On singular integral operators along surfaces

Ahmad Al-Salman; Abdulla M. Jarrah

doi:10.1515/gmj-2025-2024

Article

On singular integral operators along surfaces

Ahmad Al-Salman and Abdulla M. Jarrah

Published/Copyright: March 28, 2025

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From the journal Georgian Mathematical Journal

Abstract

Suppose that Ω ∈ L 1 ⁢ ( 𝕊 n - 1 × 𝕊 n - 1 ) is a homogeneous function of degree zero in the sense

Ω ⁢ ( t ⁢ x , s ⁢ y ) = Ω ⁢ ( x , y ) for any ⁢ t , s > 0

satisfying the cancellation property

∫ 𝕊 n - 1 Ω ⁢ ( u , ⋅ ) ⁢ 𝑑 σ n ⁢ ( u ) = ∫ 𝕊 m - 1 Ω ⁢ ( ⋅ , v ) ⁢ 𝑑 σ m ⁢ ( v ) = 0 .

Under suitable growth conditions on the mappings ϕ 1 , ϕ 2 : ( 0 , ∞ ) → ℝ , we prove that the singular integral operator

T Ω ( ϕ 1 ↫ ϕ 2 ) ⁢ ( f ) ⁢ ( x , y ) = p.v. ⁢ ∫ ℝ n × ℝ m f ⁢ ( ( x - ϕ 1 ⁢ ( | v | ) ⁢ u , y - ϕ 2 ⁢ ( | u | ) ⁢ v ) ) ⁢ Ω ⁢ ( u , v ) | u | n ⁢ | v | m ⁢ 𝑑 u ⁢ 𝑑 v

is bounded on L p ⁢ ( ℝ n × ℝ m ) , 1 < p < ∞ , provided the kernel function Ω satisfies weak size conditions. Furthermore, we prove the boundedness of the related truncated maximal singular integral operators.

Keywords: Singular integral operators; product domains; rough kernels; Hardy–Littlewood maximal function; truncated maximal singular integrals; twisted surfaces

MSC 2020: 42B20; 42B15; 42B25

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Received: 2024-09-23

Accepted: 2024-12-20

Published Online: 2025-03-28

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https://doi.org/10.1515/gmj-2025-2024

Keywords for this article

Singular integral operators; product domains; rough kernels; Hardy–Littlewood maximal function; truncated maximal singular integrals; twisted surfaces