On extremizers for certain inequalities of the k-plane transform and related topics

This dissertation is concerned with determining optimal constants and extremizers, functions which achieve them, for certain inequalities arising in harmonic analysis.;The main inequality considered is the Lp- Lq inequality for the k-plane transform. It was shown in [ii] that the k-plane transform is a bounded operator from Lp of Euclidean space to Lq of the Grassmann manifold of all affine k-planes in Rd for certain exponents depending on k and d. Specifically, for 1 ≤ q ≤ d + 1 and p = dq/n-d+dq there exists a finite positive constant A0 > 0 such that [ [special characters omitted].;Extremizers of the inequality have previously been shown to exist when q =2 by Baernstein and Loss [3] , when k = 2 and q is an integer, also in [3], when k = d - 1 and q = d + 1 by Christ [12], and when q = d + 1 for general k by Drouot [17]. In each of these cases, f 0(x) = (1= |x|2) [special characters omitted] is an extremizer. When q = 2 [3] or k = n - 1 and q = d + 1 [12] this extremizer has been shown to be unique up to composition with certain explicit symmetries of the inequality.;Chapter 3 contains two proofs that when q is an integer, there exist extremizers, functions which achieve equality in the inequality with the sharp constant.;Chapter 4 extends Christ's uniqueness result for the endpoint case from k = n - 1 to general k. In particular, we show that for q = d + 1 for k ∈ [1, d - 1], the extremizing function is unique up to composition with affine maps. This is achieved by modifying the methods of [12] to apply to functions which are only assumed to be measurable Lp functions (rather than smooth L p functions).;Chapter 6 shows that when q and (1/p - 1) are both integers, all extremizers are infinitely differentiable. This involves a family of weighted inequalities for the k-plane transform and the analysis of a nonlinear Euler-Lagrange equation.;Chapter 7, considers the related question of extremizing n-tuples of characteristic functions for certain multilinear inequalities of Hardy-Riesz-Brascamp-Lieb- Luttinger-Rogers type. Extremizing n-tuples are characterized in a special case. This chapter is joint work with Christ.