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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. Inparticular, for convex domains this settles a conjecture by Filippas, Maz'yaand Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities foreigenvalues of Schr\"odinger operators on domains.

Hardy-Sobolev-Maz'ya inequalities for arbitrary domains