Mean value properties for eigenfunctions of the Laplacian on symmetric spaces
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| Cite this: | Mean value properties for eigenfunctions of the Laplacian on symmetric spaces / V.V. Volchkov, Vit.V. Volchkov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 31-35. — Бібліогр.: 8 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-200062025-02-09T16:57:26Z Mean value properties for eigenfunctions of the Laplacian on symmetric spaces Volchkov, V.V. Volchkov, Vit.V. 2008 Article Mean value properties for eigenfunctions of the Laplacian on symmetric spaces / V.V. Volchkov, Vit.V. Volchkov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 31-35. — Бібліогр.: 8 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/20006 517.5 en Труды Института прикладной математики и механики НАН Украины application/pdf Інститут прикладної математики і механіки НАН України |
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Volchkov, V.V. Volchkov, Vit.V. |
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Volchkov, V.V. Volchkov, Vit.V. Mean value properties for eigenfunctions of the Laplacian on symmetric spaces Труды Института прикладной математики и механики НАН Украины |
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Volchkov, V.V. Volchkov, Vit.V. |
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Volchkov, V.V. |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces |
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mean value properties for eigenfunctions of the laplacian on symmetric spaces |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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Mean value properties for eigenfunctions of the Laplacian on symmetric spaces / V.V. Volchkov, Vit.V. Volchkov // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 31-35. — Бібліогр.: 8 назв. — англ. |
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Труды Института прикладной математики и механики НАН Украины |
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AT volchkovvv meanvaluepropertiesforeigenfunctionsofthelaplacianonsymmetricspaces AT volchkovvitv meanvaluepropertiesforeigenfunctionsofthelaplacianonsymmetricspaces |
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2025-11-28T06:05:01Z |
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ISSN 1683-4720 Труды ИПММ НАН Украины. 2008. Том 17
UDK 517.5
c©2008. V.V. Volchkov, Vit.V. Volchkov
MEAN VALUE PROPERTIES FOR EIGENFUNCTIONS
OF THE LAPLACIAN ON SYMMETRIC SPACES
Let X = G/K be a symmetric space of the noncompact type with complex group G, L the Laplace-
Beltrami operator on X. It is proved that eigenfunctions of L are characterized by vanishing of integrals
over all balls in X with special radii.
A well-known theorem on ball means for the Helmholtz equation asserts that a
function f ∈ C(Rn) is a solution of the equation ∆u + λ2u = 0 if and only if for all
x ∈ Rn and r > 0 ∫
|u|6r
f(x + u)du =
(
2πr
λ
)n/2
Jn/2(λr)f(x),
where Jk is the kth-order Bessel function of the first kind. In particular, this implies that
every solution of the equation
∆u + u = 0
has zero integrals over all balls in Rn with radii belonging to the zero set of Jn/2. In [1]
(also see [2], Part II, Theorem 1.12), the first author proved the following.
Theorem A. Let f ∈ L1,loc(Rn), n > 2, and let {νk}∞k=1 be the sequence of all
positive zeros of the function Jn/2. Then the integrals of f over all balls in Rn with radii
ν1, ν2, . . . are equal to zero if and only if ∆f + f = 0 (here the equality is understood in
the sense of distributions).
Analogues of Theorem A for rank one symmetric spaces were established by V.V.Vol-
chkov in [3]. These results enabled him to get a description of the Pompeiu sets in terms of
approximations of their indicator functions by linear combinations of indicator functions
of balls with special radii in L1 (see [1–3]). The purpose of this paper is to obtain an
analogue of Theorem A for symmetric spaces of arbitrary rank.
As regards basic notions and facts from the theory of symmetric spaces, see [4]–[6]. Let
X = G/K be a symmetric space of the noncompact type, G being a connected semisimple
Lie group with finite center and K a maximal compact subgroup. Let o = {K} be the
origin in X and denote the action of G on X by (g, x) → gx for g ∈ G, x ∈ X. The Lie
algebras of G and K are respectively denoted by g and k. The adjoint representations
of g and G are respectively denoted by ad and Ad. Let 〈 , 〉 be the Killing form of gC,
the complexification of g. The form 〈 , 〉 induces a G-invariant Riemannian structure on
X with the corresponding distance function d(·, ·) and the Riemannian measure dx. For
R > 0, y ∈ X we set
BR(y) = {x ∈ X : d(x, y) < R}, BR = BR(o).
31
V.V. Volchkov, Vit.V. Volchkov
Let L1(X) and L1,loc(X) be the classes of complex-valued functions on X that are dx-
integrable and locally integrable, respectively. The Laplace-Beltrami operator on X is
denoted by L.
Let p be the orthogonal complement of k in g with respect to 〈 , 〉 and let a ⊂ p
be any maximal abelian subspace (all such subspaces have the same dimension). The
dimension of a is called the real rank of G and the rank of the space X. We shall write
rankX = dim a.
Let a∗ be the dual of a, a∗C and aC their respective complexifications. Next, let Aλ ∈ aC
be determined by 〈H,Aλ〉 = λ(H) (H ∈ a) and put
〈λ, µ〉 = 〈Aλ, Aµ〉, λ, µ ∈ a∗C.
For λ ∈ a∗ put |λ| = 〈λ, λ〉1/2,
gλ = {U ∈ g : [H, U ] = λ(H)U for all H ∈ a},
where [·, ·] is the bracket operation in g. If λ 6= 0 and gλ 6= {0} then λ is called a
(restricted) root and mλ = dim gλ is called its multiplicity. The spaces gλ are called root
subspaces. The set of restricted roots will be denoted by Σ. A point H ∈ a is called
regular if λ(H) 6= 0 for all λ ∈ Σ. The subset a′ ⊂ a of regular elements consists of the
complement of finitely many hyperplanes, and its components are called Weyl chambers.
Fix a Weyl chamber a+ ⊂ a. We call a root positive if it is positive on a+. Let Σ+ denote
be the set of positive roots; for α ∈ Σ+ we will also use the notation α > 0 and put
ρ =
1
2
∑
α>0
mαα.
Let exp be the exponential mapping of g into G. As usual we set ExpP = (expP ) K ∈
X for each P ∈ p. Denote by J the corresponding Jacobian, i.e.,
∫
X
f(x)dx =
∫
p
f(ExpP )J(P )dP, f ∈ L1(X).
Define
dµ(x) = (J(Exp−1x))−1/2dx.
Let D′(X) (resp. E ′(X)) be the space of distributions (resp. distributions of compact
support) on X, E ′\(X) the space of K-invariant compactly supported distributions on
X, T̃ the spherical transform of a distribution T ∈ E ′\(X). As in [7], denote by E ′\\(X) the
set of all distributions T ∈ E ′\(X) with the following property: T ∈ E ′\\(X) if and only if
there exists a function
◦
T : [0,+∞) → C such that
T̃ (λ) =
◦
T (|λ|)
32
Mean value properties for eigenfunctions of the Laplacian on symmetric spaces
for all λ ∈ a∗. It can be shown that for each T ∈ E ′\\(X) the function
◦
T admits extension
to C so that
◦
T becomes an even entire function. For each T ∈ E ′\\(X) we set Z(
◦
T ) =
{ζ ∈ C :
◦
T (ζ) = 0}.
From the Paley-Wiener theorem for the spherical transform (see [5, Chap.4, Theo-
rem 7.1]) it follows that the class E ′\\(X) is broad enough. We point out that
E ′\\(X) = E ′\(X) provided rankX = 1.
Let × denotes the convolution on X. We recall that if f ∈ D′(X) and T ∈ E ′(X) then
〈f × T, u〉 =
〈
T (g2K),
〈
f(g1K),
∫
K
u(g1kg2K)dk
〉〉
, u ∈ D(X), (1)
where D(X) is the space of complex-valued C∞-functions of compact support on X. The
following characterizations of the class E ′\\(X) were proved in [7], [8].
Theorem B. If T ∈ E ′\(X) then the following assertions are equivalent.
(i) T ∈ E ′\\(X).
(ii) For each λ ∈ a∗ every solution f ∈ C∞(X) of the equation
Lf = − (|λ|2 + |ρ|2) f
satisfies the equality
f × T = T̃ (λ)f.
Theorem C. Let X = G/K be a symmetric space of the noncompact type with
complex group G, and let T ∈ (E ′\∩L1)(X). Then the following assertions are equivalent.
(i) T has the form
T (x) = (J(Exp−1x))−1/2u(d(o, x)), x ∈ X,
for some function u : [0, +∞) → C.
(ii) T ∈ E ′\\(X).
The main result of this paper is as follows.
Theorem 1. Let X = G/K be a symmetric space of the noncompact type with
complex group G. Let {λq}∞q=1 be the sequence of all positive zeros of Jl/2, where l =
1
2dimX, and let c > 0. Assume that f ∈ L1,loc(X) and rq = λq/c, q = 1, 2, . . .. Then the
following items are equivalent.
(i) f satisfies the equation
(L + |ρ|2 + c2)f = 0.
33
V.V. Volchkov, Vit.V. Volchkov
(ii) For all g ∈ G, q ∈ N, ∫
Brq
f(gx)dµ(x) = 0. (2)
Proof. Let g ∈ G and let χq be the characteristic function of the ball Brq . Then
χq(g−1o) = χq(go). (3)
Using Lemma 4.4 and Propositions 4.8 and 4.10 from [5, Chap. 4], one infers that
J−1/2(Exp−1(go))ψλ(Exp−1(go)) = J−1/2(Exp−1(g−1o))ψ−λ(Exp−1(g−1o)),
where
ψλ(P ) =
∫
K
ei〈Aλ,Ad(k)P 〉dk, P ∈ p.
Hence
J−1/2(Exp−1(go)) = J−1/2(Exp−1(g−1o)). (4)
In view of (3), (4) and (1) relation (2) can be written as
f × Tq = 0, (5)
where Tq(x) = J−1/2(Exp−1x)χq(x), x ∈ X. The proof of Theorem С (see [7], [8]) shows
that Tq ∈ E ′\\(X) and
T̃q(λ) = γqIl/2(rq
√
〈λ, λ〉), λ ∈ a∗C,
where Il/2(z) = Jl/2(z)z−l/2 and the constant γq is independent of λ (see [2], Part I,
Example 6.1). This together with Theorem B gives the implication (i)→(ii).
Let us prove that (ii)→(i). Define now Tq ∈ E ′\\(X) by the relation
T̃q(λ) = (〈λ, λ〉 − c2)−1Il/2
(
rq
√
〈λ, λ〉), λ ∈ a∗C, q ∈ N.
One therefore has
⋂∞
q=1Z
( ◦
T q
)
= ∅ (see [2], Part II, Lemma 1.29). In addition, equality
((L + |ρ|2 + c2)Tq )̃ (λ) = (c2 − 〈λ, λ〉)T̃q(λ)
and the argument in the proof of the implication (i)→(ii) show that (ii) can be brought
to the form
((L + |ρ|2 + c2)f)× Tq = 0 for all q.
Appealing now to [7, Theorem 4.12] we arrive at the desired statement. ¤
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– 1993. – Т.184. – №7. – С.71–78.
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shers, 2003. – 454pp.
34
Mean value properties for eigenfunctions of the Laplacian on symmetric spaces
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1978. – 633p.
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6. Helgason S. Geometric Analysis on Symmetric spaces. – Rhode Island: Amer. Math. Soc., Providence,
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spaces and on phase space associated to the Heisenberg group // J. Analyse Math. – 2008. – Vol.105.
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8. Volchkov V.V., Volchkov Vit.V., Zaraisky, D.A. The class E ′\\(X) in the theory of convolution
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Donetsk National University
volchkov@univ.donetsk.ua
Received 10.10.08
35
содержание
Том 17
Донецк, 2008
Основан в 1997г.
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