Harmonic Analysis of Differential Operators in L^1

Abstract

This thesis studies two features of the endpoint \(L^1\) failure for Calder\'{o}n-Zygmund inequalities. Part \ref{SW_part} concerns alternative, replacement inequalities in the form of Stein-Weiss inequalities. These inequalities generalise Sobolev inequalities for elliptic differential operators, which are implied by Calder\'{o}n-Zygmund inequalities in \(L^p\), \(p>1\). Part \ref{Ornstein_part} studies a mechanism describing the \(L^1\) failure of Calder\'{o}n-Zygmund known as Ornstein's non-inequality.

In Part \ref{SW_part} of this thesis, we study Stein-Weiss inequalities with \(L^1\) data. In Chapter \ref{paper} we extend the \(L^1\) Stein-Weiss inequalities studied by De N\'{a}poli and Picon \cite{NapPic} in two ways: First we address the necessity of the cocanceling condition appearing in their Stein-Weiss inequalities. We replace the cocanceling condition with a weaker vanishing moment assumption, and under this assumption extend the \(L^1\) Stein-Weiss inequalities to \(L^1\big(|x|^{a } \ dx\big)\) data for all positive, non-integer exponents \(a\). Second, in relation to integer exponents, while \cite{NapPic} showed that Stein-Weiss fails for \(L^1\big(|x| \ dx\big)\) data, we prove a weaker Korn type Hardy-Sobolev inequality. These inequalities were previously inaccessible due to the growth of \(|x|\), and we demonstrate a specific example on \(\R^2\) of where the original duality estimate by Bousquet and Van Schaftingen \cite{can} for canceling operators can be improved.

In Part \ref{Ornstein_part} we study integral bounds relating to Ornstein's non-inequality, which states that an \(L^1\) inequality between constant coefficient linear differential operators holds if and only if a linear dependence relation holds between the partial derivatives. This immediately reduces the \(L^1\) inequality to a consequence of linear algebra. We first review how positive 1-homogeneity and \textit{quasiconvexity} (in the sense of Morrey) can be used to derive Ornstein's non-inequality, following the work of \cite{KK}. Then, we explore suitable generalisations of this theorem, by weakening the 1-homogeneity assumption via a recession integrand. We also extend this result to the mixed smoothness setting originally found in \cite{stoly}. This part recounts from the author's point of view their joint work (in preparation) with Bernd Kirchheim and Jan Kristensen.