A Class of Oscillatory Singular Integrals with Rough Kernels and Fewnomials Phases

This paper is concerned with the oscillatory singular integral operator T Q defined by T Q f ( x ) = p . v . ∫ R n f ( x - y ) Ω ( y ) | y | n e i Q ( | y | ) d y , where Q ( t ) = ∑ 1 ≤ i ≤ m a i t α i is a real-valued polynomial on R , Ω is a homogenous function of degree zero on R n with mean value zero on the unit sphere S n - 1 . Under the assumption of that Ω ∈ H 1 ( S n - 1 ) , the authors show that T Q is bounded on the weighted Lebesgue spaces L p ( ω ) for 1 < p < ∞ and ω ∈ A ~ p I ( R + ) with the uniform bound only depending on m , the number of monomials in polynomial Q , not on the degree of Q as in the previous results. This result is new even in the case ω ≡ 1 , which can also be regarded as an improvement and generalization of the result obtained by Guo in [New York J. Math. 23 (2017), 1733-1738].