I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the form $$ \left\vert \int f\,\text{d}\mu \right\vert \leq \Vert \mu\Vert \cdot \Vert f\Vert $$ for a norm $\Vert f\Vert $ that could be naturally evaluated, say, for $f\in C^\infty_c(\mathbb R^n) $. This inequality should not entirely destroy the cancellations coming from the fact that $\mu $ is complex, like happens in $$\left\vert \int f\,\text{d}\mu \right\vert \leq \vert \mu\vert(X) \cdot \Vert f\Vert_\infty. $$ The total variation norm goes in that direction, but seems itself too rigid for the applications I have in mind. I'm sure it has been generalized in many ways, but have a hard time finding good references. I would be thankful for any hints!
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

You may look at the KantorovichRubinstein norm mentioned in this answer. It also goes under the name "flat norm" but I don't know a reference for this. Standard references for these things are the "Measure Theory" books by Bogachev and "Optimal Transport: Old and New" by Villani. 

