In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring ine
qualities for magnetic Schrodinger operators on multi-dimensional cylinders.
In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are
imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.
We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method
is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlsons inequalities and spectral theory of discrete operators are given.
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $R^d$, $d geq 2$. In particular, we derive upper bounds on Riesz means of order $sigma geq 3/2$, that improve the sharp Berezin inequality by a negative s
econd term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.
