# Hardy type inequalities in higher dimensions with explicit estimate of constants

## (*Lobachevskii Journal of Mathematics, Volume XXI*)

Let Ω be an open set in *R*^{n} such that
Ω ≠ *R*^{n}.
For 1 ≤ *p* < ∞, 1 < *s* <
∞ and δ = dist(*x*, ∂Ω) we estimate the Hardy
constant

*c*_{p}(*s*, Ω)= sup {||*f*/
δ^{s/p}||_{Lp(Ω)}:*
f ∈ C*_{0}^{∞}(Ω),
||(∇ f)/δ^{s/p - 1}||_{Lp(Ω)}=1}

and some related quantities.

For open sets Ω ⊂ *R*^{2} we prove the following bilateral estimates

min {2, p} M_{0}(Ω) ≤ c_{p} (2, Ω)
≤ 2 p
(π M_{0} (Ω) + a_{0})^{2}, a_{0}=4.38,

where M_{0}(Ω) is the geometrical parameter defined as the
maximum modulus of ring domains in Ω with center on
∂Ω. Since the condition M_{0} (Ω) < ∞
means the uniformly perfectness of ∂Ω, these
estimates give a direct proof of the following Ancona-Pommerenke
theorem:
c_{2}(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 ⊂ *R*^{n} (Ω
≠ *R*^{n})
and any *p* ∈ [1, ∞) the sharp
inequality
*c*_{p} (*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.