Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 21, 2006, 3–31

© F. G. Avkhadiev

Abstract. Let $\Omega$ be an open set in ${\mathbb{ℝ}}^n$ such that $\Omega \ne {\mathbb{ℝ}}^n$. For $1\le p<\infty$, $1<s<\infty$ and $\delta =dist\left(x,\partial \Omega \right)$ we estimate the Hardy constant

$c_p\left(s,\Omega \right)=sup\left\{\parallel f∕{\delta }^{s∕p}{\parallel }_{L^p\left(\Omega \right)}:f\in C_0^{\infty }\left(\Omega \right),\parallel \left(\nabla f\right)∕{\delta }^{s∕p-1}{\parallel }_{L^p\left(\Omega \right)}=1\right\}$

and some related quantities.

For open sets $\Omega \subset {\mathbb{ℝ}}^2$ we prove the following bilateral estimates

where $M_0\left(\Omega \right)$ is the geometrical parameter defined as the maximum modulus of ring domains in $\Omega$ with center on $\partial \Omega$. Since the condition $M_0\left(\Omega \right)<\infty$ means the uniformly perfectness of $\partial \Omega$, these estimates give a direct proof of the following Ancona-Pommerenke theorem: $c_2\left(2,\Omega \right)$ is finite if and only if the boundary set $\partial \Omega$ is uniformly perfect (see [2], [22] and [40]).

Moreover, we obtain the following direct extension of the one dimensional Hardy inequality to the case $n\ge 2$: if $s>n$, then for arbitrary open sets $\Omega \subset {\mathbb{ℝ}}^n$ ($\Omega \ne {\mathbb{ℝ}}^n$) and any $p\in \left[1,\infty \right)$ the sharp inequality $c_p\left(s,\Omega \right)\le p∕\left(s-n\right)$ 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.

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

for $p\ge 1$, $s\in \mathbb{ℝ}$, $s\ne 1$ and any absolutely continuous function $u:\left[0,\infty \right)\to \mathbb{ℝ}$, $u^{\prime }∕t^{s∕p-1}\in L^p\left[0,\infty \right)$ such that $u\left(0\right)=0$ in the case $s>1$ and $u\left(+\infty \right)=0$ 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 $u\not\equiv 0$ then equality is not attained, but the constant ${\left(p∕\mid s-1\mid \right)}^p$ 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 $\Omega$ is an open and proper subset of ${\mathbb{ℝ}}^n$, $u$ and $u^{\prime }$ are replaced by functions $f\in C_0^{\infty }\left(\Omega \right)$ and $\nabla f$, the gradient of $f$, and powers of $t$ are replaced by powers of

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

The classical examples which are simple consequences of the one dimensional Hardy inequalities are given by the equations : for $p\in \left[1,\infty \right)$ and $s\in \mathbb{ℝ}$

where $H$ is a half space in ${\mathbb{ℝ}}^n$ (see [10], [34], [38]). For $p>1$ and any open convex set $\Omega \subset {\mathbb{ℝ}}^n$, $\Omega \ne {\mathbb{ℝ}}^n$ it is known that

(see [17], [32], [33]). Explicit estimates of $c_p\left(s,\Omega \right)$ are also known in some particular cases when $\Omega$ is not a convex domain. Namely,

1) if $\Omega$ is a simply connected plane domain, then $c_2\left(2,\Omega \right)\le 4$ (see [2], [3], [11], [16]);

2) if $\Omega$ is a domain in ${\mathbb{ℝ}}^n$ with smooth boundary, then $c_p\left(p,\Omega \right)\ge p∕\left(p-1\right)$ (see [16] and [32]).

For $p\ge 1$ and $s>1$, it is a classical fact that there exists a finite constant $c_p\left(s,\Omega \right)$ for any domain $\Omega$ 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 $\Omega$. 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 $c_p\left(s,\Omega \right)$ in the case when $p\in \left[1,\infty \right),s\ge n$ and to estimate some related quantities.

In Sections 2 and 3 we examine the quantity $c_p\left(s,\Omega \right)$ in the case, when $p\in \left[1,\infty \right)$ and $s=n$. In Section 2 the case $p\in \left[1,\infty \right)$ and $n=s=2$ is considered. For plane domains $\Omega \subset {\mathbb{ℝ}}^2$ we prove the following bilateral estimates

where $M_0\left(\Omega \right)$ is the geometrical parameter defined as the maximum modulus of genuine annuli in $\Omega$ with center on $\partial \Omega$. Note that, by results of Ch. Pommerenke [40] and A. Ancona [2], $c_2\left(2,\Omega \right)$ for a plane domain $\Omega \subset {\mathbb{ℝ}}^2$ is finite if and only if the boundary set $\partial \Omega$ is uniformly perfect. Clearly, our estimates give $L^p$ - version and a direct proof of the Ancona - Pommerenke theorem, since the condition $M_0\left(\Omega \right)<\infty$ means the uniformly perfectness of $\partial \Omega$.

In Section 3 we extend our estimates to the quantities

and

related to Rellich’s constant of $\Omega \subset {\mathbb{ℝ}}^2$ and discuss a generalization of results to the quantity $c_p\left(n,\Omega \right)$ for space domains. In particular, we prove that

where $\Omega \subset {\mathbb{ℝ}}^n$, $n\ge 3$.

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

is valid. This completes the following known facts: J.L. Lewis [31] discovered that there is $c_p\left(p,\Omega \right)<+\infty$ for any open set $\Omega \subset {\mathbb{ℝ}}^n$, whenever $p>n$. A.Wannebo [44] proved a generalization of this assertion: if $p>n$ and $s>p-\varepsilon \left(p,n\right)$ for a convenient $\varepsilon \left(p,n\right)>0$, then $c_p\left(s,\Omega \right)<+\infty$ for any open set $\Omega \subset {\mathbb{ℝ}}^n$.

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

is considered. For open sets with finite volume $\mid \Omega \mid =mes\Omega$ we prove the inequalities

We extend the above results on $c_p\left(s,\Omega \right)$ and $c\left(p,\Omega \right)$ to some other Hardy type inequalities with explicit estimates of all constants in function of parameters and simple geometric quantities of $\Omega$. 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

for any bounded open set $\Omega \subset {\mathbb{ℝ}}^n$ with finite boundary surface area in the sense of Minkowski. On the other hand, for parameters $p\ge 1$, $s>1$ and convex open sets $\Omega \subset {\mathbb{ℝ}}^n$ we prove that any function $f\in C_0^{\infty }\left(\Omega \right)$ satisfies the inequality

where $\delta =dist\left(x,\partial \Omega \right)$, ${\delta }_0=sup\left\{\delta \left(x\right):x\in \Omega \right\}$.

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.

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 $c_2\left(2,\Omega \right)\ge 2$ for any domain with smooth boundary and $c_2\left(2,\Omega \right)=2$ for convex domains $\Omega$ (see [16]). Moreover, if $\Omega$ is a simply connected domain in $\mathbb{ℂ}$ then $c_2\left(2,\Omega \right)\le 4$ (see [2], [3], [11] and [16]). In the general case, for instance, in the case when $\Omega$ is not a finitely connected domain, explicit estimates of $c_2\left(2,\Omega \right)$ are unknown.

In this section, we shall prove $L^p$-version ($1\le p<\infty$) of Theorem 1 with bilateral explicit estimates of the Hardy constant using a simple geometrical parameter of $\Omega$. In particular, we give a direct proof of Theorem 1.

Let $\Omega$ be an open set in the complex plane $\mathbb{ℂ}$ such that $\Omega \ne \mathbb{ℂ}$. For any fixed $p\in \left[1,\infty \right)$ we consider Hardy’s inequality

where $z=x+iy$ and $c_p\left(2,\Omega \right)$ is the minimum possible constant that generalizes $c_2\left(2,\Omega \right)$.

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

where the supremum is taken over all annuli $A$ such that

We take $M_0\left(\Omega \right)=0$ by definition, when there is no circle in $\Omega$ with center on $\partial \Omega$. We say that $\partial \Omega$ is uniformly perfect if $M_0\left(\Omega \right)<\infty$.

In the sequel, we need the constant

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

The main result of this section is the following assertion.

Proof of Theorem 2. First we prove the lower estimate for $c_p\left(2,\Omega \right)$. Clearly, it is sufficient to consider the case, when $0<M_0\left(\Omega \right)\le \infty$ and $0<c_p\left(2,\Omega \right)<\infty$.

We shall examine the cases $p\ge 2$ and $p<2$ separately. Suppose first that $2\le p<\infty$ and $c_p\left(2,\Omega \right)<2M_0\left(\Omega \right)$ for an open and proper subset of $\mathbb{ℂ}$. From the definition of $M_0\left(\Omega \right)$ it follows that there is an annulus $A=\left\{z\in \mathbb{ℂ}:r\left(A\right)<\mid z-z_0\mid <R\left(A\right)\right\}\subset \Omega$ such that $z_0\in \partial \Omega$ and

Without loss of generality we can suppose that $z_0=0,R\left(A\right)=1$ and $r\left(A\right)=\varepsilon \in \left(0,1\right)$, since $c_p\left(2,\Omega \right)$ is invariant under linear transformations of $\Omega$. We have

and

Using the polar coordinates, functions $f\left(r,\theta \right)=v\left(r\right)$ with $v\in C_0^{\infty }\left(\varepsilon ,1\right)$ and the estimate $\left(z,\partial \Omega \right)\le \mid z\mid$, we obtain

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

Approximating $g_0\left(t\right)=sint$ by functions $g\in C_0^{\infty }\left(0,\pi \right)$ , we get

which contradicts to the assumption $c_p\left(2,\Omega \right)<2M_0$. Hence, $2M_0\left(\Omega \right)\le 2M_0\le c_p\left(2,\Omega \right)$ in the case $2\le p<\infty$.

In the case $1\le p<2$ and $c_p\left(2,\Omega \right)<\infty$, we combine the Hardy and Hölder inequalities in the following way

It follows that $c_2\left(2,\Omega \right)\le \frac{2}{p}{c}_p\left(2,\Omega \right)$. As is proved that $c_2\left(2,\Omega \right)\ge 2M_0\left(\Omega \right)$, we get

when $p\in \left[1,2\right)$.

Now we suppose that $M_0\left(\Omega \right)<\infty$ and we prove the upper estimate. Clearly, the condition $M_0\left(\Omega \right)<\infty$ assure that $\partial \Omega$ has no isolated point. Also, it is sufficient to obtain the upper inequality in (2) for connected components of $\Omega$.

Since $\mathbb{ℂ}\setminus \Omega \ne \varnothing$ and $\partial \Omega$ has no isolated point, any connected component of $\Omega$ is a hyperbolic domain in $\mathbb{ℂ}$, i.e. its boundary has more than one point in $\mathbb{ℂ}$. Without loss of generality we can suppose that $\Omega$ itself is a hyperbolic domain in $\mathbb{ℂ}$. Let ${\lambda }_{\Omega }$ be the density of the Poincaré metric on $\Omega$ with curvature $-4$ (see [1], [9]).

Let $p>1$ and let $f\in C_0^{\infty }\left(\Omega \right)$. Since $p>1$, we have that $\mid f{\mid }^p\in C_0^1\left(\Omega \right)$ and $\mid \nabla \mid f{\mid }^p\mid =p\mid f{\mid }^{p-1}\mid \nabla f\mid$. Using the Liouville equation in the form

and the Green formula

for $v=log{\lambda }_{\Omega }^{-1}$ and $u=\mid f{\mid }^p,f\in C_0^{\infty }\left(\Omega \right)$, we obtain

Combining this with the Hölder inequality

we immediately get

for any $f\in C_0^{\infty }\left(\Omega \right)$. Since this inequality is proved for any $p\in \left(1,\infty \right)$, letting $p\to 1$ for a fixed $f\in C_0^{\infty }\left(\Omega \right)$ we obtain that (3) is true in the case $p=1$, too.

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

one gets

Hence, for any $p\in \left[1,\infty \right)$ and any $f\in C_0^{\infty }\left(\Omega \right)$,

where

Beardon and Pommerenke [12] proved that the condition $M_0\left(\Omega \right)<\infty$ holds if and only if $\alpha \left(\Omega \right)>0$ and, in particular,

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

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 $M_0\left(\Omega \right)=0$ in (2).

Let us mention that $M_0\left(\Omega \right)=0$ for any simply connected domain, but the converse is not true. The family $\left\{\Omega :M_0\left(\Omega \right)=0\right\}$ is a large collection of open sets $\Omega \subset \mathbb{ℂ}$ and it contains domains of arbitrary connectivity. For example, the equality $M_0\left({\Omega }_0\setminus \overline{K}\right)=0$ holds for domains satisfying the following conditions:

$\left(1\right)$ ${\Omega }_0$ is an open set in $\mathbb{ℂ}$ such that $sup\left\{dist\left(z,\partial {\Omega }_0\right):z\in {\Omega }_0\right\}=1$, in particular, ${\Omega }_0$ is a stripe with width $2$;

$\left(2\right)$ $K={\bigcup }_{m=1}^{\infty }{C}_m$, where $C_m$ are continuums (connected compact sets) such that $diam{C}_m\ge 2$ for all $m\ge 1$;

$\left(3\right)$ $K\subset {\Omega }_0$ and ${\Omega }_0\setminus \overline{K}$ is nonempty.

The Pommerenke condition on the capacity density for $\mathbb{ℂ}\setminus \Omega$ has the form

where $capE$ 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

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

We complete this section by three examples considering domains of the form $B\left(0,3\right)\setminus E_j=\left\{z\in \mathbb{ℂ}:\mid z\mid <3\right\}\setminus E_j$ with perfect boundaries.

Example 1. Suppose that $E_1={\bigcup }_{m=1}^{\infty }{K}_m\bigcup \left\{0\right\}$, where

The domain ${\Omega }_1=B\left(0,3\right)\setminus E_1$ contains the annuli

and

Consequently, $M_0\left({\Omega }_1\right)=c_p\left(2,{\Omega }_1\right)=\infty$.

Example 2. Now, we consider $E_2={\bigcup }_{m=1}^{\infty }{L}_{2m-1}\bigcup \left\{0\right\}$, where

For any annulus $A$ in ${\Omega }_2=B\left(0,3\right)\setminus E_2$ with center on $\partial {\Omega }_2$ we have $R\left(A\right)∕r\left(A\right)\le 3$. It is an easy task to show that $2\pi M_0\left({\Omega }_2\right)=log3$. Accordingly, $c_p\left(2,{\Omega }_2\right)\le {\left(8.76+log3\right)}^{2}p∕2<48p$.

Example 3. Let $E_3$ be the classical Cantor set. In [12] estimates for $\alpha \left(\mathbb{ℂ}\setminus E_3\right)$ are proved. We consider

It is easy to show that $M_0\left({\Omega }_3\right)=M_0\left({\Omega }_2\right)={\left(2\pi \right)}^{-1}log3$. Thus, $c_p\left(2,{\Omega }_3\right)<48p$.

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 $\Omega$ be an open and proper subset of $\mathbb{ℂ}$. We define

and

where $z=x+iy$ and $\Delta$ is the Laplace operator.

In [3] it is proved that ${\kappa }_1\left(\Omega \right)\le 4$ and ${\kappa }_2\left(\Omega \right)\le 4$ for any simply connected domain $\Omega$. In the next theorem we give bilateral estimates of ${\kappa }_1\left(\Omega \right)$ and ${\kappa }_2\left(\Omega \right)$ for open sets with uniformly perfect boundary. Also, we obtain an improvement of the upper bound of $c_2\left(2,\Omega \right)$, ${\kappa }_1\left(\Omega \right)$ and ${\kappa }_2\left(\Omega \right)$ for doubly connected domains.

Proof of Theorem 3. Suppose that $\partial \Omega$ is uniformly perfect and $f\in C_0^{\infty }\left(\Omega \right)$. Using the Green formula and the Cauchy - Schwartz inequality one gets

This inequality and the definitions of $c_2\left(2,\Omega \right)$ and ${\kappa }_1\left(\Omega \right)$ imply

for any $f\in C_0^{\infty }\left(\Omega \right)$ satisfying

Consequently,

Inequalities (2) and (8) immediately give the upper bounds of ${\kappa }_1\left(\Omega \right)$ and ${\kappa }_2\left(\Omega \right)$ in the general case. Moreover, it is known (see, for instance, [20], Lemma 1.1) that the inequality

is valid for any simply or doubly connected hyperbolic domain $\Omega$. 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 ${\kappa }_2\left(\Omega \right)$, only. To this end we assume that ${\kappa }_2\left(\Omega \right)<2M_0\left(\Omega \right)$. Without loss of generality we can suppose that $0\in \Omega$ and there exists an annulus $A=\left\{z\in \mathbb{ℂ}:\varepsilon <\mid z\mid <1\right\}$ such that $A\subset \Omega$ and $M_0:=\frac{1}{2\pi }log\frac{1}{\varepsilon }>{\kappa }_2\left(\Omega \right)∕2$. One has

and $dist\left(z,\partial \Omega \right)<\mid z\mid$ for $z\in A$. Consequently,

for radial functions $v\left(r\right)=f\left(r,\theta \right),v\in C_0^{\infty }\left(\varepsilon ,1\right)$.

After the changes $r=\varepsilon exp\left(2M_{0}t\right)$ and $v\left(r\right)=g\left(t\right)$, the last inequality can be written as the following Wirtinger type inequality

Consequently, ${\kappa }_2\left(\Omega \right)\ge 2M_0$.

This completes the proof of Theorem 3.

Finally, we consider the spatial case. Let $\Omega$ be an open set in ${\mathbb{ℝ}}^n$ such that $\Omega \ne {\mathbb{ℝ}}^n$. Suppose that $1\le p<\infty ,1<s<\infty$ and consider the Hardy constant

where $\delta =dist\left(x,\partial \Omega \right)$.

It is known that $c_p\left(s,\Omega \right)<\infty$ for any domain $\Omega$ with Lipschitz boundary (see [38]). More general families of domains with finite $c_p\left(s,\Omega \right)$ are given in the papers by Ancona [2] (case $p=s=2$), Lewis [31] ( case $p=s>1$) and Wannebo [44] (case $p\ge 1,s>1$).

In the case $n\ge 3$, 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 $\Omega \subset {\mathbb{ℝ}}^n\left(\Omega \ne {\mathbb{ℝ}}^n,n\ge 3\right)$ we consider the quantity $M_0\left(\Omega \right)$ defined as in the planar case. More precisely, let