Abstract
In this paper we study fundamental solutions for the Laplace–Beltrami operator
defined on smooth enough functions in \(\mathbb {R}_+^{n}=\{ (x_1,\ldots ,x_n)\in \mathbb {R}^{n}: x_n>0\}\). We represent explicit formulas for the fundamental solutions. Moreover, we establish fundamental solutions using Jacobi polynomials when \(n=3,5,7,\ldots .\)
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1 Introduction
In this paper, fundamental solutions of the Laplace–Beltrami operator of the hyperbolic upper half-space are considered. This is a continuation of the previous research by the authors in [6, 8,9,10,11], where we have looked at different special cases. In [12] the first author and Vuojamo found the fundamental solution in terms of associate Legendre functions of the second kind, but explicit representations in terms of elementary functions of kernels were not presented.
The theory is closely connected to to the axially symmetric potential theory created by Weinstein [20]. Heinz Leutwiler [14] initiated the research of Laplace–Beltrami equations connected to the differential equation of Weinstein. The general theory was also researched by Ryan et al. [4] and it has also connections to research of iterated Dirac operators of Ryan [19] (see [7]). It has interesting connections to hyperbolic Brownian motion, see e.g. [5].
The Laplace–Beltrami operator is a geometric operator, i.e. its form depends on the metric of the space. In this paper we consider the conformal metric of the hyperbolic upper half-space. It is generally interesting because the operator is with non-constant coefficients and thus considerably more difficult to handle than the constant coefficient cases. In this paper, we point out that the parity of the treated space has a fundamental effect on the shape of the fundamental solutions. We also state that in odd dimensions the basic solution can be presented using Jacobian polynomials. There is no corresponding construction in the even case. We will return to this case in the future.
The structure of the article is as follows:
2 Conformal Hyperbolic Upper-Half Space and Laplace–Beltrami Operators
Consider the hyperbolic upper half-space
equipped with the metric
The geometry of the hyperbolic space \((\mathbb {R}_+^{n},g_H)\) is well known and studied. The geodesics are circular arcs perpendicular to the hyperplane \(x_{n}=0\), that is half-circles whose origin is on \(x_{n}=0,\) and straight vertical lines parallel to the \(x_{n}\)-axis.
The distance between two points \(x,y\in \mathbb {R}_+^{n}\) with respect to the metric \(g_H\) is (see e.g. Theorem 4.6.1 in [18])
where
is a symmetric invariant, where \(|x|^2=x_1^2+\cdots +x_n^2\) is the usual Euclidean quadratic form.
For \(n>2\) we define the conformal metric on the upper-half space by
where \(\alpha \in \mathbb {R}\). One reason to consider the preceding conformal metric is the simple form of the associated Laplace–Beltrami operator
where \(\Delta =\frac{\partial ^2}{\partial x_1^2}+\cdots +\frac{\partial ^2}{\partial x_n^2}\) is the Euclidean Laplacian. When \(\alpha =n-2\), we obtain the hyperbolic Laplace operator
If \(\Omega \subset \mathbb {R}_{+}^{n}\) is open, a twice continuously differentiable function \(f:\Omega \rightarrow \mathbb {R}\) is called \(\alpha \)-hyperbolic harmonic if
If \(\alpha =n-2\), we call an \(\alpha \)-hyperbolic harmonic function just hyperbolic harmonic. Heinz Leutwiler initiated the research of hyperbolic harmonic functions and their function theory in [15, 16]. It has been continued intensively by the first author, Leutwiler and the second author and there is a book in preparation [8].
3 Fundamental Solution for \(\Delta _\alpha \)
In this paper, we consider fundamental solutions of the operator \(\Delta _\alpha \). A fundamental solution is a function \(H_{\alpha ,n}(x,y)\) that satisfies the equation
in the distribution sense, where \(\delta \) is the usual Dirac delta distribution at \(y\in \mathbb {R}^n_+\). In above the \(\omega _{n-1}\) is the surface area of the unit ball \(S^{n-1}\subset \mathbb {R}^n\). The necessary condition is, that \(H_{\alpha ,n}\) is singular at the diagonal \(x=y\).
The fundamental solutions are presented in terms of associated Legendre functions of the second kind. Associated Legendre functions are defined by (see e.g. 8.703 in [13])
where \(\Gamma \) is the usual gamma function and \({}_{2}F_{1}\) is the hypergeometric function defined with power series representation (see e.g. [1, 2, 13])
for \(|z|<1\), and \(a,b,c\in \mathbb {C}\) with \(c\ne 0,-1,-2,\ldots \). In the hypergeometric function the Pochammer symbol is defined by
Hence the hypergometric function terminates if a or b is a negative integer.
A reader should observe that the preceding definition for a associated Legendre function is up to the constant \(e^{i\pi \nu }\) the usual one, see e.g. in [13].
In [5] the following theorem is verified.
Theorem 3.1
Let x and y be distinct elements in \(\mathbb {R}_{+}^{n}\) and \(\alpha \in \mathbb {R}\). Denote \(r_{h}=d_{H}\left( x,y\right) \). Define
-
(a)
If \(n\in \mathbb {N}\) and \(n\ge 3\), the fundamental solution is
$$\begin{aligned} H_{\alpha ,n}\left( x,y\right)&=\frac{x_{n}^{\frac{\alpha +2-n}{2}} y_{n}^{\frac{\alpha +2-n}{2}}}{2^{\frac{n-2}{2}}\Gamma \left( \frac{n}{2}\right) }\left( \lambda ^{2}(x,y)-1\right) ^{\frac{2-n}{4}}\hat{Q} {}_{\rho _{\alpha }}^{\frac{n-2}{2}}\left( \lambda (x,y)\right) \\&=x_{n}^{\frac{\alpha +2-n}{2}}y_{n}^{\frac{\alpha +2-n}{2}}\sinh ^{2-n} r_{h}g_{\rho _{\alpha },n}\left( r_{h}\right) , \end{aligned}$$where
$$\begin{aligned} g_{\rho _{\alpha },n}\left( r_{h}\right)&=C\left( \alpha ,n\right) \lambda ^{\frac{n-4-2\rho _{\alpha }}{2}}~_{2}F_{1}\\ {}&\quad \left( \frac{2\rho _{\alpha }-n+4}{4},\frac{2\rho _{\alpha }-n+6}{4};\frac{2\rho _{\alpha }+3}{2};\frac{1}{\cosh ^{2}r_{h}}\right) \end{aligned}$$and
$$\begin{aligned} C\left( \alpha ,n\right) =\frac{\sqrt{\pi }\Gamma \left( \frac{2\rho _{\alpha }+n}{2}\right) }{2^{\frac{2\rho _{\alpha }+n}{2}}\Gamma \left( \frac{2\rho _{\alpha }+3}{2}\right) \Gamma \left( \frac{n}{2}\right) }. \end{aligned}$$ -
(b)
If \(n=2\), the fundamental solution is
$$\begin{aligned} H_{\alpha ,2}\left( x,y\right)&=x_{n}^{\frac{\alpha }{2}}y_{n}^{\frac{\alpha }{2}}\hat{Q}_{\rho _{\alpha }}\left( \lambda (x,y)\right) \\&=x_{n}^{\frac{\alpha }{2}}y_{n}^{\frac{\alpha }{2}}{\text {arcoth}} \left( \lambda (x,y)\right) g_{\rho _{\alpha }}\left( \lambda (x,y)\right) , \end{aligned}$$where
$$\begin{aligned} g_{\rho _{\alpha }}\left( \lambda \right) =\frac{\hat{Q}_{\rho _{\alpha }}\left( \lambda \right) }{{\text {arcoth}}\left( \lambda \right) }=\frac{2\hat{Q}_{\rho _{\alpha }}\left( \lambda \right) }{\ln \left( \frac{\lambda +1}{\lambda -1}\right) }. \end{aligned}$$
We can compute the first explicit example.
Example 3.2
Consider the case \(\alpha =0\). Using the integral representation 8.712 in [13], we have
that is,
Applying (1), we conclude
where \(\widehat{y}=(y_{1},\ldots ,y_{n-1},-y_{n})\).
The previous example tells us that fundamental solutions should always be thought of as unique in the sense that some function of the operator’s kernel can be added to them. This feature can be utilized when searching for Green’s functions, in which case the added function takes care of the needed boundary values.
The second example shows that the fundamental solution may be also product of known harmonic fundamental solutions.
Example 3.3
If \(\alpha =2-n\) and \(n\ge 3\) we use the formula (see e.g. [13, 3.666])
and the Legendre dublication formula, and obtain
Example 3.4
If \(n=2\) and \(y\in \mathbb {R}_{+}^{2}\), then
Indeed, if \(n=2\) and \(\alpha =0\), we compute by virtue of 8.821 (3) in [13]
3.1 Fundamental \(\alpha \)-Hyperbolic Harmonic Functions Inductively
In the theory of harmonic functions, if you know the fundamental harmonic functions in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\), you may obtain the formula for fundamental harmonic functions in all dimensions simply by differentiating with respect to the r which is the distance from the origin. We are aiming to give a similar result for \(\alpha \)-hyperbolic harmonic functions, but the formula depends on the parity of the space.
We recall an important tool.
Lemma 3.5
[6] Let \(\Omega \) be an open set contained in \(\mathbb {R}_{+}^{n}\). A function \(f:\Omega \rightarrow \) \(\mathbb {R}\) is \(\alpha \)-hyperbolic harmonic if and only if the function \(g\left( x\right) =x_{n}^{\frac{n-\alpha -2}{2}}f\left( x\right) \) is the eigenfunction of the hyperbolic Laplace operator corresponding to the eigenvalue \(\gamma _{n,\alpha }=\frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) \), that is
We are looking for eigenfunctions of the hyperbolic Laplace operator depending only on \(\lambda \). Then the hyperbolic Laplace operator has the following representation.
Proposition 3.6
[9] Let \(a\in \mathbb {R}_{+}^{n}\). If f : \(\mathbb {R}_{+}^{n}\rightarrow \mathbb {R}\) is twice continuously differentiable and depending only on \(\lambda =\lambda \left( x,a\right) \) then
We need to reformulate the properties of associated Legendre functions for our used notations
Lemma 3.7
If \(\rho >-\frac{3}{2}\), \(\mu \ge 0\), \(z\in \mathbb {C}\) and \(\left| z\right| >1\), then
Proof
Since
we obtain the result from the corresponding formulas for the associated Legendre functions 8.732 and 8.734 given in [13]. \(\square \)
Applying the previous lemma and the induction principle, we deduce the following property.
Theorem 3.8
Let \(\rho >-\frac{3}{2}\) and \(\mu \ge 0\). If \(m\in \mathbb {N}\), then
where \(\lambda >1\).
Proof
We first prove the assertion for \(m=1\) and for any real \(\tau \ge 0\). Comparing the left side of identities of the previous corollary, we obtain
Dividing by \((\lambda ^{2}-1)^\frac{1}{2}\) both sides of the equality, we compute further
Using the preceding formula, we compute
which implies that the result hold for \(m=1\) and for all \(\tau \ge 0\). The induction hypothesis is that
holds for some \(s\in \mathbb {N}\) and all \(\mu \ge 0\). Applying (7) for \(\tau =\mu +s\), we obtain
Applying the induction hypothesis, we conclude
Consequetly, by the general induction principle the result holds for all \(m\in \mathbb {N}\). \(\square \)
The key tool is the results connecting different eigenvalues.
Proposition 3.9
Let \(\beta \) and \(\gamma \) be real numbers. If f : \(\left] 1,\infty \right[ \rightarrow \mathbb {R}\) is four times differentiable solution of the equation
then \(g\left( \lambda \right) =f^{\prime }\left( \lambda \right) \) satisfies the equation
Proof
We just compute
completing the proof. \(\square \)
Applying the previous proposition, it is relatively simple to verify the result.
Theorem 3.10
Assume that \(\beta \in \left\{ 0,1\right\} \) and \(\gamma \in \mathbb {R}\). If \(s\in \mathbb {N}\) and f : \(\left] 1,\infty \right[ \rightarrow \mathbb {R}\) is \(\left( s+2\right) \) times differentiable solution of the equation
then the function \(g\left( \lambda \right) =f^{\left( s\right) }\left( \lambda \right) \) satisfies the equation
Proof
The previous lemma implies that the result holds for \(s=1\). Assume that the result holds for some \(s\in \mathbb {N}\). Then the function \(h\left( \lambda \right) =f^{\left( s\right) }\left( \lambda \right) \) satisfies the equation
Using the previous lemma we obtain that the function \(g\left( \lambda \right) =h^{\prime }\left( \lambda \right) =f^{\left( s+1\right) }\left( \lambda \right) \) satisfies the equation
Hence the result holds for \(s+1\), completing the proof. \(\square \)
Applying the previous result and Proposition 3.6, we obtain easily the general formulas.
Theorem 3.11
Let \(\gamma _{n,\alpha }=\frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) \) and \(\alpha \in \mathbb {R}\). Let \(f\in \mathcal {C}^{\infty }\left( \left] 1,\infty \right[ \right) .\)
-
(a)
If f satisfies the differential equation
$$\begin{aligned} \left( \lambda ^{2}-1\right) f^{\prime \prime }\left( \lambda \right) +\lambda f^{\prime }\left( \lambda \right) =\gamma _{1,\alpha }f\left( \lambda \right) , \end{aligned}$$then the \(\tfrac{n-1}{2}\):th derivative \(g\left( \lambda \right) =f^{\left( \tfrac{n-1}{2}\right) }\left( \lambda \right) \) satisfies the differential equation
$$\begin{aligned} \left( \lambda ^{2}-1\right) g^{\prime \prime }\left( \lambda \right) +n\lambda g^{\prime }\left( \lambda \right) =\gamma _{n,\alpha }g\left( \lambda \right) \end{aligned}$$for all odd \(n\in \mathbb {N}\).
-
(b)
If f satisfies the differential equation
$$\begin{aligned} \left( \lambda ^{2}-1\right) f^{\prime \prime }\left( \lambda \right) =\gamma _{2,\alpha }f\left( \lambda \right) \end{aligned}$$then the \(\tfrac{n}{2}\):th derivative \(g\left( \lambda \right) =f^{\left( \tfrac{n}{2}\right) }\left( \lambda \right) \) satisfies the equation
$$\begin{aligned} \left( \lambda ^{2}-1\right) g^{\prime \prime }\left( \lambda \right) +2n\lambda g^{\prime }\left( \lambda \right) =\gamma _{2n,\alpha }g\left( \lambda \right) \end{aligned}$$for all even \(n\in \mathbb {N}\).
Applying the previous theorem, we immediately deduce the fundamental solutions depending on the parity of the space. We will denote \(\widehat{Q}_\mu :=\widehat{Q}_\mu ^0\) and observe, that they are usual Legendre functions.
Theorem 3.12
Let \(n\in \mathbb {N}\) and \(n\ge 2\).
-
(a)
If n is even, then the fundamental \(\alpha \)-hyperbolic harmonic function is given by
$$\begin{aligned} H_{\alpha .n}\left( x,a\right)&=\frac{\left( -1\right) ^{\frac{n-2}{2}}x_{n}^{\frac{\alpha +2-n}{2}} y_{n}^{\frac{\alpha +2-n}{2}}}{2^{\frac{n-2}{2}}\Gamma \left( \frac{n}{2}\right) }\frac{\partial ^{\frac{n-2}{2}}\widehat{Q}_{\rho _{\alpha }}\left( \lambda \right) }{\partial \lambda ^{\frac{n-2}{2}}} \end{aligned}$$and
$$\begin{aligned} H_{\alpha ,2}\left( x,y\right) =x_{n}^{\frac{\alpha }{2} }y_{n}^{\frac{\alpha }{2}}Q_{\rho _{\alpha }}\left( \lambda \right) . \end{aligned}$$ -
(b)
If \(n=2m+1\) is odd, then the fundamental solution has the representation
$$\begin{aligned} H_{\alpha ,n}\left( x,y\right)&=\frac{\left( -1\right) ^{m-1}x_{n}^{\frac{\alpha +2-n}{2}}y_{n} ^{\frac{\alpha +2-n}{2}}}{2^{\frac{n-2}{2}}\Gamma \left( \frac{n}{2}\right) }\frac{\partial ^{m-1}\left( \left( \lambda ^{2}-1\right) ^{-\frac{1}{4} }\widehat{Q}_{\rho _{\alpha }}^{\frac{1}{2}}\left( \lambda \right) \right) }{\partial \lambda ^{m-1}} \end{aligned}$$and
$$\begin{aligned} H_{\alpha ,3}\left( x,y\right) =\frac{x_{n}^{\frac{\alpha -1}{2}} y_{n}^{\frac{\alpha -1}{2}}e^{-\frac{2\rho _{\alpha }+1}{2}r_{h}}}{\sinh r_{h}}. \end{aligned}$$
Proof
The first assertion follows from [13, 8.752 (4)].
Let \(n=3\). Applying [13, 8.754 (4)], we obtain
Assume next that \(n=2m+1\). Applying Theorem 3.8 for \(\mu =\frac{1}{2}\) and
where
\(\square \)
For \(\alpha =n-2\) the fundamental solution \(h_{n\text { }}\) was computed by Ahlfors [3, p.57]
and in [9] it was proved by the authors that
In order to verify that the equality of the fundamental solutions \(H_{n-2,n\text { }}\) and \(h_{n}\) we need a simple observation.
Lemma 3.13
If \(\rho >-\frac{1}{2}\), \(\mu \ge 0\) and \(\lambda >1\), then
and therefore
for any \(n\in \mathbb {N}\) and \(n\ge 2.\)
Proof
Applying [13, 8.732 (2)], we deduce
Since
for all \(\ \nu >-\frac{1}{2}\) we obtain the assertion
Since \(\rho =\frac{n-2}{2}\ge 0\), substituting \(\mu =\frac{n}{2}\) and \(\rho =\frac{n-2}{2}\), we obtain the final statement
\(\square \)
Theorem 3.14
If \(n\in \mathbb {N}\) and \(n\ge 2\), then
Proof
We first prove the assertion for even n. Assume first that \(n=2\) and denote \(\lambda =\lambda \left( x,a\right) \). Then
If n is even and \(n>2\), applying [13, 8.752 (4), 8.824], we obtain
completing the proof using (8) in even case.
In odd case we first we note that
The assertion holds for \(n=3\), since
Assume that the result holds for odd \(n\ge 3\), that is
Applying the differential formula (3.8) for \(m=1\), \(\mu =\frac{n-2}{2}\) and \(\rho =\frac{n}{2}\), we obtain, we obtain
By virtue of Lemma 3.13, we infer the identity
Applying the induction hypothesis, we obtain further
Hence we have
Applying the partial integration, we deduce
whicn by solving the formula for \(\frac{n-1}{n}\int _{\lambda }^{\infty } \frac{1}{\left( t^{2}-1\right) ^{\frac{n+1}{2}}}dt\) implies that
Hence
Again applying the differential formula Theorem 3.8 and Lemma 3.13, we obtain
Finally, using the induction hypothesis, we conclude
completing the proof. \(\square \)
4 Connections to Jacobi Polynomials
Let \(\alpha \in \mathbb {R}\backslash (-\mathbb {N})\) and \(\beta \in \mathbb {R}\), and \(n\in \mathbb {N}\). The Jacobi polynomials of degree n is defined in terms of hypergeometric functions by
for \(z\in \mathbb {C}\). Jacobi polynomilas satisfy the following useful formula.
Proposition 4.1
(Rodrigues formula). Let z be real and n a non negative integer. The Jacobi polynomials have the property
The second associated Legendre function is defined by
converging \(|1-z|<2\), where \(\mu \in \mathbb {R}\backslash \mathbb {N}\). Let us first prove the following connection between the second Legendre function and Jacobi polynomials.
Proposition 4.2
If \(\alpha \in \mathbb {R}\backslash (-\mathbb {N})\) and \(m\in \mathbb {N}\), then
Proof
We write
and
Hence, we obtain
Using (4), we have
\(\square \)
Hence, we can represent the fundamental solution using a Jacobi polynomial (see also [17])
Theorem 4.3
Let \(x,y\in \mathbb {R}_{+}^{n}\) and denote \(r_{h}=d_{h}\left( x,y\right) \). If \(n\ge 3\) is an odd integer then
and
Proof
Assume \(n\ge 3\) is odd. The formula 8.739 in [13] is in our case
and using the formula 8.2.1 from [1], we have
The using Proposition (4.2), we obtain
We compute
that is
Hence
\(\square \)
We can write the Jacobi polynomials as follows.
Proposition 4.4
If \(\alpha \in \mathbb {R}\backslash (-\mathbb {N})\) and \(m\in \mathbb {N}\), then
and
Proof
Using (9) we have
and using \((-m)_j=(-1)^j(m-j+1)_j\), we have
Then applying (4) and \(\Gamma (m+1)=m!\), we obtain
and using again (4), we conclude
We compute
and obtain
\(\square \)
Let us complete the paper by computing the following examples.
Example 4.5
If \(n=3\), we have \(P_{0}^{\left( \alpha ,-\alpha \right) }\left( \coth (r_{h})\right) =1\) and
Example 4.6
If \(n=5\), we have
and
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Communicated by Daniele Struppa.
Dedicated to Prof. John Ryan, best wishes and new goals for retirement.
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Eriksson, SL., Orelma, H. Fundamental Solutions for the Laplace–Beltrami Operator Defined by the Conformal Hyperbolic Metric and Jacobi Polynomials. Complex Anal. Oper. Theory 18, 10 (2024). https://doi.org/10.1007/s11785-023-01459-0
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DOI: https://doi.org/10.1007/s11785-023-01459-0