Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 21, 2006, 3–31 \begin{matrix}c \\ ◯ \end{matrix} F. G. Avkhadiev

F. G. Avkhadiev

HARDY TYPE INEQUALITIES IN HIGHER DIMENSIONS WITH EXPLICIT ESTIMATE OF CONSTANTS

Abstract. 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}∥_{L^{p}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥(∇f)/δ^{s/p−1}∥_{L^{p}(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) ≤ 2p (πM_{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 (see [2],

[22] and [40]).

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 [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.

2000 Mathematical Subject Classification. 26C10, 30A10. Key words and phrases. Hardy type inequalities, distance to the boundary,

uniformly perfect sets, Rellich type inequalities.

Contents.

1. Introduction.

2. Bilateral estimates of Hardy’s constant for plane open sets with uniformly perfect boundary.

3. Other results connected with uniformly perfect sets, a conjecture in the spatial case.

4. Solution of a problem by J.L. Lewis and A. Wannebo.

5. Boundary moments of an open set in connection with constants in Hardy type inequalities.

6. An improved form of the Brezis-Marcus inequality and related results.

1. Introduction.

Hardy type inequalities in Sobolev spaces have many applications in Mathematical Physics.

The original Hardy theorem (see [25], Theorem 330) gives that

∫_{0}^{+∞} \frac{|u(t)|^{p}}{t^{s}} dt ≤ (\frac{p}{|s − 1|})^{p} ∫_{0}^{+∞} \frac{|u′(t)|^{p}}{t^{s−p}} dt

for p ≥ 1, s ∈ ℝ, s ≠ 1 and any absolutely continuous function u : [0, ∞) → ℝ, u′/t^{s/p−1} ∈ L^{p}[0, ∞) such that u(0) = 0 in the case s > 1 and u(+∞) = 0 in the case s < 1.

If p = 1 then equality in the Hardy inequality is valid for any monotone function u; if p > 1 and u ≢ 0 then equality is not attained, but the constant (p/|s − 1|)^{p} is still sharp.

The Hardy inequality has been generalized in many ways. Our aim is to consider its direct generalizations when the domain of integration Ω is an open and proper subset of ℝ^{n}, u and u′ are replaced by functions f ∈ C_{0}^{∞}(Ω) and ∇f, the gradient of f, and powers of t are replaced by powers of

δ = δ(x) = dist(x, ∂Ω).

Let Ω be an open and proper subset of ℝ^{n}. We first consider the following Hardy constant

c_{p}(s, Ω) := sup \{∥\frac{f}{δ^{s/p}} ∥_{L^{p}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥\frac{∇f}{δ^{s/p−1}} ∥_{L^{p}(Ω)} = 1\} .

The classical examples which are simple consequences of the one dimensional Hardy inequalities are given by the equations : for p ∈ [1, ∞) and s ∈ ℝ

c_{p}(s, ℝ^{n} \ \{0\}) = \frac{p}{|s − n|}, c_{p}(s, H) = \frac{p}{|s − 1|},

where H is a half space in ℝ^{n} (see [10], [34], [38]). For p > 1 and any open convex set Ω ⊂ ℝ^{n}, Ω ≠ ℝ^{n} it is known that

c_{p}(s, Ω) = \frac{p}{p − 1}

(see [17], [32], [33]). Explicit estimates of c_{p}(s, Ω) are also known in some particular cases when Ω is not a convex domain. Namely,

1) if Ω is a simply connected plane domain, then c_{2}(2, Ω) ≤ 4 (see [2], [3], [11], [16]);

2) if Ω is a domain in ℝ^{n} with smooth boundary, then c_{p}(p, Ω) ≥ p/(p − 1) (see [16] and [32]).

For p ≥ 1 and s > 1, it is a classical fact that there exists a finite constant c_{p}(s, Ω) for any domain Ω with Lipschitz boundary (see, for instance, [10], [16], [38]). It is known that the Lipschitz condition is not a necessary one and can be replaced by more general conditions on the boundary of Ω. In this direction there are several deep results due to A.Ancona [2], H. Brezis and M. Marcus [13], E.B.Davies [16], [17], P. Koskela and X. Zhong [30], J.L.Lewis [31], V.G. Maz’ya [34], V.M.Miklyukov and M.K.Vuorinen [35], and A.Wannebo [44].

The main aim of the present paper is to obtain explicit estimates of c_{p}(s, Ω) in the case when p ∈ [1, ∞), s ≥ n and to estimate some related quantities.

In Sections 2 and 3 we examine the quantity c_{p}(s, Ω) in the case, when p ∈ [1, ∞) and s = n. In Section 2 the case p ∈ [1, ∞) and n = s = 2 is considered. For plane domains Ω ⊂ ℝ^{2} we prove the following bilateral estimates

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

where M_{0}(Ω) is the geometrical parameter defined as the maximum modulus of genuine annuli in Ω with center on ∂Ω. Note that, by results of Ch. Pommerenke [40] and A. Ancona [2], c_{2}(2, Ω) for a plane domain Ω ⊂ ℝ^{2} is finite if and only if the boundary set ∂Ω is uniformly perfect. Clearly, our estimates give L^{p} - version and a direct proof of the Ancona - Pommerenke theorem, since the condition M_{0}(Ω) < ∞ means the uniformly perfectness of ∂Ω.

In Section 3 we extend our estimates to the quantities

κ_{1}(Ω)^{2} := sup {∥f/δ∥_{L^{2}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥δΔf∥_{L^{2}(Ω)} = 1}

and

κ_{2}(Ω) := sup {∥∇f∥_{L^{2}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥δΔf∥_{L^{2}(Ω)} = 1}

related to Rellich’s constant of Ω ⊂ ℝ^{2} and discuss a generalization of results to the quantity c_{p}(n, Ω) for space domains. In particular, we prove that

c_{p}(n, Ω) ≥ 2 min{1, p/n}M_{0}(Ω),

where Ω ⊂ ℝ^{n}, n ≥ 3.

One of the main results of the present paper is a direct extension of the original Hardy inequality to the case n ≥ 2. More precisely, in Section 4 the following assertion is proved: if s > n, then for arbitrary open sets Ω ⊂ ℝ^{n}, Ω ≠ ℝ^{n} and p ≥ 1 the sharp inequality

c_{p}(s, Ω) ≤ \frac{p}{s − n}

is valid. This completes the following known facts: J.L. Lewis [31] discovered that there is c_{p}(p, Ω) < +∞ for any open set Ω ⊂ ℝ^{n}, whenever p > n. A.Wannebo [44] proved a generalization of this assertion: if p > n and s > p − ε(p, n) for a convenient ε(p, n) > 0, then c_{p}(s, Ω) < +∞ for any open set Ω ⊂ ℝ^{n}.

We find that some Hardy type inequalities are connected with isoperimetric properties of open sets Ω ⊂ ℝ^{n}. Some theorems in this direction are given in Sections 5 and 6. For instance, in Section 5 the constant

c(p, Ω) = sup {∥f/δ∥_{L^{n}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥∇f∥_{L^{p}(Ω)} = 1} , p ∈ (n, ∞),

is considered. For open sets with finite volume |Ω| = mes Ω we prove the inequalities

|Ω|^{1/n−1/p} ≤ c(p, Ω) ≤ \frac{p}{p − n} |Ω|^{1/n−1/p}.

We extend the above results on c_{p}(s, Ω) and c(p, Ω) to some other Hardy type inequalities with explicit estimates of all constants in function of parameters and simple geometric quantities of Ω. In particular, we obtain an improved version of a Hardy type inequality by H.Brezis and M.Marcus [13] in convex domains and give its generalizations (see Sections 4, 5 and 6).

For instance, in Section 6 we prove that

c_{p}(s, Ω) ≥ \frac{p}{s − 1}, (p ≥ 1, s > 1)

for any bounded open set Ω ⊂ ℝ^{n} with finite boundary surface area in the sense of Minkowski. On the other hand, for parameters p ≥ 1, s > 1 and convex open sets Ω ⊂ ℝ^{n} we prove that any function f ∈ C_{0}^{∞}(Ω) satisfies the inequality

∫_{Ω} \frac{|f|^{p}}{δ^{s}} dx + \frac{1}{(s − 1)δ_{0}^{s}} ∫_{Ω} |f|^{p}dx ≤ (\frac{p}{s − 1})^{p} ∫_{Ω} \frac{|∇f|^{p}}{δ^{s−p}} dx,

where δ = dist(x, ∂Ω), δ_{0} = sup{δ(x) : x ∈ Ω}.

The main results of the present paper were announced in our talks in the International Conference ”Geometric Analysis and its Applications”, Volgograd State University, (2004) (see [5]), in the International Conference and workshop dedicated to the centennial of Sergei Mikhailovich Nikolskii, Russian Academy of Sciences, Moscow (2005) (see [6]) and in the 13-th Saratov winter school on the function theory and its applications, Saratov State University, (2006) (see [7]).

2. Bilateral estimates of Hardy’s constant for plane open sets with uniformly perfect boundary

In [19] Fernández observed that Pommerenke’s capacity density condition [40] is equivalent to Ancona’s condition [2] on domains with strong barrier. This leads to the following excellent fact.

Theorem 1. If Ω is a plane domain then the Hardy constant

c_{2}(2, Ω) = sup \{∥\frac{f}{dist(., ∂Ω)}∥_{L^{2}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥∇f∥_{L^{2}(Ω)} = 1\}

is finite if and only if the boundary set ∂Ω is uniformly perfect.

One can find this result and many important characterizations of uniformly perfect sets in the recent book by Garnett and Marshall [22], see Page 119 and Pages 343-345. Also, it is known that c_{2}(2, Ω) ≥ 2 for any domain with smooth boundary and c_{2}(2, Ω) = 2 for convex domains Ω (see [16]). Moreover, if Ω is a simply connected domain in ℂ then c_{2}(2, Ω) ≤ 4 (see [2], [3], [11] and [16]). In the general case, for instance, in the case when Ω is not a finitely connected domain, explicit estimates of c_{2}(2, Ω) are unknown.

In this section, we shall prove L^{p}-version (1 ≤ p < ∞) of Theorem 1 with bilateral explicit estimates of the Hardy constant using a simple geometrical parameter of Ω. In particular, we give a direct proof of Theorem 1.

Let Ω be an open set in the complex plane ℂ such that Ω ≠ ℂ. For any fixed p ∈ [1, ∞) we consider Hardy’s inequality

∫∫_{Ω} \frac{|f|^{p}}{dist(z, ∂Ω)^{2}} dxdy ≤ c_{p}(2, Ω)^{p} ∫∫_{Ω} \frac{|∇f|^{p}}{dist(z, ∂Ω)^{2−p}} dxdy, ∀f ∈ C_{0}^{∞}(Ω),

where z = x + iy and c_{p}(2, Ω) is the minimum possible constant that generalizes c_{2}(2, Ω).

Following to [12] and [40], we characterize the open set Ω by moduli of ring domains that separate ∂Ω. More precisely, we define the maximum modulus

M_{0}(Ω) := sup \frac{1}{2π} log \frac{R(A)}{r(A)},

where the supremum is taken over all annuli A such that

A = {z ∈ ℂ : r(A) < |z − z_{0}| < R(A)} ⊂ Ω and z_{0} ∈ ∂Ω.

We take M_{0}(Ω) = 0 by definition, when there is no circle in Ω with center on ∂Ω. We say that ∂Ω is uniformly perfect if M_{0}(Ω) < ∞.

In the sequel, we need the constant

a_{0} = \frac{Γ(1/4)^{1/4}}{4π^{2}} ≈ 4.38  (1)

from the sharp form of Landau’s theorem (see [26] and [29]).

The main result of this section is the following assertion.

Theorem 2. If 1 ≤ p < ∞ and Ω is an open and proper subset of ℂ, then

min{2, p} M_{0}(Ω) ≤ c_{p}(2, Ω) ≤ 2p (πM_{0}(Ω) + a_{0})^{2}.  (2)

In particular, the Hardy constant c_{p}(2, Ω) is finite if and only if ∂Ω is uniformly perfect.

Proof of Theorem 2. First we prove the lower estimate for c_{p}(2, Ω). Clearly, it is sufficient to consider the case, when 0 < M_{0}(Ω) ≤ ∞ and 0 < c_{p}(2, Ω) < ∞.

We shall examine the cases p ≥ 2 and p < 2 separately. Suppose first that 2 ≤ p < ∞ and c_{p}(2, Ω) < 2M_{0}(Ω) for an open and proper subset of ℂ. From the definition of M_{0}(Ω) it follows that there is an annulus A = {z ∈ ℂ : r(A) < |z − z_{0}| < R(A)} ⊂ Ω such that z_{0} ∈ ∂Ω and

∞ > log \frac{R(A)}{r(A)} > πc_{p}(2, Ω).

Without loss of generality we can suppose that z_{0} = 0, R(A) = 1 and r(A) = ε ∈ (0, 1), since c_{p}(2, Ω) is invariant under linear transformations of Ω. We have

M_{0} := \frac{1}{2π} log \frac{1}{ε} > c_{p}(2, Ω)/2

and

∫∫_{A} \frac{|f|^{p}}{dist(z, ∂Ω)^{2}} dxdy ≤ c_{p}(2, Ω)^{p} ∫∫_{A} \frac{|∇f|^{p}}{dist(z, ∂Ω)^{2−p}} dxdy, ∀ f ∈ C_{0}^{∞}(A).

Using the polar coordinates, functions f(r, θ) = v(r) with v ∈ C_{0}^{∞}(ε, 1) and the estimate (z, ∂Ω) ≤ |z|, we obtain

∫_{ε}^{1} \frac{|v(r)|^{p}rdr}{r^{2}} ≤ c_{p}(2, Ω)^{p} ∫_{ε}^{1} \frac{|v′(r)|^{p}rdr}{r^{2−p}}, ∀ v ∈ C_{0}^{∞}(ε, 1).

By the change r = εexp(2M_{0}t) and v(r) = g(t) of variables this is equivalent to the Wirtinger type inequality (see [25])

∫_{0}^{π} |g(t)|^{p} dt ≤ \frac{c_{p}(2, Ω)^{p}}{2^{p}M_{0}^{p}} ∫_{0}^{π} |g′(t)|^{p}dt, ∀ g ∈ C_{0}^{∞}(0, π).

Approximating g_{0}(t) = sin t by functions g ∈ C_{0}^{∞}(0, π) , we get

c_{p}(2, Ω)^{p} ≥ 2^{p}M_{0}^{p} ∫_{0}^{π} | sin t|^{p}dt/ ∫_{0}^{π} | cos t|^{p}dt = 2^{p}M_{0}^{p},

which contradicts to the assumption c_{p}(2, Ω) < 2M_{0}. Hence, 2M_{0}(Ω) ≤ 2M_{0} ≤ c_{p}(2, Ω) in the case 2 ≤ p < ∞.

In the case 1 ≤ p < 2 and c_{p}(2, Ω) < ∞, we combine the Hardy and Hölder inequalities in the following way

∫∫_{Ω} \frac{|f|^{2}}{dist(z, ∂Ω)^{2}} dxdy = ∫∫_{Ω} \frac{(|f|^{2/p})^{p}}{dist(z, ∂Ω)^{2}} dxdy

≤ (\frac{2}{p})^{p} c_{p}(2, Ω)^{p} ∫∫_{Ω} \frac{|f|^{2−p}|∇f|^{p}}{dist(z, ∂Ω)^{2−p}} dxdy

≤ (\frac{2}{p})^{p} c_{p}(2, Ω)^{p} (∫∫_{Ω} \frac{|f|^{2}}{dist(z, ∂Ω)^{2}} dxdy)^{1−p/2} (∫∫_{Ω} |∇f|^{2} dxdy)^{p/2} .

It follows that c_{2}(2, Ω) ≤ \frac{2}{p}c_{p}(2, Ω). As is proved that c_{2}(2, Ω) ≥ 2M_{0}(Ω), we get

c_{p}(2, Ω) ≥ pM_{0}(Ω),

when p ∈ [1, 2).

Now we suppose that M_{0}(Ω) < ∞ and we prove the upper estimate. Clearly, the condition M_{0}(Ω) < ∞ assure that ∂Ω has no isolated point. Also, it is sufficient to obtain the upper inequality in (2) for connected components of Ω.

Since ℂ\Ω ≠ ∅ and ∂Ω has no isolated point, any connected component of Ω is a hyperbolic domain in ℂ, i.e. its boundary has more than one point in ℂ. Without loss of generality we can suppose that Ω itself is a hyperbolic domain in ℂ. Let λ_{Ω} be the density of the Poincaré metric on Ω with curvature −4 (see [1], [9]).

Let p > 1 and let f ∈ C_{0}^{∞}(Ω). Since p > 1, we have that |f|^{p} ∈ C_{0}^{1}(Ω) and |∇|f|^{p}| = p|f|^{p−1}|∇f|. Using the Liouville equation in the form

\frac{Δ log λ_{Ω}(z)^{−1}}{λ_{Ω}(z)^{2}} = −4, z = x + iy ∈ Ω,

and the Green formula

∫∫_{Ω}[uΔv + (∇u, ∇v)]dxdy = 0

for v = log λ_{Ω}^{−1} and u = |f|^{p}, f ∈ C_{0}^{∞}(Ω), we obtain

4 ∫∫_{Ω} |f|^{p}λ_{Ω}(z)^{2} dxdy = p ∫∫_{Ω} |f|^{p−1}λ_{Ω}(z)(∇|f|, ∇λ_{Ω}(z)^{−1}) dxdy.

Combining this with the Hölder inequality

∫∫_{Ω} |f|^{p−1}λ_{Ω}|(∇|f|, ∇λ_{Ω}^{−1})| dxdy

≤ (∫∫_{Ω} |f|^{p}λ_{Ω}^{2} dxdy)^{1−1/p} (∫∫_{Ω} λ_{Ω}^{2−p}|(∇f, ∇λ_{Ω}^{−1})|^{p} dxdy)^{1/p} ,

we immediately get

∫∫_{Ω} |f|^{p}λ_{Ω}^{2} dxdy ≤ (\frac{p}{4})^{p} ∫∫_{Ω} λ_{Ω}^{2−p}|(∇f, ∇λ_{Ω}^{−1})|^{p} dxdy  (3)

for any f ∈ C_{0}^{∞}(Ω). Since this inequality is proved for any p ∈ (1, ∞), letting p → 1 for a fixed f ∈ C_{0}^{∞}(Ω) we obtain that (3) is true in the case p = 1, too.

Using (3) and Osgood’s inequality [39]

λ_{Ω}(z)|∇λ_{Ω}(z)^{−1}| ≤ \frac{2}{dist(z, ∂Ω)}, z = x + iy ∈ Ω,

one gets

∫∫_{Ω} |f|^{p}λ_{Ω}(z)^{2} dxdy ≤ (\frac{p}{2})^{p} ∫∫_{Ω} \frac{λ_{Ω}(z)^{2−2p}|∇f|^{p}}{dist(z, ∂Ω)^{p}} dxdy

Hence, for any p ∈ [1, ∞) and any f ∈ C_{0}^{∞}(Ω),

α(Ω)^{2} ∫∫_{Ω} \frac{|f|^{p}}{dist(z, ∂Ω)^{2}} dxdy ≤ \frac{(p/2)^{p}}{α(Ω)^{2p−2}} ∫∫_{Ω} \frac{|∇f|^{p}}{dist(z, ∂Ω)^{2−p}} dxdy, (4)

where

α(Ω) := inf{λ_{Ω}(z)dist(z, ∂Ω) : z ∈ Ω}.

Beardon and Pommerenke [12] proved that the condition M_{0}(Ω) < ∞ holds if and only if α(Ω) > 0 and, in particular,

α(Ω)^{−1} ≤ 2πM_{0}(Ω) + 2a_{0},  (5)

where a_{0} is the constant from Landau’s theorem. By [26] and [29] it is known that the sharp value of a_{0} is given by formula (1).

From (4) it follows that

c_{p}(2, Ω) ≤ \frac{p}{2 α(Ω)^{2}} .  (6)

Evidently, inequalities (5) and (6) imply the upper bound in (2). The proof of Theorem 2 is complete.

Consider the upper bound corresponding to the case M_{0}(Ω) = 0 in (2).

Corollary 2.1. Let Ω be an open and proper subset of ℂ. If there is no circle in Ω with center on ∂Ω, then c_{p}(2, Ω) < 38.4 p.

Let us mention that M_{0}(Ω) = 0 for any simply connected domain, but the converse is not true. The family {Ω : M_{0}(Ω) = 0} is a large collection of open sets Ω ⊂ ℂ and it contains domains of arbitrary connectivity. For example, the equality M_{0}(Ω_{0} \ \overline{K}) = 0 holds for domains satisfying the following conditions:

(1) Ω_{0} is an open set in ℂ such that sup{dist(z, ∂Ω_{0}) : z ∈ Ω_{0}} = 1, in particular, Ω_{0} is a stripe with width 2;

(2) K = ⋃_{m=1}^{∞} C_{m}, where C_{m} are continuums (connected compact sets) such that diam C_{m} ≥ 2 for all m ≥ 1;

(3) K ⊂ Ω_{0} and Ω_{0} \ \overline{K} is nonempty.

The Pommerenke condition on the capacity density for ℂ \ Ω has the form

C(Ω) := inf \{\frac{cap (\{|z − z_{o}| ≤ r\} ⋂(ℂ \ Ω))}{r} : z_{0} ∈ ∂Ω, 0 < r < ∞\} > 0 ,

where cap E is the logarithmic capacity of E. In our notations, Pommerenke’s estimates of the capacity density (see the proof of Theorem 1 in [40]) can be written as

M_{0}(Ω) ≤ \frac{1}{2π} log \frac{1}{C(Ω)} ≤ 2M_{0}(Ω) + \frac{4 log 2}{π} .  (7)

From (7) and Theorem 2, one immediately obtains the following assertion.

Corollary 2.2. If 1 ≤ p < ∞ and Ω is an open and proper subset of ℂ, then

\frac{1}{4π} log \frac{1}{C(Ω)} − \frac{log 4}{π} ≤ c_{p}(2, Ω) ≤ \frac{p}{2} ( log \frac{1}{C(Ω)} + 2a_{0})^{2} .

We complete this section by three examples considering domains of the form B(0, 3)\E_{j} = {z ∈ ℂ : |z| < 3}\E_{j} with perfect boundaries.

Example 1. Suppose that E_{1} = ⋃_{m=1}^{∞} K_{m} ⋃{0}, where

K_{m} = {z = x + iy ∈ ℂ : y = 0, m^{−m} ≤ x ≤ 2m^{−m}}.

The domain Ω_{1} = B(0, 3)\E_{1} contains the annuli

A_{m} = {z ∈ ℂ : 2(m + 1)^{−m−1} < |z| < m^{−m}}

and

\frac{R(A_{m})}{r(A_{m})} = \frac{m + 1}{2} (1 + \frac{1}{m})^{m} → ∞ as m → ∞.

Consequently, M_{0}(Ω_{1}) = c_{p}(2, Ω_{1}) = ∞.

Example 2. Now, we consider E_{2} = ⋃_{m=1}^{∞} L_{2m−1} ⋃{0}, where

L_{2m−1} = {z = x + iy ∈ ℂ : y = 0, 3^{−2m+1} ≤ x ≤ 3^{−2m+2}}.

For any annulus A in Ω_{2} = B(0, 3)\E_{2} with center on ∂Ω_{2} we have R(A)/r(A) ≤ 3. It is an easy task to show that 2πM_{0}(Ω_{2}) = log 3. Accordingly, c_{p}(2, Ω_{2}) ≤ (8.76 + log 3)^{2}p/2 < 48 p.

Example 3. Let E_{3} be the classical Cantor set. In [12] estimates for α(ℂ\E_{3}) are proved. We consider

Ω_{3} = B(0, 3)\E_{3}.

It is easy to show that M_{0}(Ω_{3}) = M_{0}(Ω_{2}) = (2π)^{−1} log 3. Thus, c_{p}(2, Ω_{3}) < 48 p.

3. Other results connected with uniformly perfect sets, a conjecture in the spatial case

We shall extend Theorem 1 to certain functionals connected with Rellich’s constants and discuss a generalization of Theorem 1 to the space domains.

First, we consider two following quantities used in [3] for simply connected domains and related to Rellich’s constants (compare [18], [36] and [42]).

Let Ω be an open and proper subset of ℂ. We define

κ_{1}(Ω)^{2} := sup \{∥\frac{f}{dist(., ∂Ω)}∥_{L^{2}(Ω)} : f ∈ C_{0}^{∞}(Ω),

∥dist(., ∂Ω)Δf∥_{L^{2}(Ω)} = 1}

and

κ_{2}(Ω) := sup {∥∇f∥_{L^{2}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥dist(., ∂Ω)Δf∥_{L^{2}(Ω)} = 1} ,

where z = x + iy and Δ is the Laplace operator.

In [3] it is proved that κ_{1}(Ω) ≤ 4 and κ_{2}(Ω) ≤ 4 for any simply connected domain Ω. In the next theorem we give bilateral estimates of κ_{1}(Ω) and κ_{2}(Ω) for open sets with uniformly perfect boundary. Also, we obtain an improvement of the upper bound of c_{2}(2, Ω), κ_{1}(Ω) and κ_{2}(Ω) for doubly connected domains.

Theorem 3. If Ω is an open and proper subset of ℂ, then the quantity κ_{j}(Ω) (j = 1, 2) is finite if and only if ∂Ω is a uniformly perfect set. Moreover,

2M_{0}(Ω) ≤ κ_{1}(Ω) ≤ 4(πM_{0}(Ω) + a_{0})^{2},
2M_{0}(Ω) ≤ κ_{2}(Ω) ≤ 4(πM_{0}(Ω) + a_{0})^{2}.

If Ω is a doubly connected domain in ℂ, then

M_{0}(Ω) ≤ c_{2}(2, Ω)/2 ≤ πM_{0}(Ω) + a_{0},
M_{0}(Ω) ≤ κ_{1}(Ω)/2 ≤ πM_{0}(Ω) + a_{0},
M_{0}(Ω) ≤ κ_{2}(Ω)/2 ≤ πM_{0}(Ω) + a_{0}.

Proof of Theorem 3. Suppose that ∂Ω is uniformly perfect and f ∈ C_{0}^{∞}(Ω). Using the Green formula and the Cauchy - Schwartz inequality one gets

∫∫_{Ω} |∇f|^{2}dxdy = − ∫∫_{Ω} fΔf dxdy

≤ (∫∫_{Ω} |f|^{2}dist(z, ∂Ω)^{−2} dxdy)^{1/2} (∫∫_{Ω} dist(z, ∂Ω)^{2}|Δf|^{2} dxdy)^{1/2} .

This inequality and the definitions of c_{2}(2, Ω) and κ_{1}(Ω) imply

∫∫_{Ω} |∇f|^{2}dxdy ≤ κ_{1}(Ω)^{2}, ∫∫_{Ω} |f|^{2}dist(z, ∂Ω)^{−2} dxdy ≤ c_{2}(2, Ω)^{4}

for any f ∈ C_{0}^{∞}(Ω) satisfying

∫∫_{Ω} dist(z, ∂Ω)^{2}|Δf|^{2} dxdy = 1.

Consequently,

κ_{2}(Ω) ≤ κ_{1}(Ω) ≤ c_{2}(2, Ω).  (8)

Inequalities (2) and (8) immediately give the upper bounds of κ_{1}(Ω) and κ_{2}(Ω) in the general case. Moreover, it is known (see, for instance, [20], Lemma 1.1) that the inequality

∫∫_{Ω} |f|^{2}λ_{Ω}^{2} dxdy ≤ ∫∫_{Ω} |∇f|^{2} dxdy, ∀f ∈ C_{0}^{∞}(Ω),  (9)

is valid for any simply or doubly connected hyperbolic domain Ω. It is obvious that inequalities (5), (8) and (9) imply the upper bounds in Theorem 3 for doubly connected domains.

Thank to (8), we have to prove the lower estimates of Theorem 3 for κ_{2}(Ω), only. To this end we assume that κ_{2}(Ω) < 2M_{0}(Ω). Without loss of generality we can suppose that 0 ∈ Ω and there exists an annulus A = \{z ∈ ℂ : ε < |z| < 1\} such that A ⊂ Ω and M_{0} := \frac{1}{2π} log \frac{1}{ε} > κ_{2}(Ω)/2. One has

∫∫_{A} |∇f|^{2} dxdy ≤ κ_{2}(Ω)^{2} ∫∫_{A} dist(z, ∂Ω)^{2}|Δf|^{2} dxdy, ∀ f ∈ C_{0}^{∞}(A)

and dist(z, ∂Ω) < |z| for z ∈ A. Consequently,

∫_{ε}^{1} v′(r)^{2}r dr ≤ κ_{2}(Ω)^{2} ∫_{ε}^{1}(rv′′(r) + v′(r))^{2}r dr

for radial functions v(r) = f(r, θ), v ∈ C_{0}^{∞}(ε, 1).

After the changes r = εexp(2M_{0}t) and v(r) = g(t), the last inequality can be written as the following Wirtinger type inequality

∫_{0}^{π} |g′(t)|^{2} dt ≤ \frac{κ_{2}(Ω)^{2}}{(2M_{0})^{2}} ∫_{0}^{π} |g′′(t)|^{2}dt, ∀ g ∈ C_{0}^{∞}(0, π).

Consequently, κ_{2}(Ω) ≥ 2M_{0}.

This completes the proof of Theorem 3.

Finally, we consider the spatial case. Let Ω be an open set in ℝ^{n} such that Ω ≠ ℝ^{n}. Suppose that 1 ≤ p < ∞, 1 < s < ∞ and consider the Hardy constant

c_{p}(s, Ω) = sup \{∥\frac{f}{δ^{s/p}} ∥_{L^{p}(Ω)} : f ∈ C_{0}^{∞}(Ω), ∥\frac{∇f}{δ^{s/p−1}} ∥_{L^{p}(Ω)} = 1\} ,

where δ = dist(x, ∂Ω).

It is known that c_{p}(s, Ω) < ∞ for any domain Ω with Lipschitz boundary (see [38]). More general families of domains with finite c_{p}(s, Ω) are given in the papers by Ancona [2] (case p = s = 2), Lewis [31] ( case p = s > 1) and Wannebo [44] (case p ≥ 1, s > 1).

In the case n ≥ 3, as is indicated in the paper [28] of Järvi and Vuorinen, the concept of uniformly perfect sets is not equivalent to the known density concepts used in the theory of Hardy’s inequalities in higher dimensions.

For an open set Ω ⊂ ℝ^{n} (Ω ≠ ℝ^{n}, n ≥ 3) we consider the quantity M_{0}(Ω) defined as in the planar case. More precisely, let

M_{0}(Ω) := sup \frac{1}{2π} log \frac{R(A)}{r(A)},

where the supremum is taken over all ring domains A such that A = {x ∈ ℝ^{n} : r(A) < |x − x_{0}| < R(A)} ⊂ Ω and x_{0} ∈ ∂Ω . If such a ring domain A doesn’t exist, then M_{0}(Ω) = 0. If M_{0}(Ω) < ∞ then the set ∂Ω is said to be uniformly perfect (compare [28]).

It is clear that the conditions M_{0}(Ω) < ∞ and c_{p}(s, Ω) < ∞ are not equivalent in the case s ≠ n. For instance, if s > n, then c_{p}(s, Ω) ≤ p/(n − s) for any open set Ω ⊂ ℝ^{n} (Ω ≠ ℝ^{n}) (see Theorem 5, below). If 1 < s < n and B_{0} is a punctured ball, then it is easy to show that M_{0}(B_{0}) = ∞ but c_{p}(s, B_{0}) is finite for any p ∈ [1, ∞).

We conjecture that a direct comparison of M_{0}(Ω) and the Hardy constant is possible in the case s = n. At least, if c_{p}(n, Ω) is finite, then ∂Ω is uniformly perfect. Evidently, the last assertion is a consequence of the following theorem.

Theorem 4. If p ∈ [1, ∞) , n ≥ 3 and Ω is an open and proper subset of ℝ^{n}, then

c_{p}(n, Ω) ≥ 2 min{1, p/n}M_{0}(Ω).  (10)

Proof of Theorem 4. We will follow the proof of lower estimates in Theorem 2 with some necessary changes. In particular, we consider the cases p ≥ n and p < n separately.

Assume that n ≤ p < ∞ , 0 < M_{0}(Ω) ≤ ∞ and c_{p}(n, Ω) < 2M_{0}(Ω). Without loss of generality we can conclude that there is a positive constant ε such that

c_{p}(n, Ω) < 2M_{0} := \frac{1}{π} log \frac{1}{ε} < ∞

and

A = {x ∈ ℝ^{n} : ε < |x| < 1} ⊂ Ω, (0, ..., 0) ∈ ∂Ω.

Using radial functions, the spherical coordinates and the obvious inequality dist(x, ∂Ω) ≤ |x| for x ∈ A, we can write (compare the proof of Theorem 2 for details)

∫_{ε}^{1} \frac{|v(r)|^{p}r^{n−1}dr}{r^{n}} ≤ c_{p}(n, Ω)^{p} ∫_{ε}^{1} \frac{|v′(r)|^{p}r^{n−1}dr}{r^{n−p}}, ∀ v ∈ C_{0}^{∞}(ε, 1).

Using the changes r = ε exp(2M_{0}t), u(r) = g(t) and straightforward computations, we obtain

∫_{0}^{π} |g(t)|^{p} dt ≤ \frac{c_{p}(n, Ω)^{p}}{2^{p}M_{0}^{p}} ∫_{0}^{π} |g′(t)|^{p} dt, ∀ g ∈ C_{0}^{∞}(0, π).

Hence, c_{p}(n, Ω)/(2M_{0}) ≥ 1. This completes the proof of (10) in the case n ≤ p.

In the case 1 ≤ p < n and c_{p}(n, Ω) < ∞, we again combine the Hardy and Hölder inequalities in the following way

∫_{Ω} \frac{|f|^{n}}{δ^{n}} dx = ∫_{Ω} \frac{(|f|^{n/p})^{p}}{δ^{n}} dx

≤ (\frac{n}{p})^{p} c_{p}(n, Ω)^{p} ∫_{Ω} \frac{|f|^{n−p}|∇f|^{p}}{δ^{n−p}} dx

≤ (\frac{n}{p})^{p} c_{p}(n, Ω)^{p} (∫_{Ω} \frac{|f|^{n}}{δ^{n}} dx)^{1−p/n} (∫_{Ω} |∇f|^{n} dx)^{p/n} ,

where δ = dist(x, ∂Ω). It follows that c_{n}(n, Ω) ≤ \frac{n}{p}c_{p}(n, Ω). Since c_{n}(n, Ω) ≥ 2M_{0}(Ω), we get

c_{p}(n, Ω) ≥ \frac{p}{n}M_{0}(Ω),

when p ∈ [1, n).

The proof of Theorem 4 is complete.

It seems to be natural the following generalization of Theorem 1.

Conjecture. The equivalence {c_{p}(n, Ω) < ∞ ⟺ M_{0}(Ω) < ∞} is true in the spatial case, too.

Remark. In the literature on uniformly perfect sets one can find several definitions of maximum modulus of Ω. To define M_{0}(Ω) we have used ring domains in Ω with centers on ∂Ω. This directly is connected with the basic definition of Ch. Pommerenke in [40]. One can consider a slightly different parameter M(Ω) defined as the maximum modulus of ring domains which are in Ω and separate ∂Ω ⋃{∞} (compare [12] and [28]). It is easy to show that

M_{0}(Ω) ≤ M(Ω) ≤ M_{0}(Ω) + \frac{1}{2π} log 3.

4. Solution of a problem by J.L. Lewis and A. Wannebo

In [31], J. L. Lewis proved that there is c_{p}(p, Ω) < +∞ for any open set Ω ⊂ ℝ^{n} if p > n. A. Wannebo [44] proved a generalization of this assertion: if p > n and s > p − ε(p, n) for a convenient ε(p, n) > 0, then c_{p}(s, Ω) < +∞ for any open set Ω ⊂ ℝ^{n}.

We find that the single condition s > n assure that c_{p}(s, Ω) < +∞ for any open set Ω ⊂ ℝ^{n} and any p ≥ 1. Surprisingly, the constant has a simple upper bound in this case. More precisely, we obtain the following extension of the one dimensional Hardy inequality.

Theorem 5. Let Ω be an open and proper subset of ℝ^{n}. If p ≥ 1 and s > n then

∫_{Ω} \frac{|f|^{p}}{δ^{s}} dx ≤ (\frac{p}{s − n})^{p} ∫_{Ω} \frac{|∇f|^{p}}{δ^{s−p}} dx, ∀f ∈ C_{0}^{∞}(Ω),  (11)

where δ = δ(x) = dist(x, ∂Ω).

The constant p^{p}(s − n)^{−p} in (11) is the best one for many sets Ω. For example, it is sharp for every Ω of the form Ω_{0} \ {x_{0}}, where Ω_{0} is an open set in ℝ^{n} and x_{0} ∈ Ω_{0}.

From Theorem 5 it follows that the basic inequality of Hardy

∫_{0}^{+∞} \frac{|u(t)|^{2}}{t^{2}} dt ≤ 4 ∫_{0}^{+∞} |u′(t)|^{2}dt, u′ ∈ L^{2}, u(0) = 0,

has a sharp analog in ℝ^{n}:

∫_{Ω} \frac{|f|^{2n}}{δ^{2n}} dx ≤ 4^{n} ∫_{Ω} |∇f|^{2n}dx, ∀f ∈ C_{0}^{∞}(Ω),

which is valid for any open set Ω ⊂ ℝ^{n} (Ω ≠ ℝ^{n}).

We will prove that equality in (11) is not attained in the corresponding Sobolev space if f ≢ 0 and

δ_{0} = δ_{0}(Ω) := sup{dist(x, ∂Ω) : x ∈ Ω} < +∞.

More precisely, we prove the following refined version of Theorem 5.

Theorem 6. Let Ω be an open and proper subset of ℝ^{n}. If p ≥ 1 and s > n, then

∫_{Ω} \frac{|f|^{p}}{δ^{s}} dx+ \frac{1}{(s − 1)δ_{0}^{s}} ∫_{Ω} |f|^{p}dx ≤ (\frac{p}{s − n})^{p} ∫_{Ω} \frac{|∇f|^{p}}{δ^{s−p}} dx, ∀f ∈ C_{0}^{∞}(Ω),

(12)

where δ = dist(x, ∂Ω), δ_{0} = sup{δ(x) : x ∈ Ω}.

Proof of Theorems 5 and 6. Let Ω be an open and proper subset of ℝ^{n}. For given f ∈ C_{0}^{∞}(Ω) we use the following approximation of Ω.

For h ∈ (0, 1) we consider the simplest covering of ℝ^{n} by cubes

Q_{h,z} = [0, h]^{n} + hz, z ∈ Z^{n},

and define the finite set

Z^{n}(Ω, h) = {z ∈ Z^{n} : Q_{h,z} ⊂ Ω ∩ {x ∈ ℝ^{n} : |x| < 1/h}}

and the following approximation of Ω:

Ω_{h} = int ∪_{z∈Z^{n}(Ω,h)} Q_{h,z}.

For a given f ∈ C_{0}^{∞}(Ω) it is clear that it suffices to prove (11) and (12) with Ω = Ω_{h} and any h ∈ (0, 1). By the change y = x/h, x ∈ Ω_{h}, of variables we also see that (11) and (12) for Ω = Ω_{h} and Ω = Ω_{1} are equivalent. Thus, we have to prove (11) and (12) for a set of the form

Ω_{1} = int ∪_{j=1}^{m} ([0, 1]^{n} + z_{j}), z_{j} ∈ Z^{n}.

Let S be a q-face of Q_{1,z_{j}}. Suppose that S ⊂ ∂Ω_{1} and define the following subset of Ω_{1}:

Q(S) = {x ∈ ℝ^{n} : x′ ∈ int S, B(x, |x − x′|) ⊂ Ω_{1}},

where B(x, |x − x′|) is the ball {y ∈ ℝ^{n} : |y − x| < |x′ − x|}. We have to note that the interior of S is taken in ℝ^{q} ⊃ S and, by definition, int S = S if S is a 0-face i.e. a point.

Suppose that Q(S) ≠ ∅, this is always the case if S is a (n − 1)- face and S ⊂ ∂Ω_{1}. The set Q(S) ≠ ∅ is starlike with respect to S, i.e. x′ + t(x − x′) ∈ Q(S) for every x′ ∈ int S and all t ∈ (0, 1) if |x−x′| = dist(x, ∂Ω_{1}) and x ∈ Q(S). Up to a rotation, Q(S) ⊂ S ×ℝ_{+}^{n−q}, and

\overline{Q}(S) = S × \{t ∈ ℝ_{+}^{n−q} : 0 ≤ |t| ≤ φ(\frac{t}{|t|}; S, Q)\},

where φ is a positive function satisfying the inequality

sup φ ≤ sup{dist(x, ∂Ω_{1}) : x ∈ Ω_{1}}.

If S′ is a cubic j-face (j = 0, 1, …, n − 1) and S′ ⊂ (∂Ω_{1}) \ S then the set

(S, S′) := {x ∈ Ω : dist(x, S) = dist(x, S′) ≤ dist(x, ∂Ω_{1})}

is a bounded subset of a (n − 1)-plane or a (n − 1)-surface of order 2. Since mes_{n}(S, S′) = 0 and

(∂Q(S)) \ S ⊂ ∪_{S′}(S, S′),

we obtain that mes_{n}∂Q(S) = 0. Consequently, for any g ∈ C(\overline{Ω}_{1})

∫_{Ω_{1}} g(x) dx = ∑_{S⊂∂Ω_{1}} ∫_{Q(S)} g(x) dx.  (13)

In the sequel we will need the notations:

S_{+}^{n−q} = {ω ∈ R_{+}^{n−q} : |ω| = 1},
φ_{q} = φ(.; S, Q_{1}).

Let S be a cubic (n − k)-face such that S ⊂ ∂Ω_{1} and Q(S) ≠ ∅, where k = 1, 2, …, n. By Fubini’s theorem, we get the following formulas, depending of the dimension of S:

if k = 1, then

∫_{Q(S)} g(x) dx = ∫_{S} dx′ ∫_{0}^{φ_{n−1}(x′)} g(x′ + rν(x′))dr;  (14)

if 2 ≤ k ≤ n − 1, then

∫_{Q(S)} g(x) dx = ∫_{S} dx′ ∫_{S_{+}^{k}} dω ∫_{0}^{φ_{n−k}(ω)} g(x′ + ωr)r^{k−1}dr; (15)

if k = n and S = {x′}, then

∫_{Q(S)} g(x) dx = ∫_{S_{+}^{n}} dω ∫_{0}^{φ_{0}(ω)} g(x′ + ωr)r^{n−1}dr.  (16)

Suppose that f ∈ C_{0}^{∞}(Ω_{1}), p ≥ 1, s > n, δ = δ(x) = dist(x, ∂Ω_{1}), δ_{0} = sup{δ(x) : x ∈ Ω_{1}}. We will use (14), (15) and (16) for the function

g(x) = |f(x)|^{p} (\frac{1}{δ^{s}(x)} + \frac{1}{(s − 1)δ_{0}^{s}} ) .

Since δ(x) = r, 1 ≤ k ≤ n in (14)–(16), we have

∫_{0}^{φ_{n−k}} |f|^{p} (\frac{1}{r^{s}} + \frac{1}{(s − 1)δ_{0}^{s}} ) r^{k−1}dr

≤ ∫_{0}^{φ_{n−k}} (t^{k−s−1} + \frac{t^{k−1}}{(s − 1)δ_{0}^{s}} ) dt ∫_{0}^{t} |f|^{p−1}|∇f|dr

= p ∫_{0}^{φ_{n−k}} |f|^{p−1}|∇f|A(r, φ_{n−k})dr,

where

A(r, φ_{n−k}) = \frac{1}{s − k} (\frac{1}{r^{s−k}} − \frac{1}{φ_{n−k}^{s−k}} ) + \frac{1}{k(s − 1)δ_{0}^{s}} (φ_{n−k}^{k} − r^{k})

≤ \frac{1}{s − k} (\frac{1}{r^{s−k}} − \frac{r^{k}}{δ_{0}^{s}} )

≤ \frac{r^{k−1}}{(s − n)r^{s−1}} .

Therefore, one gets

∫_{Q(S)} |f|^{p} (\frac{1}{δ^{s}(x)} + \frac{1}{(s − 1)δ_{0}^{s}} ) dx ≤ \frac{p}{s − n} ∫_{Q(S)} \frac{|f|^{p−1}|∇f|}{δ^{s−1}(x)}dx

for all k = 1, 2, …, n. By using this and formula (13) we obtain

∫_{Ω_{1}} |f|^{p} [\frac{1}{δ^{s}(x)} + \frac{1}{(s − 1)δ_{0}^{s}} ] dx ≤ \frac{p}{s − n} ∫_{Ω_{1}} \frac{|f|^{p−1}|∇f|}{δ^{s−1}(x)}dx.

This is the inequality to prove in the case p = 1. If p > 1 then we apply the Hölder inequality to get (12) for Ω = Ω_{1}.

The proof is complete.

Finally, we consider a simple example to get that the upper bound (p/(s − n))^{p} in Theorems 5 and 6 is sharp (compare with the example of Hardy [25] for one dimensional case).

Let Ω_{0} be an open set in ℝ^{n} such that

0 ∈ ∂Ω_{0}, {x ∈ ℝ^{n} : 0 < |x| < 3} ⊂ Ω_{0}.

Let us denote

X = ∫_{Ω_{0}} \frac{|u|^{p}}{δ^{s}} dx, Y = ∫_{Ω_{0}} \frac{|∇u|^{p}}{δ^{s−p}} dx, δ = dist(x, ∂Ω_{0}).

If p ≥ 1, s > n, ε > 0 and

u_{ε}(x) = |x|^{(s−n+ε)/p}, 0 < |x| ≤ 1,
u_{ε}(x) = 2 − |x| , 1 < |x| ≤ 2,
u_{ε}(x) = 0 , 2 < |x| < ∞,

then straightforward computations give (ω_{n−1} = |∂B_{1}|)

X(u_{ε}) = \frac{ω_{n−1}}{ε} + O(1), Y (u_{ε}) = \frac{ω_{n−1}}{ε} (\frac{s − n + ε}{p})^{p} + O(1).

Approximating u_{ε} by radial functions that belong to C_{0}^{∞}(B(0, 3) \ {0}) and letting ε → 0 we obtain that c_{p}(s, Ω_{0}) ≥ p(s − n)^{−1}. Consequently, c_{p}(s, Ω_{0}) = p(s − n)^{−1} for any p ∈ [1, ∞) and any s ∈ (n, ∞). In particular, the constant from Theorem 5 is sharp for the punctured ball B(0, 3)\{0}. Since the Hardy constant is invariant under linear transformations of Ω, there exist extremal domains with given δ_{0} ∈ (0, ∞]. For instance, if Ω_{0} = B(0, 2δ_{0})\{0} and s ∈ (n, ∞), then c_{p}(s, Ω_{0}) = p(s−n)^{−1} and δ_{0}(Ω_{0}) = δ_{0}. Hence, the constant p^{p}(s − n)^{−p} in Theorem 6 is sharp, too.

5. Boundary moments of an open set in connection with constants in Hardy type inequalities

In [3] and [4], we used α-moment of Ω about its boundary i.e. the quantity

∫_{Ω} dist(x, ∂Ω)^{α}dx

to get bilateral estimates of constants in some inequalities of Mathematical Physics. The aim of this section is to show that these moments are also connected with constants in some Hardy type inequalities.

In the sequel we will use the following consequences of formulas (14),

(15) and (16):

∫_{Q(S)} \frac{|f|^{p}}{δ^{s}} dx ≤ p ∫_{Q(S)} \frac{|f|^{p−1}}{δ^{s−1}} |∇f|Φ_{s}(x, S)dx,  (17)

where p ≥ 1, s ∈ ℝ, S is a cubic (n − k)-face and

Φ_{s}(x, S) = δ^{s−k} ∫_{δ}^{ψ_{n−k}} \frac{dt}{t^{s−k+1}} (1 ≤ k ≤ n).  (18)

Evidently, by using (17) and (18) and following the proof of Theorem 5, one can give generalizations of Theorem 5 for admissible values of parameters in the inequality

(_{∫Ω} \frac{|f|^{q}}{δ^{α}} dx)^{1/q} ≤ c (_{∫Ω} \frac{|∇f|^{p}}{δ^{β}} dx)^{1/p} , ∀f ∈ C_{0}^{∞}(Ω).

We illustrate this idea by some particular cases, only. Consider first a case, when the constant in a Hardy type inequality is connected with the volume of Ω, i. e. with the 0-moment of Ω. Let us denote

c(p, Ω) = sup \{∥\frac{f}{δ} ∥_{L^{n}(Ω)} : ∥∇f∥_{L^{p}(Ω)} = 1, f ∈ C_{0}^{∞}(Ω)\} , p ∈ (n, ∞).

Theorem 7. Let Ω be an open set in ℝ^{n}(n ≥ 1) with finite volume |Ω| = mes Ω. If p > n, then

|Ω|^{1/n−1/p} ≤ c(p, Ω) ≤ \frac{p}{p − n}|Ω|^{1/n−1/p},  (19)

i.e. the following inequality

(\frac{1}{|Ω|} ∫_{Ω} \frac{|f|^{n}}{δ^{n}} dx)^{1/n} ≤ λ (\frac{1}{|Ω|} ∫_{Ω} |∇f|^{p}dx)^{1/p} , ∀f ∈ C_{0}^{∞}(Ω), (20)

is valid with a constant λ such that

1 ≤ λ ≤ \frac{p}{p − n} .

Proof of Theorem 7. Let f ∈ C_{0}^{∞}(Ω). Applying the Hölder inequality with the exponents p/n and p/(p − n) and Theorem 5, we easily obtain

(∫_{Ω} \frac{|f|^{n}}{δ^{n}} dx)^{1/n} ≤ |Ω|^{1/n−1/p} (∫_{Ω} \frac{|f|^{p}}{δ^{p}} dx)^{1/p}

≤ \frac{p}{p − n}|Ω|^{1/n−1/p} (∫_{Ω} |∇f|^{p} dx)^{1/p} .

From this one immediately obtains the upper bounds for c(p, Ω) and λ.

Now we prove the lower estimate for c(p, Ω). According to the theorem on regularized distance functions (see V.I. Burenkov [14], P. 78, compare A.P. Calderon and A. Zigmund [15] and L.E. Fraenkel [21]), for any open set Ω ⊂ ℝ^{n} (Ω ≠ ℝ^{n}) and for any β ∈ (0, 1) there exists a C^{∞}(Ω)-function δ_{β}(., Ω) such that

βδ(x, Ω) ≤ δ_{β}(x, Ω) ≤ δ(x, Ω),  |∇δ_{β}(x, Ω)| ≤ 1,  x ∈ Ω.

Consider the functions

f_{αβε}(x) = \{ \begin{matrix}(δ_{β}(x, Ω) − ε)^{α}, & if x ∈ Ω(β, ε), \\ 0, & if x ∈ Ω \ Ω(β, ε), \end{matrix}

where 0 < β ≤ 1, 1 ≤ α < ∞, 0 < ε < βδ_{0}(Ω) and

Ω(β, ε) = {x ∈ Ω : δ_{β}(x, Ω)| > ε}, Ω(1, ε) = {x ∈ Ω : δ(x, Ω)| > ε}.

The set Ω(β, ε) is bounded since the volume of Ω is finite. It is clear that f_{αβε} ∈ C_{0}^{1}(Ω) for α > 1 and β < 1. Since C_{0}^{∞}(Ω) is dense in C_{0}^{1}(Ω) (see, for instance, [14] ), one can write

c(p, Ω) ≥ (∫_{Ω(β,ε)} \frac{|f_{αβε}|^{n}}{δ^{n}} dx)^{1/n} (∫_{Ω(β,ε)} |∇f_{αβε}|^{p} dx)^{−1/p} ,

where α > 1 and β < 1. Letting α → 1 and β → 1 and using |∇δ_{β}(x, Ω)| ≤ 1, we get

c(p, Ω) ≥ (∫_{Ω(1,ε)} \frac{|f_{11ε}|^{n}}{δ^{n}} dx)^{1/n} (∫_{Ω(1,ε)} dx)^{−1/p}

= (∫_{Ω(1,ε)} \frac{(δ − ε)^{n}}{δ^{n}} dx)^{1/n} |Ω(1, ε)|^{−1/p},

where δ = dist(x, ∂Ω). Lebesgue’s theorem on majorized convergence applied to the last inequality as ε → 0 gives

c(p, Ω) ≥ |Ω|^{1/n−1/p}.

This completes the proof of Theorem 7.

In the next theorem we consider the following inequality

∫_{Ω} \frac{|f|}{δ^{s}}dx ≤ c (_{∫Ω} \frac{|∇f|^{p}}{δ^{s}} dx)^{1/p} ,  ∀f ∈ C_{0}^{∞}(Ω).  (21)

Theorem 8. Suppose that Ω is an open set in ℝ^{n} (n ≥ 1) such that

M = ∫_{Ω} δ^{\frac{p}{p−1} −s}dx < +∞.

If p > 1, s ≥ n and 1/p+1/s > 1, then the best constant in (21) satisfies the inequalities

1 − \frac{1}{p} ≤ \frac{c}{M^{1−1/p}} ≤ Γ^{1−1/p} (\frac{2p}{p − 1}) (1 − \frac{1}{p}) ,  (22)

where Γ is Euler’s gamma function.

Proof of Theorem 8. Following the proof of Theorem 6 with a little change we obtain the upper estimate in (22). Namely, by using (17) and (18) for s > n and p = 1, summing over all S with Q(S) ≠ ∅, and applying the Hölder inequality with the exponents p and q = \frac{p}{p−1}, we easily obtain

∫_{Ω_{1}} \frac{|f|}{δ^{s}}dx ≤ (∫_{Ω_{1}} \frac{|∇f|^{p}}{δ^{s}} dx)^{1/p} (∫_{Ω_{1}} Φ^{q}dx)^{1/q} ,

where

Φ|Q(S) = \frac{δ^{s/p}}{δ^{s−1}} ∫_{δ}^{ψ_{n−k}} \frac{dt}{t^{s−k+1}} (k = 1, 2, …, n)

for any cubic (n − k)-face S. From this it follows that

∫_{Ω_{1}} Φ^{q}dx ≤ \frac{1}{q^{q}} Γ(q + 1) · M

since

∫_{0}^{φ} (\frac{r^{s/p}}{r^{s−1}} ∫_{r}^{φ} \frac{dt}{t^{s−k+1}} )^{q} r^{k−1}dr

= \frac{φ^{q−s+k}}{(s − k)^{q}} ∫_{0}^{1} τ^{q−s}(1 − τ^{s−k})^{q}τ^{k−1}dτ

= \frac{Γ(\frac{q}{s−k})Γ(q + 1)}{(s − k)^{q}Γ(\frac{q}{s−k} + q)} ∫_{0}^{φ} r^{q−s}r^{k−1}dr

≤ q^{−q}Γ(q + 1) ∫_{0}^{φ} r^{q−s}r^{k−1}dr.

To obtain the last inequality we have used that

\frac{Γ(\frac{q}{s−k})}{(s − k)^{q}Γ(\frac{q}{s−k} + q)} ≤ q^{−q}.

This inequality is a simple consequence of the identity Γ(α + 1) = αΓ(α) and the sharp estimates by W. Gautschi for the gamma function (see [23] or the book by D. S. Mitrinovic [37]).

To obtain the lower estimate in (22), we first observe that

∫_{Ω} \frac{|f_{0}|}{δ^{s}}dx (_{∫Ω} \frac{|∇f_{0}|^{p}}{δ^{s}} dx)^{−1/p} = (1 − 1/p)M^{1−1/p}

for the function f_{0} = δ^{\frac{p}{p−1}} .

To complete the proof, we apply the Calderon - Zigmund - Burenkov theorem on regularized distance functions ([14], P. 78) as in the proof of Theorem 7.

6. An improved form of the Brezis-Marcus inequality and related results.

We shall obtain the following generalization of the cited equation

c_{p}(p, Ω) = p/(p − 1)

for convex domains.

Theorem 9. Let Ω be an open, convex and proper subset of ℝ^{n}. If p ≥ 1 and s > 1 then

∫_{Ω} \frac{|f|^{p}}{δ^{s}} dx ≤ (\frac{p}{s − 1})^{p} ∫_{Ω} \frac{|∇f|^{p}}{δ^{s−p}} dx, ∀f ∈ C_{0}^{∞}(Ω),  (23)

where δ = δ(x) = dist(x, ∂Ω).

Also, we shall prove the following lower estimate (compare the case α = n − 1 of Theorem 5.1 in [16]). Let Ω be a bounded open set in ℝ^{n}. Consider its boundary surface area by Minkowski (see [24]):

σ(Ω) = lim_{t→+0} sup \frac{A(t)}{t},

where A(t) = mes{x ∈ Ω : dist(x, ∂Ω) < t}.

Theorem 10. If p ≥ 1 and s > 1 and Ω is a bounded open set in ℝ^{n} with finite surface area σ(Ω), then

c_{p}(s, Ω) ≥ \frac{p}{s − 1}.

From Theorems 9 and 10 it follows that c_{p}(s, Ω) = p/(s − 1) for p ≥ 1 and s > 1 and any bounded convex domain Ω ⊂ ℝ^{n}. The main aim of this section is to improve this result using additional terms in the inequality (23). To this end, examine first the following theorem of H.Brezis and M.Marcus [13]: if Ω is a bounded open and convex subset of ℝ^{n} and λ = 1/diam(Ω)^{2} then

∫_{Ω} \frac{|f|^{2}}{δ^{2}} dx + λ ∫_{Ω} |f|^{2}dx ≤ 4 ∫_{Ω} |∇f|^{2}dx, ∀f ∈ C_{0}^{∞}(Ω). (24)

In [27] M.Hoffmann–Ostenhof, T.Hoffmann–Ostenhof and A.Laptev proved that λ in (24) can be replaced by c(n)/|Ω|^{2/n}, where |Ω| = mes Ω.

It is natural to ask whether inequality (24) is valid with some λ > 0 for an unbounded convex domain. It is clear that the validity of (24) implies

λ ≤ 4λ_{1}(Ω),

where λ_{1}(Ω) is the first Dirichlet eigenvalue for the Laplace equation. According to the theory of Isoperimetric Inequalities in Mathematical Physics (see [8], [10], [41]) we have

λ ≤ \frac{const.}{δ_{0}^{2}(Ω)}.

This argument shows that (24) is not true with λ > 0 for any unbounded convex domain Ω in the case δ_{0} = δ_{0}(Ω) = +∞. It is also clear that there are unbounded convex domains Ω ⊂ ℝ^{n} with δ_{0}(Ω) < +∞ in the case n ≥ 2, only.

We extend the Brezis - Marcus inequality to certain unbounded convex domains. More precisely, we prove that (24) is true with λ = 1/δ_{0}^{2}, and that a similar improved version of (23) is valid.

Theorem 11. Let Ω be an open, convex and proper subset of ℝ^{n}. If p ≥ 1 and s > 1, then for any f ∈ C_{0}^{∞}(Ω)

∫_{Ω} \frac{|f|^{p}}{δ^{s}} dx + \frac{1}{(s − 1)δ_{0}^{s}} ∫_{Ω} |f|^{p}dx ≤ (\frac{p}{s − 1})^{p} ∫_{Ω} \frac{|∇f|^{p}}{δ^{s−p}} dx,  (25)

where δ = dist(x, ∂Ω), δ_{0} = sup{δ(x) : x ∈ Ω}.

Proof of Theorems 9 and 11. Let Ω be an open, convex and proper subset of ℝ^{n}. It is known that for any compact set K ⊂ Ω there exists a convex n-dimensional polytope Q such that K ⊂ intQ ⊂ Ω (see [24]). Hence, for given f ∈ C_{0}^{∞}(Ω) it is sufficient to prove inequalities (23) and (25) for every convex, n-dimensional polytope Q such that

supp f ⊂ intQ ⊂ Ω.

Let Q be such a polytope, and let S_{1}, S_{2}, …, S_{m} be the collection of all (n − 1)-faces of Q. First we will construct a special decomposition of Q:

Q = ∪_{j=1}^{m}Q_{j}, intQ_{j} ∩ Q_{k} = ∅ for j ≠ k, (26)

where Q_{j} are convex and compact sets. Namely, for each x′ ∈ intS_{j} we define

φ_{j}(x′) = max{t ∈ ℝ_{+} : B(x′ + tν(x′), t) ⊂ Q},

where ν(x′) is the interior normal to S_{j} at the point x′, B(x, t) is the ball {y ∈ ℝ^{n} : |y − x| ≤ t}, ℝ_{+} = [0, ∞). We easily obtain that

Q_{j} := S_{j} ∪ {x = x′ + tν(x′) : 0 < t ≤ φ_{j}(x′)}

is a closed, n-dimensional and convex set, and Q_{1}, Q_{2}, …, Q_{m} satisfy (26). Due to convexity of Q_{j}, mes_{n}∪_{j=1}^{m}∂Q_{j} = ∅. Hence, for any function g ∈ L^{1}(Q)

∫_{Q} g(x) dx = ∑_{j=1}^{m} ∫_{Q_{j}} g(x) dx

and, by Fubini’s theorem

∫_{Q_{j}} g(x) dx = ∫_{S_{j}} dx′ ∫_{0}^{φ_{j}(x′)} g(x′ + tν(x′))dt.  (27)

For any x = x′ + tν(x′) ∈ Q_{j} one has

δ(x) = t,  δ(x) ≤ φ_{j}(x′) ≤ δ_{0},  (28)

where δ = δ(x) = dist(x, ∂Q) and δ_{0} is its maximum in Q.

Suppose that p ≥ 1, s > 1 and f ∈ C_{0}^{∞}(Q). By using (27) and (28) for the function

g(x′ + tν(x′)) = |f(x′ + tν(x′))|^{p} (\frac{1}{t^{s}} + \frac{1}{(s − 1)δ_{0}^{s}} ) ,

we get

∫_{Q_{j}} |f(x)|^{p} [\frac{1}{δ^{s}(x)} + \frac{1}{(s − 1)δ_{0}^{s}} ] dx

≤ p ∫_{S_{j}} dx′ ∫_{0}^{φ_{j}(x′)} (\frac{1}{t^{s}} + \frac{1}{(s − 1)δ_{0}^{s}} ) dt ∫_{0}^{t} |f(y)|^{p−1} |\frac{∂f(y)}{∂τ}| dτ

= \frac{p}{s − 1} ∫_{S_{j}} dx′ ∫_{0}^{φ_{j}(x′)} \frac{|f(y)|^{p−1}}{τ^{s−1}} |\frac{∂f(y)}{∂τ}| A(x′, τ)dτ,

where y = x′ + τν(x′), and

A(x′, τ) = 1 − \frac{τ^{s−1}}{φ_{j}^{s−1}} + \frac{τ^{s−1}}{δ_{0}^{s}} [φ_{j}(x′) − τ] ≤ 1,

By using this and the inequality |\frac{∂f}{∂τ} | ≤ |∇f| and by summing over j = 1, 2, …, m we get

I_{Q} := ∫_{Q} |f|^{p} (\frac{1}{δ^{s}} + \frac{1}{(s − 1)δ_{0}^{s}} ) dx ≤ \frac{p}{s − 1} ∫_{Q} \frac{|f|^{p−1}}{δ^{s−1}} |∇f|dx. (29)

If p = 1, then (29) is the inequality (25) for Ω = Q. Following Hardy (see [25], Theorem 330), in the case p > 1 we apply Hölder’s inequality in (29) to get

I_{Q} ≤ \frac{p}{s − 1} (∫_{Q} (\frac{|f|^{p−1}}{δ^{s−s/p}} )^{p′} dx)^{1/p′} (∫_{Q} (\frac{|∇f|}{δ^{s/p−1}} )^{p} dx)^{1/p}

= \frac{p}{s − 1} (∫_{Q} \frac{|f|^{p}}{δ^{s}} dx)^{1/p′} (∫_{Q} \frac{|∇f|^{p}}{δ^{s−p}} dx)^{1/p} ,

where 1/p′ = 1 − 1/p. Consequently,

I_{Q} ≤ (\frac{p}{s − 1})^{p} ∫_{Q} \frac{|∇f|^{p}}{δ^{s−p}} dx.

This completes the proof of Theorems 9 and 11.

Proof of Theorem 10. Suppose that p ≥ 1, s > 1 and σ(Ω) is finite. Let us denote

X = ∫_{Ω} \frac{|u|^{p}}{δ^{s}} dx, Y = ∫_{Ω} \frac{|∇u|^{p}}{δ^{s−p}} dx, δ = dist(x, ∂Ω).

For ε ∈ (0, 1) and u = u_{ε}(x) = δ^{(s−1+ε)/p} we have

X = M_{−1+ε}(Ω),  Y = (\frac{s − 1 + ε}{p})^{p} M_{−1+ε}(Ω),  (30)

where M_{−1+ε}(Ω) is the following moment of Ω about its boundary

M_{−1+ε}(Ω) = ∫_{Ω} δ^{−1+ε}dx.

Using (30), the equation

lim_{ε→+0} sup εM_{−1+ε}(Ω) ≤ σ(Ω)  (31)

and the Calderon - Zigmund - Burenkov theorem (see [14], P. 78), we obtain that

c_{p}(s, Ω) ≥ p(s − 1)^{−1}.

To prove (31) we remark that

γ_{k} := sup \{\frac{A(t)}{t} : 0 < t < \frac{δ_{0}}{k} \} − σ(Ω) → 0

as k → ∞. Consequently,

b := lim_{ε→+0} sup ε ∫_{0}^{δ_{0}} \frac{A(t)}{t^{2−ε}}dt = lim_{k→∞} lim_{ε→+0} sup ε ∫_{0}^{δ_{0}/k} \frac{A(t)}{t^{2−ε}}dt

≤ lim_{k→∞}[σ(Ω) + γ_{k}] lim_{εø0} (\frac{δ_{0}}{k} )^{ε} = σ(Ω).

Integrating by parts and using the known formulas (see [43], Chapter 1), we have

(1 − ε) ∫_{0}^{δ_{0}} \frac{A(t)}{t^{2−ε}}dt = \frac{A(δ_{0})}{δ_{0}^{1−ε}} + ∫_{0}^{δ_{0}} \frac{dA(t)}{t^{1−ε}} = M_{−1+ε}(Ω).

for any ε ∈ (0, 1). Consequently,

lim_{ε→+0} sup εM_{−1+ε}(Ω) = b ≤ σ(Ω)

which proves (31).

The proof of Theorem 10 is complete.

Example 4. Suppose that p ≥ 1 and s > 1. Our aim is to obtain an upper estimate for λ > 0 in the inequality

∫_{Ω} \frac{|u|^{p}}{δ^{s}} dx + \frac{λ}{δ_{0}^{s}} ∫_{Ω} |u|^{p}dx ≤ (\frac{p}{s − 1})^{p} ∫_{Ω} \frac{|∇u|^{p}}{δ^{s−p}} dx, ∀u ∈ C_{0}^{∞}(Ω), (32)

for convex domains (compare formula (25)).

We will examine the domains

Ω_{ε} = (−ε, 1 + ε)^{n−1} × (−ε, ε) ⊂ R^{n}

and functions u_{ε} defined by

u_{ε}(x) = δ^{(s−1+ε)/p}, 0 < ε < 1,

in the case n = 2 only. Note that δ_{0} = sup δ = ε for Ω_{ε}.

For n = 2, straightforward computations give

X = \frac{2ε^{ε}}{ε} + \frac{8ε^{ε}}{1 + ε}, Y = (\frac{s − 1 + ε}{p})^{p} X,

Z := \frac{λ}{δ_{0}^{s}} ∫_{Ω_{ε}} u_{ε}^{p}dx = \frac{2λε^{ε}}{s + ε} (1 + \frac{4ε}{s + 1 + ε})

and

\frac{X + Z − (\frac{p}{s−1})^{p} Y}{2ε^{ε}} = \frac{λ}{s} − \frac{p}{s − 1} + O(ε).

If (32) is true then

λ ≤ \frac{ps}{s − 1}.

HARDY TYPE INEQUALITIES 29

Hence, the best possible value of λ in (32) satisfies the inequalities

\frac{1}{s − 1} ≤ λ ≤ \frac{ps}{s − 1} (∀p ≥ 1, s > 1).

Acknowledgments. I wish to express my gratitude to D. Kh. Mushtary, K.-J.Wirths, W. Sander, and T. Carroll for their help with the bibliographical sources.

This research was supported by a grant of the Deutsche Forschungsgemeinschaft and by the Russian Fund of Basic Research (Grant 05-01-00523).

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Kazan State University, Universitetskaya Str.17, 420008, Kazan, Russia E-mail address: favhadiev@ksu.ru

Received March 9, 2006