We prove a decomposition theorem for the equivariant $K$-theory of actions of affine group schemes $G$ of finite type over a field on regular separated Noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition that holds, for example, for $G$ abelian or smooth. We reduce the problem to the case of a ${\rm GL}\sb n$-action and finally to a split torus action.

For a large class of smooth dagger spaces–rigid spaces with overconvergent structure sheaf–we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P. Berthelot's rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology in situations of semistable reduction.

We prove a multiple integral inequality for convex domains in $\mathbf {R}\sp n$ of finite inradius. This inequality is a version of the classical inequality of H. Brascamp, E. Lieb, and J. Luttinger, but here, instead of fixing the volume of the domain, one fixes its inradius $r\sb D$ and the ball is replaced by $(-r\sb D, r\sb D)\times $\mathbf {R}\sp {n-1}$. We also obtain a sharper version of our multiple integral inequality, which generalizes the results in [6], for two-dimensional bounded convex domains where we replace infinite strips by rectangles. It is well known by now that the Brascamp-Lieb-Luttinger inequality provides a powerful and elegant method for obtaining and extending many of the classical geometric and physical isoperimetric inequalities of G. Pólya and G. Szegö. In a similar fashion, the new multiple integral inequalities in this paper yield various new isoperimetric-type inequalities for Brownian motion and symmetric stable processes in convex domains of fixed inradius which refine in various ways the results in [2], [3], [4], [5], and [20]. These include extensions to heat kernels, heat content, and torsional rigidity. Finally, our results also apply to the processes studied in [10] whose generators are relativistic Schrödinger operators.

Let $k$ be a local field of characteristic not $2$, and let $G$ be the group of $k$-rational points of a connected reductive linear algebraic group defined over $k$ with a simple derived group of $k$-rank at least $2$. We construct new uniform pointwise bounds for the matrix coefficients of all infinite-dimensional irreducible unitary representations of $G$. These bounds turn out to be optimal for ${\rm SL}\sb n(k), n\geq 3$, and ${\rm Sp}\sb {2n}(k),n\geq 2$. As an application, we discuss a simple method of calculating Kazhdan constants for various compact subsets of semisimple $G$.