Farit G. Avkhadiev

Hardy type inequalities in higher dimensions with explicit estimate of constants

(Lobachevskii Journal of Mathematics, Volume XXI)

Let Ω be an open set in Rn such that Ω ≠ Rn. For 1 ≤ p < ∞, 1 < s < ∞ and δ = dist(x, ∂Ω) we estimate the Hardy constant
cp(s, Ω)= sup {||f/ δs/p||Lp(Ω): f ∈ C0(Ω), ||(∇ f)/δs/p - 1||Lp(Ω)=1}
and some related quantities.

For open sets Ω ⊂ R2 we prove the following bilateral estimates
min {2, p} M0(Ω) ≤ cp (2, Ω) ≤ 2 p (π M0 (Ω) + a0)2, a0=4.38,
where M0(Ω) is the geometrical parameter defined as the maximum modulus of ring domains in Ω with center on ∂Ω. Since the condition M0 (Ω) < ∞ means the uniformly perfectness of ∂Ω, these estimates give a direct proof of the following Ancona-Pommerenke theorem: c2(2, ∂Ω) is finite if and only if the boundary set ∂Ω is uniformly perfect. Moreover, we obtain the following direct extension of the one-dimensional Hardy inequality to the case n ≥ 2: if s > 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 and A.Wannebo. 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 for convex domains and give its generalizations.



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