Methods for the analysis of oscillatory integrals and Bochner-Riesz operators

Abstract

For a smooth surface Γ of arbitrary codimension, one can consider the Lp mapping properties of the Bochner-Riesz multiplier m(ζ) = dist(ζ,Γ)^α φ(ζ), where α > 0 and φ is an appropriate smooth cutoff function. Even for the sphere, the exact Lp boundedness range remains a central open problem in Euclidean harmonic analysis. We consider the Lp integrability of the Bochner-Riesz convolution kernel for a particular class of surfaces (of any codimension). For a subclass of these surfaces the range of Lp integrability of the kernels differs substantially from the Lp boundedness range of the corresponding Bochner-Riesz multiplier operator. Extending work of Mockenhoupt, we then establish a range of operator bounds, which are sharp in the α exponent, under the assumption of an appropriate L2 restriction estimate. Hickman and Wright established sharp oscillatory integral estimates, associated with a particular class of surfaces, and derived restriction estimates. We extend this work to certain curves of standard type and corresponding surfaces of revolution. These surfaces are discussed as an explicit class for which we have Lp → Lp boundedness of the corresponding Bochner-Riesz operators.

Understanding the structure of the roots of real polynomials is important in obtaining stable bounds for oscillatory integrals with polynomial phases. For real polynomials with exponents in some fixed set, Ψ(t)=x+y1 t^{k1} +...+yL t^{kL}, we analyse the different possible root structures that can occur as the coefficients vary. We first establish a stratification of roots into tiers containing roots of comparable sizes. We then show that at most L non-zero roots can cluster about a point. Supposing additional restrictions on the coefficients, we derive structural refinements. These structural results extend work of Kowalski and Wright and provide a characteristic picture of root structure at coarse scales. As an application, these results are used to recover the sharp oscillatory integral estimates of Hickman and Wright, using bounds for oscillatory integrals of Phong and Stein.