Oscillatory integrals and weighted gradient flows

We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far off from being sharp. In addition, the theorem provides a new proof, up to endpoints, that the well-known oscillatory integral estimates of Varchenko [V] when the Newton polyhedron of the phase function is nondegenerate extend to corresponding bounds for surface measure Fourier transforms when the index is less than $\frac{1}{2}$. A sharp version of this was originally proven in [G2].