UC Berkeley

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 $L^p$-$L^q$ inequality for the $k$-plane transform. It was shown in cite{C84} that the $k$-plane transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the Grassmann manifold of all affine $k$-planes in $R^d$ for certain exponents depending on $k$ and $d$. Specifically, for $1leq qleq d+1$ and $p=frac{dq}{n-d+dq}$ there exists a finite positive constant $A_0>0$ such that

[

|T f|_{L^q(M)} leq A_0 |f|_{L^{p}(R^d)}.

]

Extremizers of the inequality have previously been shown to exist when $q=2$ by Baernstein and Loss cite{BL97}, when $k=2$ and $q$ is an integer, also in cite{BL97}, when $k=d-1$ and $q=d+1$ by Christ cite{C11}, and when $q=d+1$ for general $k$ by Drouot cite{D11}. In each of these cases, $f_0(x)= (1=|x|^2)^{frac{-(d-k)}{2(p-1)}}$ is an extremizer. When $q=2$ cite{BL97} or $k=n-1$ and $q=d+1$ cite{C11} this extremizer has been shown to be unique up to composition with certain explicit symmetries of the inequality.

Chapter ref{chap:exist} contains two proofs that when $q$ is an integer, there exist extremizers, functions which achieve equality in the inequality with the sharp constant.

Chapter ref{chap:unique} 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 $kin[1,d-1]$, the extremizing function is unique up to composition with affine maps. This is achieved by modifying the methods of cite{C11} to apply to functions which are only assumed to be measurable $L^p$ functions (rather than smooth $L^p$ functions).

Chapter ref{chap:smooth} shows that when $q$ and $frac{1}{p-1}$ 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 ref{chap:IBL}, 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.