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

Abstract:

We consider the oscillatory integral operator defined by \[ T_\lambda f(x)=\int _{\mathbb R} e^{i\lambda \phi (x,t)}a(x,t) f(t)dt\] where $\lambda >1$, $a\in C_c^\infty (\mathbb {R}^n\times \mathbb {R})$ and $\phi$ is a real-valued function in $C^\infty (\mathbb {R}^n\times \mathbb {R})$. 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 $\|T_\lambda \|_{L^p\to L^q}$. This generalizes a result of Hörmander for the plane to higher dimensions.