:10.3934/math.2025650

AIMS Mathematics

2025, Volume 10, Issue 6: 14459-14471. doi: 10.3934/math.2025650

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A class of rough generalized Marcinkiewicz integrals along twisted surfaces

Hussain Al-Qassem ¹,
Mohammed Ali ^{2,3
,
,}

1.
Department of Mathematics and Statistics, Qatar University, Doha, Qatar
2.
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan
3.
College of Integrative Studies, Abdullah Al-Salem University, Khaldiya, Kuwait

Received: 26 April 2025 Revised: 16 June 2025 Accepted: 18 June 2025 Published: 23 June 2025
MSC : 42B20, 42B25

In this paper, we investigated the mapping properties of generalized Marcinkiewicz integral operators associated with twisted surfaces. Under certain conditions on these surfaces, we established suitable $ L^p $ estimates for these operators, assuming the kernel functions belong to $ L^{q}\left(\mathbb{U}^{{{j}}-1}\times \mathbb{U} ^{{{k}}-1}\right) $. By combining these estimates with Yano's extrapolation technique, we further established the boundedness of these operators from the homogeneous Triebel-Lizorkin space $ \overset{.}{F}_{p}^{0, \tau}(\mathbb{R}^j\times\mathbb{R}^k) $ to the space $ L^p(\mathbb{R}^j\times\mathbb{R}^k) $ under significantly weaker assumptions on the kernels. Our results extended and improved many previously known results.
- boundedness,
- rough kernels,
- generalized Marcinkiewicz integrals,
- twisted surfaces
Citation: Hussain Al-Qassem, Mohammed Ali. A class of rough generalized Marcinkiewicz integrals along twisted surfaces[J]. AIMS Mathematics, 2025, 10(6): 14459-14471. doi: 10.3934/math.2025650

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Abstract

In this paper, we investigated the mapping properties of generalized Marcinkiewicz integral operators associated with twisted surfaces. Under certain conditions on these surfaces, we established suitable $ L^p $ estimates for these operators, assuming the kernel functions belong to $ L^{q}\left(\mathbb{U}^{{{j}}-1}\times \mathbb{U} ^{{{k}}-1}\right) $. By combining these estimates with Yano's extrapolation technique, we further established the boundedness of these operators from the homogeneous Triebel-Lizorkin space $ \overset{.}{F}_{p}^{0, \tau}(\mathbb{R}^j\times\mathbb{R}^k) $ to the space $ L^p(\mathbb{R}^j\times\mathbb{R}^k) $ under significantly weaker assumptions on the kernels. Our results extended and improved many previously known results.

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