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

Naik, Muna; Ray, Swagato K.; Sarkar, Rudra P.

Trans. Am. Math. Soc. 373, No. 7, 4735-4756 (2020).

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 \(\Delta g = -(\lambda^2+\rho^2)g\).

Reviewer: Vladimir F. Molchanov (Tambov)

MSC:

43A85	Harmonic analysis on homogeneous spaces
22E30	Analysis on real and complex Lie groups

Keywords:

mean value property; Riemannian symmetric spaces; hyperbolic spaces; eigenfunction of Laplacian

References:

[1]	Anker, Jean-Philippe, La forme exacte de l’estimation fondamentale de Harish-Chandra, C. R. Acad. Sci. Paris S\'{e}r. I Math., 305, 9, 371-374 (1987) · Zbl 0636.22005
[2]	Anker, Jean-Philippe; Damek, Ewa; Yacoub, Chokri, Spherical analysis on harmonic \(AN\) groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23, 4, 643-679 (1997) (1996) · Zbl 0881.22008
[3]	Ballmann, Werner, Lectures on spaces of nonpositive curvature, DMV Seminar 25, viii+112 pp. (1995), Birkh\"{a}user Verlag, Basel · Zbl 0834.53003
[4]	Blas1 W. Blaschke, Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logaritmischen potentials, Ber. Ver. S\"achs. Akad. Wiss. Leipzig 68 (1916), 3-7.
[5]	Reichardt, Hans, Wilhelm Blaschke, Jber. Deutsch. Math.-Verein., 69, Abt., Abt. 1, 1-8 (1966) · Zbl 0138.00804
[6]	Brelot, M., \'{E}l\'{e}ments de la th\'{e}orie classique du potentiel, Les Cours de Sorbonne. 3e cycle, 191 pp. Paperbound pp. (1959), Centre de Documentation Universitaire, Paris · Zbl 0084.30903
[7]	Benyamini, Yoav; Weit, Yitzhak, Functions satisfying the mean value property in the limit, J. Analyse Math., 52, 167-198 (1989) · Zbl 0675.31002
[8]	Bridson, Martin R.; Haefliger, Andr\'{e}, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319, xxii+643 pp. (1999), Springer-Verlag, Berlin · Zbl 0988.53001
[9]	Cheng, S. Y.; Yau, S. T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28, 3, 333-354 (1975) · Zbl 0312.53031
[10]	Colding, Tobias H.; Minicozzi, William P., II, Harmonic functions with polynomial growth, J. Differential Geom., 46, 1, 1-77 (1997) · Zbl 0914.53027
[11]	Hansen, W.; Netuka, Ivan, Successive averages and harmonic functions, J. Anal. Math., 71, 159-171 (1997) · Zbl 0879.31002
[12]	Helgason, Sigurdur, Groups and geometric analysis, Integral geometry, invariant differential operators, and spherical functions. Pure and Applied Mathematics 113, xix+654 pp. (1984), Academic Press, Inc., Orlando, FL · Zbl 0543.58001
[13]	Helgason, Sigurdur, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs 39, xviii+637 pp. (2008), American Mathematical Society, Providence, RI · Zbl 1157.43003
[14]	Helgason, Sigur\dj ur, A duality for symmetric spaces with applications to group representations, Advances in Math., 5, 1-154 (1970) (1970) · Zbl 0209.25403
[15]	Hua, Bobo, Generalized Liouville theorem in nonnegatively curved Alexandrov spaces, Chin. Ann. Math. Ser. B, 30, 2, 111-128 (2009) · Zbl 1190.46059
[16]	Kashiwara, M.; Kowata, A.; Minemura, K.; Okamoto, K.; \={O}shima, T.; Tanaka, M., Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. (2), 107, 1, 1-39 (1978) · Zbl 0377.43012
[17]	Koornwinder, Tom H., Jacobi functions and analysis on noncompact semisimple Lie groups. Special functions: group theoretical aspects and applications, Math. Appl., 1-85 (1984), Reidel, Dordrecht
[18]	Li, Peter, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2), 124, 1, 1-21 (1986) · Zbl 0613.58032
[19]	Netuka, Ivan; Vesel\'{y}, Ji\v{r}\'{\i}, Mean value property and harmonic functions. Classical and modern potential theory and applications, Chateau de Bonas, 1993, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 430, 359-398 (1994), Kluwer Acad. Publ., Dordrecht · Zbl 0863.31012
[20]	Peyerimhoff, Norbert; Samiou, Evangelia, Spherical spectral synthesis and two-radius theorems on Damek-Ricci spaces, Ark. Mat., 48, 1, 131-147 (2010) · Zbl 1189.43005
[21]	Plancherel, M.; P\'{o}lya, G., Sur les valeurs moyennes des fonctions r\'{e}elles d\'{e}finies pour toutes les valeurs de la variable, Comment. Math. Helv., 3, 1, 114-121 (1931) · Zbl 0002.02102
[22]	RE1 Repnikov, V. D.; \`Eidel’man, S. D. Necessary and sufficient conditions for establishing a solution to the Cauchy problem. Soviet Math. Dokl. 7 (1966), 388-391.
[23]	Repnikov, V. D.; \`E\u{\i}del\cprime man, S. D., A new proof of the theorem on the stabilization of the solution of the Cauchy problem for the heat equation, Mat. Sb. (N.S.), 73 (115), 155-159 (1967)
[24]	Taylor, Michael E., \(L^p\)-estimates on functions of the Laplace operator, Duke Math. J., 58, 3, 773-793 (1989) · Zbl 0691.58043
[25]	Yau, Shing Tung, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28, 201-228 (1975) · Zbl 0291.31002
[26]	Weit, Yitzhak, An asymptotic characterization of harmonic functions. Commutative harmonic analysis, Canton, NY, 1987, Contemp. Math. 91, 295-297 (1989), Amer. Math. Soc., Providence, RI · Zbl 0714.43011
[27]	Weit, Yitzhak, On a generalized asymptotic mean value property, Aequationes Math., 41, 2-3, 242-247 (1991) · Zbl 0745.30041
[28]	Willmore, T. J., Riemannian geometry, Oxford Science Publications, xii+318 pp. (1993), The Clarendon Press, Oxford University Press, New York · Zbl 0797.53002
[29]	Wong, R.; Wang, Q.-Q., On the asymptotics of the Jacobi function and its zeros, SIAM J. Math. Anal., 23, 6, 1637-1649 (1992) · Zbl 0762.33006