Oscillatory singular integrals with radial rough kernel and fewnomials phases

Boyu Jiang

doi:10.1515/gmj-2025-2041

Article

Oscillatory singular integrals with radial rough kernel and fewnomials phases

Boyu Jiang

Published/Copyright: June 17, 2025

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From the journal Georgian Mathematical Journal

Abstract

In this paper, we study oscillatory singular integrals with radial rough kernels:

T Q , h ⁢ f ⁢ ( x ) = p.v. ⁢ ∫ ℝ n f ⁢ ( x - y ) ⁢ Ω ⁢ ( y ) ⁢ h ⁢ ( | y | ) | y | n ⁢ e i ⁢ Q ⁢ ( | y | ) ⁢ d y ,

where Q ⁢ ( t ) is a real polynomial on ℝ , Ω ∈ H ⁢ ( S n - 1 ) is a homogeneous function of degree zero on the unit sphere S n - 1 with vanishing mean value, and h satisfies the following conditions:

1 R ⁢ ∫ 0 R | h ⁢ ( t ) | q ⁢ d t ≤ C , | 1 p - 1 2 | < min ⁡ { 1 2 , 1 q ′ } ,

where C is a constant independent of R > 0 , and h is the radial rough kernel. Under the assumption Ω ∈ H 1 ⁢ ( S n - 1 ) , we prove that T Q , h is bounded on the Lebesgue space L p for 1 < p < ∞ , provided that h satisfies aforementioned condition. The methods we mainly use are created by S. Guo.

Keywords: Fewnomials phases; van der Corput lemma; oscillatory singular integrals

MSC 2020: 42B20; 42B30

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Received: 2024-10-09

Revised: 2025-03-09

Accepted: 2025-03-21

Published Online: 2025-06-17

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https://doi.org/10.1515/gmj-2025-2041

Keywords for this article

Fewnomials phases; van der Corput lemma; oscillatory singular integrals