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) ,\dots, 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 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 1/2 . A sharp version of this was originally proven by the author [J. Funct. Anal. 262 (2012), no. 5, 2314–2348].