AIMS Mathematics (Feb 2024)

Weighted L<sup>p</sup> norms of Marcinkiewicz functions on product domains along surfaces

  • Badriya Al-Azri,
  • Ahmad Al-Salman

DOI
https://doi.org/10.3934/math.2024408
Journal volume & issue
Vol. 9, no. 4
pp. 8386 – 8405

Abstract

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We prove a weighted $ L^{p} $ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space $ L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{m}, \, \omega _{1}(x)dx, \, \omega_{2}(y)dy) $, provided that the weights $ \omega_{1} $ and $ \omega_{2} $ are certain radial weights and that the kernels are rough in the optimal space $ L(\log L)(\mathbb{S}^{n-1}\times\mathbb{S}^{m-1}) $. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $ L^{p} $ boundedness of the related square and maximal functions. Our weighted $ L^{p} $ inequalities extend as well as generalize previously known $ L^{p} $ boundedness results.

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