L-p inequalities for certain generalized Radon transforms.

摘要

This thesis concerns related to optimal or near-optimal L p → Lq mapping properties of generalized Radon transforms. We prove the following results.;In Part I, we establish strong-type endpoint LpRd →Lq Rd bounds for the operator given by convolution with affine arclength measure on polynomial curves. We also obtain a nearly optimal improvement in Lorentz spaces. The bounds established depend only on the dimension d and the degree of the polynomial. This is a generalization of work carried out by Dendrinos, Laghi, and Wright in dimensions 2 and 3. In Chapter 2, we establish the theorem in the model case of convolution with affine arclength along (t,t2,...,td). In Chapter 3, we extend the arguments in the previous chapter to the general polynomial case.;In Part II, we characterize (up to endpoints) the k-tuples (p1,...,pk) for which certain k-linear generalized Radon transforms map Lp1 x · · · x Lpk boundedly into R . This generalizes a result of Tao and Wright.;Part III concerns operators T defined by integrating along the zero sets of a function phi which satisfies the rotational curvature condition of Phong and Stein. Such operators are well-known to be bounded from Ld+1/d Rd→ Ld+1Rd . We say that (E, F) is a quasi-extremal pair of sets for T if ⟨TchiE, chi F⟩ ≳ |E|d/( d+1)|F|d /(d+1). In Part III, we begin work to characterize the quasi-extremal pairs of sets for operators of this form, extending work carried out by M. Christ for convolution with surface measure on the paraboloid.