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On Rough Parametric Marcinkiewicz Integrals Along Certain Surfaces

Author

Listed:
  • Mohammed Ali

    (College of Integrative Studies, Abdullah Al-Salem University, Firdous Street, Khaldiya 72303, Kuwait
    Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan)

  • Hussain Al-Qassem

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

Abstract

In this paper, we study rough Marcinkiewicz integrals associated with surfaces defined by Ψ P , ϕ = { ( ˜ P ( w ) , ϕ ( w ) ) : w ∈ R m }. We establish the L p -boundedness of these integrals when the kernel functions lie in the L q ( S m − 1 ) space. Combining this result with Yano’s extrapolation technique, we further obtain the L p -boundedness under weaker kernel conditions—specifically, when the kernels belong to either the block space B q ( 0 , − 1 / 2 ) ( S m − 1 ) or L ( log L ) 1 / 2 ( S m − 1 ) . Our results extend and refine several previously known results on Marcinkiewicz integrals, offering broader applicability and sharper conclusions.

Suggested Citation

  • Mohammed Ali & Hussain Al-Qassem, 2025. "On Rough Parametric Marcinkiewicz Integrals Along Certain Surfaces," Mathematics, MDPI, vol. 13(8), pages 1-10, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1287-:d:1634573
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