Mean value property in limit for eigenfunctions of the Laplace-Beltrami operator. (English) Zbl 1466.43005

Let \(X=G/K\) be a Riemannian symmetric space of noncompact type with rank one. Here \(G\) is a connected noncompact semisimple Lie group with finite center with real rank one and \(K\) is a maximal compact subgroup of \(G\). Let \(o\) be the origin of \(X\), let \(B(x, r)\) be the geodesic ball centered at \(x \in X\) and of radius \(r\), let \(\Delta\) be the Laplace-Beltrami operator on \(X\), and let \(\rho\) be the half sum of positive roots. The elementary spherical function \(\varphi_\lambda\), \(\lambda\in \mathbb C\), on \(X\) is the unique radial eigenfunction of \(\Delta\) with eigenvalue \(-(\lambda^2+\rho^2)\) and \(\varphi_\lambda(o)=1\). Let us introduce the convolution \[ (f * m_r^\lambda)(x) = (V_r^\lambda)^{-1} \, \int_{B(x, r)} \, f(y) \, dy,\ \ {\text {where}}\ \ V_r^\lambda=\int_{B(x, r)} \, \varphi_\lambda (x) \, dx. \] This definition is correct for \(r>0\) except for the set \(D_0^\lambda=\{r>0 \, | \, V_r^\lambda=0\}\).
It is known that \(f\) is an eigenfunction of \(\Delta\) with eigenvalue \(-(\lambda^2+\rho^2)\) only if \(f\) satisfies the generalized mean value property: \((f * m_r^\lambda)(x)=f(x)\).
The substance of main results of the paper can be expressed in short as follows: the limit of the convolution with \(m_r^\lambda\) when \(r\to \infty\) moves arbitrary functions to \(-(\lambda^2+\rho^2)\)-eigenfunctions of the Laplacian.
Here are exact statements.
(i) Suppose that for a function \(f \in L^1_{\mathrm{loc}}(X)\) and a fixed \(\lambda\in\mathbb C\), \[ (f* m_r^\lambda)(x) \to g(r) \] for every \(x\in X\) for some function \(g\) on \(X\), as \(r\) tends to \(\infty\) on \(\mathbb R^+ \setminus D_0^\lambda\). If there is a positive function \(\psi \in L^1_{\mathrm{loc}}(X)\) and a positive function \(r_0 \in L^\infty_{\mathrm{loc}}(X)\) such that for almost every \(x \in X\), \(|(f *m_r^\lambda)(x)| \leqslant \psi(x)\) whenever \(r \geqslant r_0(x)\), then \(\Delta g = -(\lambda^2+\rho^2)g\).
(ii) Let \(f\) and \(g\) be two continuous functions on \(X\). If for a fixed \(\lambda\in\mathbb C\), \[ (f *m_r^\lambda)(x) \to g(x) \] for every \(x \in X\), uniformly on compact sets, as \(r \to \infty\) on \(\mathbb R^+ \setminus D_0^\lambda\), then \(\Delta g = -(\lambda^2+\rho^2)g\).


43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
Full Text: DOI


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