1 Introduction

In a series of works [3, 4, 6, 7, 9, 13], Dall’Acqua, Meister, Pulst, Robert, Sweers, and the present author have studied the question whether (suitably modified) Green functions in bounded smooth domains \(\Omega \subset {\mathbb {R}}^n\) under Dirichlet boundary conditions for elliptic operators L of higher order \(2m\ge 4\) may be estimated from above and below by the same positive prototype function. Such kind of result may be considered as a kind of substitute or relaxation of the maximum principle which in its strong form is true only for second order operators. Employing perturbative and blow-up arguments, all these works are based on Boggio’s explicit formula for the polyharmonic Green function under Dirichlet boundary conditions in balls (see [2, p. 126] and also [5, Lemma 2.27]) and similarly in half spaces (see [5, Remark 2.28]). This does not only show its positivity in these special domains but does also allow for precise estimates from above and below, see [5, Theorem 4.6] and Remark 2(a) below.

In order to keep this note short and as simple as possible, I shall consider only the (simplest) generic case

$$\begin{aligned} n>2m. \end{aligned}$$

The “small” dimensions \(n=2,\ldots ,2m\) require more effort, but related results will also hold there. The most general class of operators, which could be treated so far, was considered by Pulst [13]. He could show that in general bounded smooth domains the estimate (2) below holds for operators of the form

$$\begin{aligned} L=(-\Delta )^m +\sum _{j=0}^{m-1}\sum _{|\alpha |=|\beta |=j} D^\alpha (a_{\alpha \beta }(\, \cdot \, ) D^\beta ), \end{aligned}$$

where the coefficients obey symmetry, smoothness, and coercivity conditions.

The goal here is to show that the same estimates can be proved for a principal part with variable coefficients, as long as this stays in the class of powers of second order elliptic operators.

Theorem 1

For \(n,m\in \mathbb {N}\) with

$$\begin{aligned} n>2m, \end{aligned}$$

let \(\Omega \subset {\mathbb {R}}^n\) be a bounded \(C^{\infty }\)-smooth domain and L be a uniformly elliptic symmetric operator of the form

$$\begin{aligned} L=\left( -\sum ^n_{i,j=1} \partial _i \left( a_{ij}(\, \cdot \,)\partial _j \right) \right) ^m. \end{aligned}$$

For the coefficients, we assume that

$$\begin{aligned} a_{ij}(\, \cdot \,)=a_{ji}(\, \cdot \,)\in C^{\infty } (\overline{\Omega } ) \end{aligned}$$

and that there exist numbers

$$\begin{aligned} 0<\lambda \le \Lambda :\quad \forall x\in \overline{\Omega },\ \forall \xi \in {\mathbb {R}}^n:\quad \lambda |\xi |^2 \le \sum ^n_{i,j=1} a_{ij}(x)\xi _i \xi _j \le \Lambda |\xi |^2. \end{aligned}$$

Let \(G_{\Omega ,L}\) denote the Green function in \(\Omega \) for the following Dirichlet problem

$$\begin{aligned} Lu=f \text{ in } \Omega , \quad u=|\nabla u|=\cdots =|\nabla ^m u|=0 \text{ on } \partial \Omega . \end{aligned}$$
(1)

Then there exist constants \(c_1\ge 0\), \(c_2 >0\), depending on the domain \(\Omega \) and the elliptic operator L, such that we have the following Green function estimate:

$$\begin{aligned} c_2^{-1}~ H_{\Omega } (x,y)\le G_{\Omega ,L} (x,y)+c_1 d_\Omega (x)^m d_\Omega (y)^m\le c_2~ H_{\Omega } (x,y) \end{aligned}$$
(2)

for all \( x,y\in \Omega \), where

$$\begin{aligned} H_{\Omega } (x,y):= \displaystyle |x-y|^{2m-n} \min \left\{ 1,\frac{d_\Omega (x)^m d_\Omega (y)^m}{|x-y|^{2m}}\right\} \end{aligned}$$
(3)

and

$$\begin{aligned} d_\Omega (x):={\text {dist}}(x,\partial \Omega ). \end{aligned}$$

Remark 2

  1. (a)

    If \(L=(-\Delta )^m\) and \(\Omega \) is a ball or a half space in \({\mathbb {R}}^n\), (2) holds true with \(c_1=0\). This follows from Boggio’s formula as already mentioned at the beginning of the introduction. Since \(n>2m\), one has a unique Green function even in the whole space and this is given by the positive fundamental solution \(G_{(-\Delta )^m,{\mathbb {R}}^n}(x,y)=c_{m,n}|x-y|^{2m-n}\) with a suitable positive normalisation constant \(c_{m,n}>0\).

  2. (b)

    The existence of a Green function is in the case of a principal part with nonconstant coefficients by no means obvious and is due to Krasovskiĭ [11, Theorem 3.3]. Thanks to the symmetry of the operator L, this is symmetric, i.e. \( G_{\Omega ,L} (x,y)= G_{\Omega ,L} (y,x)\).

  3. (c)

    Krasovskiĭ’s work requires high regularity of the coefficients of L and of the domain. To keep the exposition as simple as possible, I just assume everything to be \(C^\infty \)-smooth.

  4. (d)

    The estimate from above follows from [4, 10, 11], see also [6], so that only the estimate from below has to be proved here.

  5. (e)

    With the techniques used here and developed in the mentioned works, one may admit also lower order terms as long as symmetry and coercivity assumptions are obeyed.

  6. (f)

    As it was investigated in [8], the situation becomes completely different when L is a general uniformly elliptic operator (even with constant coefficients), which cannot be written as a composition of second order operators, and when the dimension is large, i.e. \(n\ge 2m+2\). Here, in general, even the fundamental solution is sign changing near its singularity. This means that close to its singularity the negative part of any corresponding Green function becomes unbounded of the same order of magnitude as the positive part.

  7. (g)

    For implications of two-sided estimates as in Theorem 1, one may see [14,15,16].

An interesting consequence of Theorem 1 is a uniform local positivity result, which was shown in the mentioned works before for the more special class of operators.

Theorem 3

Let the assumptions of Theorem 1 be satisfied. Then there exist a constant \(r_{\Omega ,L}>0\) such that

$$\begin{aligned} G_{\Omega ,L}(x,y)>0 \text { for all } x,y\in \Omega \text { with } \left| x-y\right| <r_{\Omega ,L}. \end{aligned}$$
(4)

2 Local estimates from below

Lemma 4

Suppose that the assumptions of Theorem 1 are satisfied. Then for each \(x_0\in \overline{\Omega }\), there exists a radius \(r=r_{x_0}>0\) and a constant \(C=C_{x_0}>0\) such that for all \(x,y\in \Omega _{x_0,r}:={\Omega }\cap B_r(x_0)\), \(x\not =y\), one has

$$\begin{aligned} G_{\Omega ,L} (x,y) \ge C H_\Omega (x,y), \end{aligned}$$
(5)

where \(H_\Omega \) is defined in (3).

Proof

We assume by contradiction that no such positive numbers exist such that (5) is satisfied. This means that we may find sequences \((x_k)_{k\in \mathbb {N}},\ (y_k)_{k\in \mathbb {N}}\subset \Omega \) with

$$\begin{aligned} x_k\not = y_k, \ x_k,y_k \rightarrow x_0 \end{aligned}$$
(6)

and

$$\begin{aligned} G_{\Omega ,L} (x_k,y_k) <\frac{1}{k} H_\Omega (x,y). \end{aligned}$$
(7)

We shall rescale the domain and the Green function such that \(|x_k-y_k|\) becomes the new length unit. In the scaling limit, we shall end up with a Green function for a constant coefficient elliptic operator in the half space or in the whole space which is known to enjoy an estimate like (5) so that we end up with a contradiction. This key argument relies on knowing the Green functions in these prototype situations explicitly, see Remark 2(a) above.

To this end, we need to distinguish the cases whether the \(x_k,y_k\) are as close together as to the boundary or much closer.

First case: The sequence \(d_\Omega (x_k)/|x_k-y_k|\) remains bounded.

This means that there exists a constant \(c_3>0\) such that

$$\begin{aligned} \frac{d_\Omega (x_k)}{|x_k-y_k|}\le c_3. \end{aligned}$$
(8)

Thanks to the triangle inequality \(d_\Omega (y_k)\le d_\Omega (x_k)+|x_k-y_k|\), we also have

$$\begin{aligned} \frac{d_\Omega (y_k)}{|x_k-y_k|}\le c_3+1. \end{aligned}$$
(9)

This yields in view of (6) that

$$\begin{aligned} d_\Omega (x_k), d_\Omega (y_k)\rightarrow 0 \text{ and } x_0\in \partial \Omega \end{aligned}$$

and due to our assumption (7) that

$$\begin{aligned} G_{\Omega ,L} (x_k,y_k) <\frac{1}{k} \frac{d_\Omega (x_k)^m d_\Omega (y_k)^m}{|x_k-y_k|^n}. \end{aligned}$$
(10)

Without loss of generality, we may assume that \(x_0=0\) and that the first unit vector \(\mathbf {e}_1\) is the exterior unit normal to \(\partial \Omega \) at \(x_0\).

For k large enough, we may define \(\tilde{x}_k\in \partial \Omega \) as the closest boundary point to \(x_k\). We introduce the rescaled Green functions

$$\begin{aligned} G_k (\xi ,\eta ) :=|x_k-y_k|^{n-2m} G_{\Omega ,L}(\tilde{x}_k+|x_k-y_k|\xi ,\tilde{x}_k+|x_k-y_k|\eta ) \end{aligned}$$

for

$$\begin{aligned} \xi ,\eta \in \Omega _k:=\frac{1}{|x_k-y_k|}\left( -\tilde{x}_k+\Omega \right) . \end{aligned}$$

These belong to the elliptic operators

$$\begin{aligned} L_k=\left( -\sum ^n_{i,j=1} \partial _i \left( a^{(k)}_{ij}(\, \cdot \,)\partial _j \right) \right) ^m \end{aligned}$$

with

$$\begin{aligned} a^{(k)}_{ij}(\xi ):=a_{ij}(\tilde{x}_k+|x_k-y_k|\xi ) \end{aligned}$$
(11)

in \(\Omega _k\). This can be seen as follows. For smooth \(v:\overline{\Omega _k}\rightarrow {\mathbb {R}}\), we define

$$\begin{aligned} u:\overline{\Omega }\rightarrow {\mathbb {R}},\quad u(x):=v\left( \frac{x-\tilde{x}_k}{|x_k-y_k|}\right) , \end{aligned}$$

and find

$$\begin{aligned} \partial _i \left( a_{ij}(x)\partial _j u(x)\right)&= \frac{1}{|x_k-y_k|^2}\partial _i \left( a^{(k)}_{ij}(\, \cdot \,)\partial _j v(\, \cdot \,)\right) \left( \frac{x-\tilde{x}_k}{|x_k-y_k|} \right) ,\\ Lu(x)&= \frac{1}{|x_k-y_k|^{2m}} (L_kv)\left( \frac{x-\tilde{x}_k}{|x_k-y_k|} \right) . \end{aligned}$$

Since \(\tilde{x}_k\rightarrow x_0= 0\), the exterior unit normal at \(\partial \Omega \) converges to the first unit vector and so we conclude that

$$\begin{aligned} \Omega _k\rightarrow {\mathcal {H}}:=\{x: x_1<0\} \text{ locally } \text{ uniformly } \text{ for } k\rightarrow \infty . \end{aligned}$$

In order to understand the limit of the \(G_k\), we observe first that Krasovskiĭ’s estimates [11, Theorem 3.3] yield uniformly in k that

$$\begin{aligned} | G_k (\xi ,\eta )|\le C|\xi - \eta |^{2m-n}. \end{aligned}$$

Employing local elliptic estimates [1], we find a limit

$$\begin{aligned} \overline{G} :\overline{\mathcal {H}}\times \overline{\mathcal {H}}\rightarrow {\mathbb {R}}\cup \{\infty \} \end{aligned}$$

in \(L^1_{loc}\) and locally uniformly in \(\overline{\mathcal {H}}\times \overline{\mathcal {H}}\setminus \{(\xi ,\xi ):\xi \in \overline{\mathcal {H}}\}\). This is symmetric, i.e. \(\overline{G}(\xi ,\eta ) = \overline{G} (\eta ,\xi )\), obeys homogeneous Dirichlet boundary conditions on \(\partial {\mathcal {H}}\), and satisfies

$$\begin{aligned} | \overline{G} (\xi ,\eta )|\le C |\xi - \eta |^{2m-n}. \end{aligned}$$
(12)

We introduce the limit operator

$$\begin{aligned} L_\infty =\left( -\sum ^n_{i,j=1} \partial _i \left( a^{\infty }_{ij}\partial _j \right) \right) ^m \end{aligned}$$

with the constant coefficients

$$\begin{aligned} a_{ij}^{\infty }=a_{ij}(x_0). \end{aligned}$$

One should observe that this operator can easily be transformed by means of a linear transformation into the polyharmonic \((-\Delta )^m\). To be more precise, let \(C_1\in SO(n)\) diagonalise \(\left( a^{\infty }_{ij} \right) _{i,j=1,\ldots ,n}\). We then introduce a diagonal dilation matrix \(C_2\) with positive diagonal entries such that \(\left( C_1 \cdot C_2 \right) ^T \cdot \left( a^{\infty }_{ij} \right) _{i,j=1,\ldots ,n} \cdot \left( C_1 \cdot C_2 \right) \) becomes the n-dimensional identity matrix. Finally we introduce a further “rotation” \(C_3\in SO(n)\) such that \(C:=C_1 \cdot C_2 \cdot C_3:{\mathcal {H}} \rightarrow {\mathcal {H}} \) maps again the half space onto itself. Using this transformation, we find that

$$\begin{aligned} G_{{\mathcal {H}},L_\infty } (\xi ,\eta )={\text {det}} (C) G_{{\mathcal {H}},(-\Delta )^m} (C\cdot \xi , C\cdot \eta ). \end{aligned}$$

The latter is given by Boggio’s formula

$$\begin{aligned} G_{{\mathcal {H}},(-\Delta )^m} (\xi , \eta )=k_{m,n}|\xi -\eta |^{2m-n} \int \limits _1^{|\xi ^*-\eta |/|\xi -\eta |} (v^2-1)^{m-1}v^{1-n}\, dv, \end{aligned}$$

where \(k_{m,n}> 0\) is a suitable constant and \(\xi ^*=(-\xi _1,\xi _2,\ldots ,\xi _n)\). So we conclude from [5, Theorem 4.6] that

$$\begin{aligned} G_{{\mathcal {H}},L_\infty } (\xi ,\eta )\ge c_4 H_{\mathcal {H}}(\xi ,\eta ) \end{aligned}$$
(13)

with a strictly positive constant \(c_4\). \(H_{\mathcal {H}}\) is as in (3) with \(d_{\mathcal {H}}(\xi )=|\xi _1|\). Since Boggio’s formula has the same form in \({\mathcal {H}}\) as in the ball, (13) is deduced in \({\mathcal {H}}\) in the same way as in the ball. Moreover, as explained in [6, 7], \(G_{{\mathcal {H}},L_\infty }\) is unique under the assumption

$$\begin{aligned} |G_{{\mathcal {H}},L_\infty } (\xi ,\eta )|\le C |\xi - \eta |^{2m-n}. \end{aligned}$$

In order to conclude that \(\overline{G}=G_{{\mathcal {H}},L_\infty }\), we only need to show that

$$\begin{aligned} L_\infty \overline{G} (\xi ,\, \cdot \,)=\delta _\xi (\, \cdot \,) \end{aligned}$$
(14)

in the distributional sense. To this end, we consider any function \(\psi \in C^\infty _0 ({\mathcal {H}})\) and find from the fact that \(G_k\) is the Green function for \(L_k\) that one has for any \(\xi \in {\mathcal {H}}\) and k large enough,

$$\begin{aligned} \psi (\xi ) = \int \limits _{\mathcal {H}} G_k (\xi ,\eta ) (L_k \psi )(\eta )\, d\eta . \end{aligned}$$

The smoothness assumptions on the coefficients yield that \(a_{ij}^{(k)} \rightarrow a_{ij}^{\infty } \) in \(C^{2m-1}_{loc}({\mathcal {H}})\) and hence

$$\begin{aligned} L_k \psi \rightarrow L_\infty \psi \end{aligned}$$

uniformly in \(C^0_0 ({\mathcal {H}})\). We come up with

$$\begin{aligned} \psi (\xi ) = \int \limits _{\mathcal {H}} \overline{G} (\xi ,\eta ) (L_\infty \psi )(\eta )\, d\eta . \end{aligned}$$

This shows that \(\overline{G}\) is a Green function for \(L_\infty \) in \({\mathcal {H}}\), which in view of its uniqueness explained above yields that

$$\begin{aligned} \overline{G}=G_{{\mathcal {H}},L_\infty }. \end{aligned}$$

Defining

$$\begin{aligned} \xi _k :=\frac{x_k-\tilde{x}_k}{|x_k-y_k|}, \quad \eta _k :=\frac{y_k-\tilde{x}_k}{|x_k-y_k|}, \end{aligned}$$

on the one hand, the assumption gives via (10) that

$$\begin{aligned} G_k(\xi _k,\eta _k )=|x_k-y_k|^{n-2m}G_{\Omega ,L}(x_k,y_k )<\frac{1}{k} \frac{d_\Omega (x_k)^m d_\Omega (y_k)^m}{|x_k-y_k|^{2m}}. \end{aligned}$$
(15)

On the other hand, we have

$$\begin{aligned} |\xi _k|= \frac{d_\Omega (x_k)}{|x_k-y_k|}\le c_3, \quad | \xi _k-\eta _k |=1. \end{aligned}$$

After passing to a further subsequence, we find \(\xi ,\eta \in \overline{\mathcal {H}}\) with \(\xi =\lim _{k\rightarrow \infty }\xi _k \), \(\eta =\lim _{k\rightarrow \infty }\eta _k\). In view of the local smooth convergence of \(G_k\) to the Green function \(G_{{\mathcal {H}},L_\infty }\), we see from (13) that there exists a positive constant \(c_5>0\) such that for k large enough,

$$\begin{aligned} G_k(\xi _k,\eta _k)&\ge c_5 d_{\Omega _k}(\xi _k)^m d_{\Omega _k}(\eta _k)^m = c_5\left( \frac{d_{\Omega } (x_k)}{|x_k-y_k|} \right) ^m \left( \frac{d_{\Omega }(y_k)}{|x_k-y_k|} \right) ^m\\&=c_5 \frac{d_{\Omega }(x_k)^m d_{\Omega }(y_k)^m}{|x_k-y_k|^{2m}} . \end{aligned}$$

This contradicts (15) and the proof of the lemma is complete in the first case.

Second case: The sequence \(d_\Omega (x_k)/|x_k-y_k|\) becomes unbounded.

The reasoning here is similar to the first case and I outline only the main steps. After passing to a subsequence, we may assume that

$$\begin{aligned} \frac{d_\Omega (x_k)}{|x_k-y_k|}\rightarrow \infty . \end{aligned}$$

This also implies that

$$\begin{aligned} \frac{d_\Omega (y_k)}{|x_k-y_k|}\ge \frac{d_\Omega (x_k)}{|x_k-y_k|} - 1\rightarrow \infty . \end{aligned}$$

We obtain from the assumption (7) that in this case

$$\begin{aligned} G_{\Omega ,L} (x_k,y_k) <\frac{1}{k} |x_k-y_k|^{2m-n}. \end{aligned}$$
(16)

We rescale again such that \(|x_k-y_k|\) becomes the new length unit; however, here it is the \(x_k\) which become the new origins.

$$\begin{aligned} G_k (\xi ,\eta ) :=|x_k-y_k|^{n-2m} G_{\Omega ,L}({x}_k+|x_k-y_k|\xi ,{x}_k+|x_k-y_k|\eta ) \end{aligned}$$

for

$$\begin{aligned} \xi ,\eta \in \Omega _k:=\frac{1}{|x_k-y_k|}\left( -{x}_k+\Omega \right) . \end{aligned}$$

These belong to the elliptic operators

$$\begin{aligned} L_k=\left( -\sum ^n_{i,j=1} \partial _i \left( a^{(k)}_{ij}(\, \cdot \,)\partial _j \right) \right) ^m \end{aligned}$$

with

$$\begin{aligned} a^{(k)}_{ij}(\xi ):=a_{ij}({x}_k+|x_k-y_k|\xi ) \end{aligned}$$
(17)

in \(\Omega _k\). In view of the assumptions in this case, we conclude that

$$\begin{aligned} \Omega _k\rightarrow {\mathbb {R}}^n. \end{aligned}$$

For the sequence \(G_k\), we find a limit in \(L^1_{loc}\) and locally uniformly in \({\mathbb {R}}^n\times {\mathbb {R}}^n\setminus \{(\xi ,\xi ):\xi \in {\mathbb {R}}^n\}\), which is symmetric and decays like \((\xi ,\eta )\mapsto |\xi -\eta |^{2m-n}\) at infinity. Thanks to the uniqueness of such kind of Green function, we conclude that in the sense just described we have

$$\begin{aligned} G_k (\, \cdot \, , \, \cdot \, ) \rightarrow G_{{\mathbb {R}}^n,L_\infty } (\, \cdot \, , \, \cdot \, ), \end{aligned}$$
(18)

where the constant coefficient operator \(L_\infty \) is obtained precisely as in the first part. I emphasise that here the assumption \(n>2m\) simplifies the proof a lot because the uniqueness conclusion follows directly from Liouville’s theorem for polyharmonic operators, cf. e.g. [12, p. 19]. Observe that a linear transformation changes \(L_\infty \) into the polyharmonic operator.

Defining

$$\begin{aligned} \eta _k :=\frac{y_k-{x}_k}{|x_k-y_k|}, \end{aligned}$$

our assumption gives via (16) that

$$\begin{aligned} G_k(0,\eta _k )=|x_k-y_k|^{n-2m}G_{\Omega ,L}(x_k,y_k )<\frac{1}{k}. \end{aligned}$$
(19)

Thanks to \(| \eta _k |=1\), we may pass to a subsequence and find \(\eta _k\rightarrow \eta _\infty \) with \(|\eta _\infty |=1\) and

$$\begin{aligned} G_{{\mathbb {R}}^n,L_\infty }(0,\eta _\infty )\le 0. \end{aligned}$$

We proceed now precisely as explained in some detail above in the first case. After applying a linear transformation, we may change \(L_\infty \) into \((-\Delta )^m\) whose Green function in \({\mathbb {R}}^n\) is given by the fundamental solution with zero boundary conditions at infinity, i.e. a positive multiple of \((\xi ,\eta )\mapsto |\xi -\eta |^{2m-n}\). We conclude that \(G_{{\mathbb {R}}^n,L_\infty }(0,\eta _\infty )>0\) and achieve a contradiction. So the proof of the lemma is complete also in this case. \(\square \)

3 Proof of the main results: a compactness argument

3.1 Proof of Theorem 1

Applying a compactness argument to

$$\begin{aligned} \overline{\Omega }=\bigcup _{x_0\in \overline{\Omega }} \Omega _{x_0,r_{x_0}/2}, \end{aligned}$$

we see that there exist positive numbers \(r>0\), \(c_6>0\), such that \(|x-y|\le r\) implies that \(G_{\Omega ,L} (x,y) \ge c_6 H_\Omega (x,y)\). If \(|x-y|\ge r\), we take from [4, 11], cf. also [5], that \( G_{\Omega ,L} (x,y) \ge -c_7 H_\Omega (x,y) \) so that

$$\begin{aligned} G_{\Omega ,L} (x,y) +2 c_7 H_\Omega (x,y) \ge c_7 H_\Omega (x,y). \end{aligned}$$

Since \(H_\Omega (x,y)\le |x-y|^{-n}d_\Omega (x)^m d_\Omega (y)^m\le r^{-n}d_\Omega (x)^m d_\Omega (y)^m\), we end up with

$$\begin{aligned} G_{\Omega ,L} (x,y) + c_8 d_\Omega (x)^m d_\Omega (y)^m \ge c_7 H_\Omega (x,y) \end{aligned}$$

and positive constants \(c_7,c_8>0\). The proof of Theorem 1 is complete. \(\square \)

3.2 Proof of Theorem 3

This theorem was proved for \(L=\Delta ^2\) in [6] when \(n\ge 3\) and in [5, Theorem 6.15] when \(n =2\). See also [7, Theorem 2] for a unified and simpler proof, which I shall adapt to the present situation.

Case I: \(d_\Omega (x)d_\Omega (y)\le \left| x-y\right| ^2\). For this situation, we have

$$\begin{aligned} H_{\Omega } (x,y)= |x-y|^{-n} d_\Omega (x)^m d_\Omega (y)^m. \end{aligned}$$
(20)

Then there is \(c>0\) such that we find \(c_2^{-1} H_{\Omega }(x,y)\ge 2 c_1 d_\Omega (x)^md_\Omega (y)^m\) for \(\left| x-y\right| <c\).

Case II: \(d_\Omega (x)d_\Omega (y) > \left| x-y\right| ^2\). Now we have

$$\begin{aligned} H_{\Omega } (x,y)=|x-y|^{2m-n}. \end{aligned}$$
(21)

Since \(n>2m\) by assumption and since \(d_\Omega (x)d_\Omega (y)\) is bounded on \(\Omega \), one finds a constant \(c>0\) such that \(c_2^{-1} H_{\Omega }(x,y)\ge 2 c_1 d_\Omega (x)^m d_\Omega (y)^m\) for \(\left| x-y\right| <c\). The proof of Theorem 3 is complete. \(\square \)