Reports of the Academy of Sciences of the USSR
1963. Volume 151, No. 6

MATHEMATICS

F. I. KARPELEVICH

NONNEGATIVE EIGENFUNCTIONS OF THE BELTRAMI–LAPLACE OPERATOR ON SYMMETRIC SPACES OF NONPOSITIVE CURVATURE

(Presented by Academician I. G. Petrovskii on March 14, 1963)

1. Let \(G\) be a connected semisimple Lie group with finite center, \(U\) its maximal compact subgroup, and \(\mathscr E\) the homogeneous space \(G/U\). In the space \(\mathscr E\) there exists an invariant Riemannian metric with respect to which \(\mathscr E\) is a symmetric space of nonpositive curvature. Denote by \(B\) the Beltrami–Laplace operator constructed from this metric, and consider the equation

\[ Bf-cf=0, \tag{1} \]

where \(f\) is a twice continuously differentiable function on \(\mathscr E\), and \(c\) is a constant.

Our aim is to study the cone \(C_c\) of nonnegative solutions of equation (1). For simplicity of exposition we restrict ourselves here to the case where \(G\) is the group of real unimodular matrices of order \(n\). However, the results obtained are of a general nature and are valid for any semisimple group. In the case where \(G\) is a complex group, results close to ours were obtained in \((^1)\).

1. If \(G\) is the group of real unimodular matrices of order \(n\), then the group of orthogonal matrices may serve as the subgroup \(U\). The Lie algebra of the group \(G\) coincides with the algebra \(\mathfrak G\) of real matrices of order \(n\) with trace zero. One may assume that the Riemannian metric in \(\mathscr E\) is induced by the invariant scalar product \((\mathfrak g_1,\mathfrak g_2)\) \((\mathfrak g_1,\mathfrak g_2\in\mathfrak G)\), given by the formula \((\mathfrak g_1,\mathfrak g_2)=\operatorname{sp}\mathfrak g_1\mathfrak g_2\). In the \(n\)-dimensional vector space consider the hyperplane \(\mathfrak h\) defined by the equation \(\sum_1^n \xi^i=0\). This hyperplane is naturally identified with the Lie algebra of the group \(H\subset G\) of diagonal matrices with positive diagonal elements. For the scalar product introduced, \((\xi_1,\xi_2)\) \((\xi_1,\xi_2\in\mathfrak h)\), we have

\[ (\xi_1,\xi_2)=\sum_1^n \xi_1^i \xi_2^i . \]

An element \(\xi\in\mathfrak h\) will be called an element of general position if, for \(i\ne j\), \(\xi^i\ne \xi^j\). Denote by \(\Lambda\) the domain in \(\mathfrak h\) defined by the inequalities \(\xi^1\geq \xi^2\geq \cdots \geq \xi^n\), and let \(\rho\) be the vector in \(\mathfrak h\) determined by the relation

\[ (\rho,\xi)=\frac12\sum_{i<j}(\xi^i-\xi^j). \]

Theorem 1. For \(c<-(\rho,\rho)\), equation (1) has no nonnegative solutions.

Along with equation (1), consider the equation

\[ \frac{\partial p}{\partial t}=Bp, \tag{2} \]

and let \(p_y(t,x)=p(t,x,y)\) be the fundamental solution of this equation, i.e. the function satisfying, as a function of \(x\), equation (2) for \(t>0\) and the initial condition \(p(0,x,y)=\delta_y(x)\), where \(\delta_y(x)\) is the \(\delta\)-function in \(\mathscr E\) with respect to the invariant volume, concentrated at the point \(y\).

Theorem 2. If \(c \geq -(\rho,\rho)\) and \(x \ne y\), then

\[ \int_0^\infty e^{-ct}p(t,x,y)\,dt \]

converges.

Theorem 2 ensures the existence (for \(c \geq -(\rho,\rho)\)) of the Green’s function of equation (1).

1. Let \(Z\) denote the group of upper triangular matrices whose diagonal elements are equal to 1. As is known, every matrix \(g \in G\) is represented, and in a unique way, in the form \(g=zhu\), where \(z \in Z\), \(h \in H\), \(u \in U\). Therefore every point \(x \in \mathscr E\) is represented in the form \(x=zhx_0\); here \(x_0\) is the point of the space \(\mathscr E\) whose stationary subgroup coincides with the subgroup \(U\). (By \(gx\) \((g\in G,\ x\in \mathscr E)\) we denote the point obtained from the point \(x\) by the motion \(g\).) For any vector \(\lambda \in \mathfrak h\) define on the group \(H\) the function \(h^\lambda\) by the formula \(h^\lambda=\exp(\lambda,\mathfrak h)\), where \(\mathfrak h=\ln h\) is the vector whose coordinates are equal to the logarithms of the diagonal elements of the matrix \(h\). Introduce on the space \(\mathscr E\) the function \(\varphi(x,\lambda)\) \((x\in \mathscr E,\ \lambda\in \mathfrak h)\), setting \(\varphi(x,\lambda)=h^{\lambda+\rho}\) for \(x=zhx_0\).

A nonnegative solution \(f(x)\) of equation (1) is called minimal if every nonnegative solution of this equation not exceeding \(f(x)\) can differ from \(f(x)\) only by a constant factor.

Theorem 3. Every minimal solution \(f(x)\) of equation (1) (for \(c \geq -(\rho,\rho)\)) is proportional to one of the functions \(\varphi(ux,\lambda)\), where \(u\in U\), \(\lambda\in \Lambda\), \((\lambda,\lambda)=c+(\rho,\rho)\).

From Theorem 3 it follows that

Theorem 4. Every nonnegative solution \(f(x)\) of equation (1) can be represented in the form

\[ f(x)=\int_{U\times\Lambda_c}\varphi(ux,\lambda)\,\mu(du\,d\lambda), \]

where \(\Lambda_c\) is the intersection of the domain \(\Lambda\) with the sphere \((\lambda,\lambda)=c+(\rho,\rho)\), and \(\mu(du\,d\lambda)\) is a finite Borel measure on \(U\times\Lambda_c\).

1. A twice continuously differentiable function \(f(x)\) on \(\mathscr E\) will be called harmonic if it satisfies the equation \(Bf(x)=0\).

Let \(K\subset G\) denote the group of all upper triangular matrices and let \(\Xi\) denote the homogeneous space \(G/K\). Since every matrix \(g\in G\) is represented in the form \(g=uk\) \((u\in U,\ k\in K)\), the group \(U\) acts transitively on \(\Xi\). Therefore the stationary subgroup \(U_x\) of any point \(x\in \mathscr E\) acts transitively on \(\Xi\). By compactness of \(U_x\), on \(\Xi\) there exists, and is unique, a measure invariant with respect to \(U_x\) and normalized so that the measure of the whole space \(\Xi\) is equal to 1. This measure depends on the point \(x\in \mathscr E\), and we denote it by \(\mu(x,d\xi)\).

Theorem 5. Every bounded harmonic function \(f(x)\) on \(\mathscr E\) can be represented in the form

\[ f(x)=\int_{\Xi}p(\xi)\,\mu(x,d\xi), \tag{3} \]

where \(p(\xi)\) is an arbitrary function on \(\Xi\) integrable with respect to the measures \(\mu(x,d\xi)\). The function \(p(\xi)\) is determined by the function \(f(x)\) uniquely almost everywhere on \(\Xi\)*.

From Theorem 5 it follows easily that

Theorem 6. Every bounded harmonic function is annihilated by all Laplace operators on the space \(\mathscr E\) by which the function \(f(x)\equiv 1\) is annihilated.

* Cf. Theorem 10 of (2).

Theorem 6 was obtained by other methods in (³).

Let \(\mathfrak{h}\) be an arbitrary element in general position belonging to \(\Lambda\). Denote by \(h(t)\) the corresponding one-parameter subgroup in the group \(H\).

Theorem 7. If the function \(f(x)\) is given by formula (3) with a continuous function \(p(\xi)\), then
\[ \lim_{t\to+\infty} f(h(t)x_0)=p(\xi_0), \]
where \(\xi_0\) is the point \(\Xi\) whose stationary subgroup coincides with the group \(K\).

Theorem 7 makes it possible to solve the question of the behavior of the function \(f(x)\) as \(x\to\infty\) along a geodesic \(\gamma\) in general position in the space \(\mathscr{E}\). It turns out that the values of the function \(f(x)\) then tend to a certain value \(p(\xi)\). (The point \(\xi\) depends on the geodesic \(\gamma\).) It is therefore natural to call the values \(p(\xi)\) of the function \(p\) the boundary values of the function \(f(x)\).

Let now \(\gamma\) be an arbitrary directed geodesic of the space \(\mathscr{E}\), and let \(F\) be a finite pencil of geodesics containing \(\gamma\) (see (²)). According to (²), the aggregate of null pencils \(\Gamma \subset F\) forms a certain Riemannian symmetric space \(\mathscr{F}\) of nonpositive curvature. Like every such space, \(\mathscr{F}\) decomposes into the direct product \(E\times \mathscr{E}'\) of a Euclidean space \(E\) and a Riemannian symmetric space \(\mathscr{E}'\) of nonpositive curvature with a semisimple group of motions. Denote by
\[ \rho:\mathscr{F}\to \mathscr{E}' \]
the projection of the space \(\mathscr{F}\) onto the space \(\mathscr{E}'\). To each geodesic \(\gamma\in F\) there corresponds a point \(x'=\pi(\gamma)\in \mathscr{E}'\) by the rule \(\pi(\gamma)=\rho(\Gamma)\), where \(\Gamma\) is the null pencil containing the geodesic \(\gamma\). We shall write \(x\to+\infty\), \(x\in\gamma\), if \(x\) tends to infinity along the directed geodesic \(\gamma\) in the positive direction.

Theorem 8. If \(f(x)\) is a bounded harmonic function with continuous boundary values and if, for two geodesics \(\gamma_1\) and \(\gamma_2\) from a finite pencil \(F\), \(\pi(\gamma_1)=\pi(\gamma_2)\), then
\[ \lim_{x\to+\infty,\ x\in\gamma_1} f(x) = \lim_{x\to+\infty,\ x\in\gamma_2} f(x). \]

Theorem 8 shows that the limiting values of the function \(f(x)\) induce a certain function \(f'(x')\) on the space \(\mathscr{E}'\).

Theorem 9. If \(f(x)\) is a bounded harmonic function with continuous boundary values, then its limiting values \(f'(x')\) on the space \(\mathscr{E}'\) form a function harmonic on \(\mathscr{E}'\).

Theorems 7–9 are immediate consequences of Theorems 11, 9, and 12 from (²). We note that the basis for obtaining the results set forth is Martin’s method (⁴) (see also (⁵)) and the method of horispherical radial parts of Laplace operators on symmetric spaces (⁶).

Moscow Institute of Railway Transport Engineers

Received
3 III 1963

REFERENCES

¹ E. B. Dynkin, DAN, 141, No. 2 (1961).
² F. I. Karpelevich, DAN, 124, No. 6 (1959).
³ F. A. Berezin, DAN, 125, No. 6 (1959).
⁴ R. S. Martin, Trans. Am. Math. Soc., 49, No. 1 (1941).
⁵ M. G. Shur, DAN, 144, No. 2 (1962).
⁶ F. I. Karpelevich, DAN, 143, No. 5 (1962).