Certain oscillatory integrals on unit square and their applications

Abstract

National Natural Science Foundation of China [10571122, 10371046]; Natural Science Foundation of Fujian Province of China [Z0511004]Let Q(2) = [0, 1](2) be the unit square in two dimension Euclidean space R(2). We study the L(p) boundedness properties of the oscillatory integral operators T(alpha,beta) defined on the set S(R(3)) of Schwartz test functions f by T(alpha,beta)f(x, y, z) = integral(Q2) f(x - t, y - s, z - t(k) s(j))e(-it-beta 1s-beta 2) t(-1-alpha 1)s(-1-alpha 2)dtds, where beta(1) > alpha(1) >= 0, beta(2) > alpha(2) >= 0 and (k, j) is an element of R(2). As applications, we obtain some L(p) boundedness results of rough singular integral operators on the product spaces