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

We consider the oscillatory integral operator defined by

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 $\Vert T_\lambda \Vert _{L^p\to L^q}$. This generalizes a result of Hörmander for the plane to higher dimensions.