Hardy type inequalities in higher dimensions with explicit estimate of constants

Let Ω be an open set in ℝn such that Ω ≠ ℝn. For 1 ≤ p n, then for arbitrary open sets Ω ⊂ ℝn (Ω ≠ ℝn) and any p ∈ [1, ∞) the sharp inequality cp(s, Ω) ≤ p/(s - n) is valid. This gives a solution of a known problem due to J.L.Lewis [31] and A.Wannebo [44]. Estimates of constants in certain other Hardy and Rellich type inequalities are also considered. In particular, we obtain an improved version of a Hardy type inequality by H.Brezis and M.Marcus [13] for convex domains and give its generalizations