Estimates for an oscillatory integral operator related to restriction to space curves

Abstract

We consider the oscillatory integral operator de fined by T(lambda)f(x) = integral(R)e(ilambdaphi(x,t))a(x, t)f(t)dt where lambda > 1, a 2 C-c(infinity) (R-n x R) and phi is a real-valued function in C-infinity(R-n x R). This operator may be thought of as a variable-curve version of the adjoint of the Fourier restriction operator for space curves. Under a certain nondegeneracy condition on phi, we obtain L-p - L-q estimates for T-lambda with a suitable bound for the operator norm parallel toT(lambda)parallel to(Lp)-->(Lq). This generalizes a result of Hormander for the plane to higher dimensions.