Hardy inequalities for fractional integrals on general domains

We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.Comment: LaTeX, 9 pages, a number of errors have been fixed and the bibliography has been update

A simple proof of a theorem of Laptev and Weidl

A new and elementary proof of a recent result of Laptev and Weidl is given. It is a sharp Lieb-Thirring inequality for one dimensional Schroedinger operators with matrix valued potentials.Comment: Replaces the version of June 28, 1999. A technical error in the proof has been correcte

Hardy-Sobolev-Maz'ya inequalities for arbitrary domains

We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains \Omega\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schr\"odinger operators on domains.Comment: 19 page

Lieb-Thirring inequalities with improved constants

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators