Wenqi will provide an overview of his PhD thesis by surveying relevant nearby results, and touch briefly on his own thesis work. We will explore two viewpoints arising from the failure of Calderon Zygmund inequalities in L^1, and this will divide our talk into two parts. 

 

In the first part we explore one way to repair the failure of these L^1 inequalities, via weaker Sobolev inequalities. We study this problem through a convenient abstraction and arrive at the Stein-Weiss inequalities, in which Wenqi's thesis work describes how the classical results can be extended into ranges which are not expected, under some mild additional assumptions. 

 

Moving to the second part, we discuss a mechanism for why the Calderon Zygmund inequalities fail. It turns out that an integral inequality featuring differential operators provides convexity information for an associated functional. The rigidity of this functional then prevents non-trivial L^1 inequalities for differential operators from occurring. Wenqi's thesis work (joint with Bernd Kirchheim and Jan Kristensen) studies generalisations of these results to functionals which are not positively 1-homogeneous. Our framework studies functionals of linear growth, which possesses a regular recession integrand.