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!
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1$\begingroup$ I suggest you look at the book "Real analysis" by G. Folland. $\endgroup$– user23860Commented Jul 1, 2013 at 16:00
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1$\begingroup$ Without knowing more about $\mu$ and $f$ I don't think you can say too much -- even if you restrict to smooth compactly supported $f$, such $f$ can always see the variation in $\mu$. More precisely, in your second inequality, you can always choose $f$ smooth and compactly supported so as to make the inequality an "equality up to epsilon" $\endgroup$– Yemon ChoiCommented Jul 1, 2013 at 19:37
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1$\begingroup$ Ah, I misunderstod - you want to choose a different norm on the space of measures. $\endgroup$– Yemon ChoiCommented Jul 2, 2013 at 6:07
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You may look at the Kantorovich-Rubinstein 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.
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$\begingroup$ Thanks for the reference! I don't see why what is called Kantorovich-Rubinstein norm in your link would not destroy cancellations in a similar way as does the usual "total mass times supremum" inequality, since it is basically also only a dual to the sup norm (never mind if it's the sup on Hölder space or anything else). But I found some interesting inequalities in the Villani book and will look at them more carefully. $\endgroup$– martinCommented Jul 5, 2013 at 9:46
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$\begingroup$ Ah, I see what you are after. In fact the Kantorovich-Rubinstein norm may lead to a little less cancellation as the test function $f$ has to be Lipschitz, and hence, can not "track jumps in $\mu$" very quickly. $\endgroup$– DirkCommented Jul 6, 2013 at 11:06
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$\begingroup$ Not sure what you mean with "jumps in $\mu$", but if you mean "discontinuities in the density wrt. Lebesgue" (assuming that it has one), then, as Yemon Choi pointed out, even $C^\infty_c $ functions satisfy the "total mass times supremum" inequality arbitrarily well, so restricting to Hölder functions is of no help. I was hoping for something inspired by the total variation norm, perhaps using dyadic decompositions of Fourier space or something like that. Anyway, thanks for your comments, I'll do more literature review. $\endgroup$– martinCommented Jul 7, 2013 at 16:56