Singular oscillatory integrals on ?~n

摘要

Let P_(d,n) denote the space of all real polynomials of degree at most d on ?_n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ∈ P_(d,1). Using this estimate, we prove that for some absolute positive constant c and every function Ω with zero mean value on the unit sphere S~(n-1). This improves a result of Stein (Ann Math Stud 112:307-355, 1986).