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. integral(Rn) f (x - y) Omega(y)/|y|(n) e(iQ(| y|)) dy, where Q(t) = Sigma(1 <= i <= m) a(i)t(alpha i) is a real-valued polynomial on R, Omega is a homogenous function of degree zero on R-n with mean value zero on the unit sphere Sn-1. Under the assumption of that Omega is an element of H-1(Sn-1), the authors show that T-Q is bounded on the weighted Lebesgue spaces L-p(omega) for 1 < p < infinity and omega is an element of (A) over tilde (I)(p)(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 omega = 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].