Annales de l’institut Fourier (AIF)

González Vieli, Francisco Javier

Intégrales trigonométriques et pseudofonctions

Pirashvili, Teimuraz

ANNALES DE L’INSTITUT FOURIER
TEIMURAZ PIRASHVILI
On Leibniz homology
Annales de l’institut Fourier, tome 44, no 2 (1994), p. 401-411
<http://www.numdam.org/item?id=AIF_1994__44_2_401_0>
© Annales de l’institut Fourier, 1994, tous droits réservés.
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Ann. Inst. Fourier, Grenoble
44, 2 (1994), 401-411
ON LEIBNIZ HOMOLOGY
by Teimuraz PIRASHVILI
Introduction.
Let g be a Lie algebra over a field k and let M be a right g-module.
Leibniz homology HL^(^M) was defined by Jean-Louis Loday (see 10.6
in [CH]). Moreover, HL^ is well-defined on a much bigger category —
the so called Leibniz algebras and their corepresentations (see [LP]). The
purpose of this paper is to construct a few spectral sequences. These give
a relation between Leibniz and ordinary homologies for Lie algebras and
relate Leibniz homology of a given Leibniz algebra h with its Liezation
hue- As a consequence we compute HL^{^ /c), where g is a semi-simple Lie
algebra or has the dimension 2 or 3 (char k = 0), or is free Lie algebra. We
also prove that in some cases (see Proposition 4.3) one has an isomorphism
HL^h,k)^HL^h^k).
The author would like to thank Friedhelm Waldhausen for hospitality
at the University of Bielefeld while this paper was being written. The author
is also indebted to Jean-Louis Loday for many useful discussions.
1. Relation between Chevalley-Eilenberg homology
and Leibniz homology for Lie algebras.
Let M be a right module over the Lie algebra g. We denote
a,(g,M)=(M^A*g,d')
CL,(g,M)=(M0T*g,d),
Key words : Lie algebras - Leibniz algebras - Homology.
A.M.S. Classification : 17B55 - 17D - 18G40.
402 TEIMURAZ PIRASHVILI
where T*g is the tensor algebra over g, d' is the Chevalley-Eilenberg
boundary map, and d is the Loday boundary map (see 10.6.2.1 in [CH]).
The natural epimorphism
CL.(g,M)-^(g,M)
is a morphism of chain complexes, which is an isomorphism in dimensions
0 and 1. We define C^g.M) such that C^\g,M)[2} is the kernel of the
above epimorphism. Let
H:el(s,M)=H^C?\s,M)).
We recall that HL, (g, M) = H, (CL^ (g, M)) and ff, (g, M) = H, (C* (g, M)).
Thus, by definition we have
PROPOSITION 1.1. — One Aas an exact sequence
0 ̂ - ff2(g, M) ̂ - HL^g, M) ̂ - H^(s, M)
^- H3(g,M) ̂ - HLs(s,M) ̂ - ...
and isomorphisms
HLi(g,M)^Hi(s,M), for i=0,l.
The product map m : g <g) A"g —> A"+lg in the exterior algebra
yields an epimorphism of chain complexes
c*(g.g)-^a(g,fc)[-i].
Here C^ (g, k) is the reduced chain complex :
Co^k) = 0,G,(g,/c) = Q(g,A;),z > 0.
We define CR^g) such that C'J?*(g)[l] is the kernel of C^(g,g) —>
G*(g,A;)[-l]. We denote the homology of CR^(g) by HR^g). Then, by
definition we have
PROPOSITION 1.2. — One has an exact sequence
0 <— ff2(g, k) — ffi(g, g) ̂ - HRo(s) ̂ - ff3(g, k)
^- ff2(g,g) ̂ - HRi(g) ̂ - ...
ON LEIBNIZ HOMOLOGY 403
As usual, we denote the kernel of (^g —> A^g by Fg. Thus Fg is
spanned by elements x 0 x. Moreover, Fg is a g-submodule o fg0g and
the restriction of
1 (g) m : g (g) g 0 A'g —^ g (g) A^g
to Fg (g) A^ gives an epimorphism of chain complexes
^*(g,rg)^c^(g),
which is an isomorphism in dimension 0. Hence HRo(g) ^ ^(g^Fg) and
for the cohomological version we have
HR°(g) ̂  {invariant quadratic forms on g}.
It follows from Theorem A that the same is also true for H^(g^k) and
H^{g, k). Moreover, the corresponding map
{invariant quadratic forms on g} —> H3^ k)
coincides with the classical homomorphism ofKoszul (see section 1 1 of [K]).
THEOREM A. — Let g be a Lie algebra and M be a right g-module.
Then there exists a spectral sequence
E^ = HRp(g) 0 HL^ M) =^ J^(g, M).
Proof. — We consider the following filtration on C^(g,M). Let
(j € CLn{g,M) and x € g. Then it follows from 10.6.3.1 of [CH] that
d((j,x,x) = (do;, x, x). Here we write (a,y) instead of a (g) y . Thus the
subspaces F^ = Cn(g, M) (g) Fg C C^g, M) give us a subcomplex and
^(^)^^L,(g,M)0rg.
More generally, for p = (a;i,...,^) and pj = (a-i, ...,.Tj, ...^), 1 < j <: i,
one has
d{^,x,x,p)={(kj,x,x,p) + {-l)^^,x,x,dp}
+^(_l)deg^(^^.^^^^
^^_^deg^^((^^^^(^^^)^^.^
Since { y , z ) 4- (z,y) € Fg, y , z C g we conclude that J^ C T}. C ... are
subcomplexes of C^g, M) with ̂  = C^\g, M) and
404 TEIMURAZ PIRASHVILI
^/^-^CL^M^-s}
^(Ker^^g —— A^g^g 0 Ker^^g __ A^g))).
Here ^ = C^_,(g,M) 0 Ker^^g -^ A^g) is spanned by
(^^,.^1,...,^) with z ^ s. Thus the corresponding spectral sequence
has the form :
E^=HL^M)^CRp{g)
because there is a natural isomorphism
ker^^g —— A^gVker^^g —— g 0 A^g)
^ke^g^A^g——A^g).
Hence the theorem follows from Proposition 10.1.7 of [CH] and from the
above formula for d.
COROLLARY 1.3. — IfH^M) = 0, then HL^M) = 0.
Proof. — It follows from Proposition 1.1 that HL^M) = 0 for
i = 0,1 and HL^{^ M) ̂  ̂ (g, M) for n > 0. We prove by induction
that H^\^M) = 0. It holds when n = 0 because E^ = 0 for q = 0,1
and Theorem A implies ^(g,M) = 0 for n = ^1. If we assume
H^\g,M) = 0 for all i < n, then HL^M) = 0 for z < n + 2. Thus
£^ = 0 for g < n + 2 and ̂ (g, M) = 0 for i < n + 2.
The cohomological versions of Theorem A and Corollary 3.3 are still
true with the same proofs (of course now one will consider C* (g, g*) instead
^ C*(g,g), where g* is the dual of g). There are well-known examples
which satisfy the assumption of Corollary 1.3 (see for example Theorem 1
of [D] and the main result of [T]).
In the next section we consider the case when g is a finite dimensional
semi-simple Lie algebra over a characteristic 0 field. Now as a sample
application of Theorem A we compute
HL^:=HL^k)
where g is two or three dimensional Lie algebra defined over the complex
numbers.
Examples 1.4. — i) Let g = r2,rs or r^, where A ^ -1, 1. Here
r2 is the nonabelian two dimensional Lie algebra, r^ and r^ are three
ON LEIBNIZ HOMOLOGY 405
dimensional Lie algebras with
r3 : [x, y\ == y , [x,z\ = y + z
r3,A : [ x , y } = y , [x,z}=\z, 0 <| A |< 1.
Then Hi(g) == 0 for i > 1. Thus ^fLn+2(g) ^ H^\g),n > 0. By the direct
calculations we obtain : HRi(g) = 0 for i > 0 and dim HRo^g) = 1. It
follows from Theorem A that H^(g) = HLn(g). Hence dim HLn(g) =
1, n>, 0.
ii) Let g = rs^ with A = —1. In this case we still have HRi{g)=0
for i > 0 and dimHRo(g) = 1, but now dim^(g) = 1, for 0^z^3, and
Hi(g) = 0 , i > 3. Easy calculation shows that the connected homomor-
phism Hs (g) —^ H^(g) = HRo{g) is zero. Thus for a;n:= dim HLn{g)
one has a;o = 1, ̂ i = 1, a^n = 2, n ^ 2.
iii) Let g = r^^ with A == . In this case Hi(g) = 0 for i > 1,
dim HRi{g) = 1 for z = 0,1,2 and ^^(g) = 0 for z > 2. Thus
^L,+2(g) = ̂ (g), ̂  = 0 for p > 2 and E^ = HL,{g) for p = 0,1,2.
We claim that ^2 and hence all differentials are zero. Indeed, this follows
from the fact that the class of z 0 (z A x A y) in HR'z{g) is a nonzero
element and if uj is some element of (7L*(g) with duj = 0 then dp == 0.
Here p = (a;, ̂ , z, x, y) - (a;, x, z, z, y) + (a;, ^/, 2;, 2;, a;) - (a;, y , x, z, z). Thus
XQ = 1,0-1 = l,a;2 = 1^3 = 2,a;n = Xn-2 + ^n-3 + ^n-4^ > 4, where
a;n = dim HLn(g).
iv) Let g = 113 be the three dimensional nilpotent Lie algebra :
[x,y\ = z. Then dim ^i(g) = 2, dim H^g) = 2, dim H^g) = 1 and
dim HRo{g) = 3, dim HR^(g) = 3, dim HR^(g) = 1. The connected
homomorphism ^(g) —^ HRQ^g) as well as all differentials of the
spectral sequence are still zero. To verify the last fact we remark that
z 0 (z A x A y) gives a nonzero element in HR^(g) and if uj € CL^{g)
is an element with duj == 0, then dp = 0, where p = ( u j , z , z , x , y ) —
(a;, x, z, z, y) + (uj, y , z, z, x) + (c^, re, ^/, z, z}. Hence for .Tn: = dim HLn(g) we
have o-o = 1, a'i = 2, a;2 = 5, x^ == 10 and Xn = 3xn-2+^Xn-3+Xn-4^ n > 4.
406 TEIMURAZ PIRASHVILI
2. Semi-simple Lie algebras as Leibniz algebras.
In this section we assume that char k = 0 and g is a finite dimensional
semi-simple Lie algebra. Proposition 2.1 was proved independently by Ntolo
[N]. She used Casimir element to construct the explicit homotopy.
PROPOSITION 2.1. — Let M be a finite dimensional right g-moduJe
and A be a finite dimensional corepresentation of g (see 1.5 and 1 . 1 1 of
[LP]). Then
HLi{g,M) =0, if % > 0
JLL,(g,A)=0, if i>l.
Proof. — It follows from the dual of Proposition 5.6 of [HS] and
Corollary 1.3 that HL^(g^M) = 0 if M is a nontrivial simple g-module.
Thus, for arbitrary M, one has
HL^M)=HL^k)^M^
But JLL,+i(g, k) ̂  HLi(g, g) (see Exercise E. 10.6.1 of [CH] or take C = k
in Lemma 2.2) and gg = 0. Thus HL^g, k) = 0 for i > 1. This completes
the proof in the case of right modules. Let A be a corepresentation. Then
there exists an extension (compare with the corresponding extension for
representations in 1.10 of [LP])
0 —>B8 —>A —^a —>0
such that B8 (resp. C01) is an (anti) symmetric corepresentation and the
proposition follows from Lemma 2.2.
LEMMA 2.2. — Let h be a Leibniz algebra. Let B8 (resp. C01) be an
(anti) symmetric corepresentation ofh. Let B and C be the underlying right
h-modules. Then HL^h.B8) = HL^(h,B) , HLo^C^ = C and
HLi^(h, C^ ̂  HLi(h, C (g) h),
where C ̂  h is considered as right h-module by the action :
[c<^)h,x] = —[a;,c](g)/i + c<S>[h,x], h, x € h, c e C.
ON LEIBNIZ HOMOLOGY 407
Proof. — One checks that the isomorphisms exist on the level of chain
complexes.
The cohomological version of Proposition 2.1 is still true. As for
Lie algebras one verifies that for an arbitrary Leibniz algebra h the
space H'^Oi^'h) in the sense of 1.8 of [LP] is a space of obstructions for
deformations in the category of Leibniz algebras. Thus we deduce from
Proposition 2.1, Corollary 1.3 and [T] the following
PROPOSITION 2.3. — Let p be a parabolic subalgebra ofg. Then p as
well as g is rigid in the category of all Leibniz algebras.
PROPOSITION 2.4. — Let f : h —> g be an epimorphism from an
arbitrary finite dimensional Leibniz algebra h to the semi-simple Lie algebra
g. Then / has a section.
Proof. — Let hLie = h/([;z',;r] = 0) be the Liezation of h. Let
ch(h) € ^(hue, h^") be the characteristic element of h (see 1.10 in [LP]),
corresponding to the extension
0 —— h^ —— h —— hLie —— 0.
By definition / factors through /i : hLie —> g- It follows from Levi theorem
that /i has a section s. By Proposition 2.1 we have
s^dl{h))eHLf2^haTln)=o.
Thus s has a lifting g —> h.
3. Free Lie algebra approach.
Let f be a free Lie algebra and M be a right f-module. Since
Hi(f^—) = 0, if i > 1, we deduce from Propositions 1.1 and 1.2 that
HLn^{f,M) ^ H^\f,M),n > 0; HR^f) = 0,z > 0 and HRo(f) ^
T^i (f ,f) . Then Theorem A gives
^(f, M) ̂  ^i(f, f) 0 HLn(f^ M).
Thus we prove the following proposition.
408 TEIMURAZ PIRASHVILI
PROPOSITION 3.1. — Let f be a free Lie algebra and M be a right
f-module. Then
^L,(f,M)^^l(f,f)0fc0^o(f,M), if n=2k
^^(f.f^^^f.M), if n = 2 A ; + l .
THEOREM B. — Let g be a Lie algebra and M be a right g-module.
Then there exists a spectral sequence £^ ====> HLp^.q(g, M) with
^k = Q^i(g) ̂  - ̂  HR^(g) 0 ̂ o(g,M)
<2fc+i = (B ̂ i (g) ̂  ... ̂  HR,, (g) 0 J^+i (g, M)
where the sum in the first (resp. second) formula is taken over allii + ... +
ik = p (resp. i-i + ... + z^+i = ?)•
Proof. — Let f* be a componentwise free simplicial Lie algebra, such
that TTof* = g and TT^ = 0, i > 0. Such an object exists because of [Q] and
we obtain a spectral sequence
E^ = HL^M) ̂  HLp^M)
because ^(Tot C'I^(f^M)) ^ HL^M). Thus Theorem B is a conse-
quence of Proposition 3.1, Kunneth relation and Lemma 3.2.
LEMMA 3.2. — One has
7TiHo{{^M) ^^o(g,M), if i=0 and =0, if % > 0
7T^i(f*,M)^^+i(g,M)
7r^i(f,,f,)^JLR,(g).
Proof. — The first part follows from the fact that Ho{f^,M) is a
constant simplicial object. The second one is well-known (see for example
[Q]). Since ^(Tot CR^(f^)) ^ H^CR^{g)^ the last part is a consequence
of Proposition 3.1.
ON LEIBNIZ HOMOLOGY 409
4. The relation between HL^h and HL^hue'
Let / : h —> b be an epimorphism of Leibniz algebras and let M be
a right b-module. We may consider it also as an h-module through /. Let
C*(h; b, M) be a chain complex, such that
GL,(h;b,M)[l] =Ker(GL,(h,M) ——CL,(b,M))
and JfL*(h;b,M) := ^(C'L*(h;b,M)). Then, by definition, we have
PROPOSITION 4.1. — There exists an exact sequence
0 <— HL^(b,M) <— ^Li(h,M) ^— HLo(h',b,M)
<— HL^(b,M) <— HL^(h,M) <— ...
THEOREM C. — Let
0—>SL—>h—>b—>0
be a short exact sequence of Leibniz algebras. Assume [h, a] = 0, h e h,
a € a. Then there exists a spectral sequence
E^ = HLp(b, a) 0 HLq{h, M) =^ HLp^q(h; b, M)
where a right ^-module structure on a is given by [a^x] := [a,/"1 (re)],
x € b,a € a.
Proof. — Let a C a and uj C <7L»<(h,M). Then d(o;,a) = (da;, a),
because [c<;,a] = 0. Thus
^° = CL^ (h, M) 0 a C CL^ (h; b, M)
is a subcomplex and H^(J^) = H^(h^M) (g) a. More generally : for
p= (x^,...,Xi) and^- = (a;i, ...,^-, ...,^), 1 ̂  j < z, one has
^.a.^^^a.^+^-^^a^^],^^)
+ ̂ (-l)^^, [a, ̂ .], ̂ ) + (-1)^(0;, a, ̂ ).
Since [a, a-] 6 a we conclude that F^ C F}. C ... are subcomplexes of
CL>,(h;b,M) with ̂  = CL^(h;b,M). Here ̂  is spanned by (a;, a,/?)
with degp < 5. One checks that there exists an isomorphism
^/jr^-i ^ GL,(h; b, M)[-5] 0 a 0 b05
and the theorem follows from 10.1.7. of [CH] and the above formula for d.
410 TEIMURAZ PIRASHVILI
Remark 4.2. — The assumptions of Theorem C hold for
0 —— h^ —— h —— hLie —— 0.
Hence Theorem C gives a relation between HL^h and HL^h^e. Since
HL\(\\.^ k) = HL-^(b.^[e^ k)^ we obtain an exact sequence
0 ̂ - (h^)^ ̂ - HL^h^k) ̂ - HL^k).
PROPOSITION 4.3. — Let h be a finite dimensional Leibniz algebra,
such that g = h^g is a semi-simple Lie algebra over characteristic zero field
or is a nilpotent Lie algebra with ch(h) == 0. Then HL^(h^ k) = JfL^(g, k).
Proof. — By Proposition 2.1 ch(h) = 0 in both cases. Hence the
epimorphism h —> g has a section. It follows from Proposition 4.1 and
Remark 4.2 that
^L,(h,A;)=^L,(g^)C^L,_i(h;g,fc), and (h^g = 0.
Theorem C gives the spectral sequence
E2^ = HL^ h^) 0 HL^ k) ==^ HL^^ g, k).
By Theorem 1 of [D], Corollary 1.3 and Proposition 2.1 we have
HL^g.h^) = 0. Thus HL^g.k) = 0, because E^ = 0 and
HL^k)=HL^k).
BIBLIOGRAPHY
[D] J. DIXMIER, Cohomologie des algebres de Lie nilpotentes. Acta Sci. Math.
Szeged., 16 (1955), 246-250.
[HS] P.J. HILTON, and U. STAMMBACH, A course in homological algebra (Grad.
Texts in Math., vol. 4). Berlin, Heidelberg, New York, Springer Verlag, 1971.
[K] J.L. KOSZUL, Homologie et cohomologie des algebres de Lie. Bull. Soc. Math.
France, 78 (1950), 65-127.
[CH] J.L. LODAY, Cyclic homology. (Grund. Math. Wiss., Bd. 301), Berlin,
Heidelberg, New York, Springer Verlag, 1992.
[LP] J.L. LODAY, and T. PIRASHVILI, Universal enveloping algebras of Leibniz
algebras and (co)homology, Math. Ann., 296 (1993), 139-158.
ON LEIBNIZ HOMOLOGY 411
[N] P. NTOLO, in preparation.
[Q] D. QUILLEN, On the (co)-homology of commutative rings. Proc. Symp. Pure
Math., 17 (1970), 65-87.
[T] A.K. TOLPYGO, Cohomologies of parabolic Lie algebras, Mathematical notes
of the Academy of Sciences of the USSR (Translations), 12 (1972), 585-588.
Manuscrit recu Ie 10 septembre 1993.
Teimuraz PIRASHVILI,
Mathematical Institute
Rukhadze Str.l
Tbilisi, 380093 (Rep. of Georgia).


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ANNALES DE L’INSTITUT FOURIER
NIKOLAI S. NADIRASHVILI
The Martin compactification of a plane domain
Annales de l’institut Fourier, tome 44, no 5 (1994), p. 1351-1354
<http://www.numdam.org/item?id=AIF_1994__44_5_1351_0>
© Annales de l’institut Fourier, 1994, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
44, 5 (1994), 1351-1354
THE MARTIN COMPACTIFICATION
OF A PLANE DOMAIN
by Nikolai S. NADIRASHVILI
In this note we prove the following
THEOREM. — The Martin compact! fication of a plane domain is
homeomorphic to a subset of the two-dimensional sphere.
ASSUMPTIONS. — IfQ. be a plane domain and R2^ is polar then any
positive harmonic function on Q is a constant. In this case we define the
Martin compactification of f2 as a one point set. So we assume from now
on that R2^ is non-polar. We may also assume without loss of generality
that BI C ^2 where Bi is the unit disk in R2 with the center at 0.
Remark. — If a simply connected domain is a proper subset of the
plane then by Riemann mapping theorem its Martin compactification is
homeomorphic to a closed disk.
CONJECTURE 1. — The Martin compactification of a subdomain of a
compact Riemannian surface is homeomorphic to a subset of this surface.
CONJECTURE 2. — Any compact metrizable space can be represented
as the Martin boundary of a certain (generaly of infinite genus) Riemannian
surface.
1. The Martin compactification.
Let G(rc, y) be the Green function of the Dirichlet Laplacian on ^2,
. with the pole at x. Let us denote gx(y) = G(x,y)/G(x,0) for x ^ 0 and
Key words : Martin boundary.
A.M.S. Classification : 31A20.
1352 NIKOLAI S. NADIRASHVILI
^o=0. Let gx(y) be the restriction of the function Qx{y) on y 6 B\. So we
have a map
7 : a;-^ e L^Bi).
The Martin metric on 0 can be defined as the metric inducted on Q, by the
map 7 : n —>• L^Bi), (cf. [1]). Compactification of f^ in the Martin metric
we denote as ^M.
Canonical map.
We set
/ : ̂  V^(0)
and /(O) = ex) by the definition. We claim that the introduced canonical
map / has the uniformly continuous inverse map from /(^) to ^M.
Proof of the theorem.
1.1. Let G C K2 be a domain and Q a disk such that Q C G. Also,
let di G QQ^i == l,...,2n, be distinct points on 9Q. We assume that the
di are indexed in the order in which they are encountered when traversing
9Q. Let / be a continuous function in G\Q such that /(a^)/(a^i) < 0 for
all i = 1,..., 2n — 1. We denote by Gi C C^Q the domain where / does not
change sign, such that di G Gi.
Lemma ([2]). — At least n + 1 of the domains Gi, i = 1,..., 2n, are
distinct.
1.2. Let rci,a;2 G ^- We prove that if f{x\) = /(a^) then x\ = a-2.
Let ^ = gx^-9x^ Then A^A = 0 in ^\({a;i}U{a-2}), ^(0) = Vtt(O) = 0.
Let r be the nodal set of ZA, r = {x 6 ^,^(.r) = 0}. If u ^ 0 then in a
neighborhood ofO, F consists ofn smooth curves intersected at the point 0,
where n is an order of vanishing of the function u at 0 (cf. [2]). By Lemma
r splits the domain fl, at least on three distinct subdomains. By maximum
principle each of those subdomains should contain a pole of the function u.
Since function u has only two poles x 1,^25 it follows that u = 0.
2. Now we prove that the map
F : z = f ( x ) e f W ^ g x
is uniformly continuous.
THE MARTIN COMPACTIFICATION OF A PLANE DOMAIN 1353
2.1. Let Bi C B, B C Q,. By Harnak inequality for any x € ^\B,
(jx < C, where C > 0 is some constant.
2.2. Let Xn, Zn € 0, n = 1,2,..., and ̂  —^ ^i, 9zr, —^ ^2 on any
compact in Q, as n —>• oo, fai ^ faa. Its required to prove that V/ii(0) 7^
V/i2(0). Let us assume the contrary, namely that V/ii(0) = V!/i2(0). We
denote h = /ii — ^2 and let fc be an order of vanishing of the function h at
0, k > 2.
2.3. Let r be the nodal set of the function h. There exists such a
small p > 0 that on Sp = 9Bp, |Vfa| > 0 and the cardinality of the set
Sp n r is equal to 2k.
2.4. We prove the existence of two bounded non-constant harmonic
functions -yi, v^ in ̂ , such that V^i(O) ^ 0, Vz»2(0) -^ 0, V2;i(0) 7^ aV-^O),
for any a e R.
Let us choose discs J9i, D^^D^ C R2 such that -D^\^ non-polar,
i = 1,2,3, and for any points Xi € -D% the quadrangle 0, rci, a;2, x^ is convex.
Let /^^ be a probability measure on Di\fl, such that the convolution In |^|*/^
is bounded from below. We set v\ = In \x\ * (/^i —^2)5 ^2 = ̂  1^1 * (/^3 ~ 1^2)-
Then ^i, v^ are bounded harmonic functions in fl, and the V^i (0), Vz'2 (0)
have the required property.
For any a 6 R2 there exists a unique linear combination
Wa = 0lVl + /?2^2 - /3l^l(0) - /32^2(0)
such that Vwa(O) = a, Wa(0) = 0. Further, if |a| —> 0 then |wa| -^ 0
uniformly in f2.
2.5. Let us denote
V^(0)-V^(0)=an,
9n =^, -9zr, -^ar,'
Then gn(0) = Vgn(O) = 0 for all n = 1,2,....
From (2.1), (2.2), (2.4) it follows that qn —>• h in B\ and hence also
qn —^ h in C^Bp) as n —^ oo. Therefore, if Fyi is a nodal set of qn then
1354 NIKOLAI S. NADIRASHVILI
for a sufficiently large n >_ N, 5p\I\ is a union of 2k distinct intervals
^...,^and
sup inf sup qn > a > 0
n>N l<J<2fc p
with some constant a. Since Wa^ —> 0 uniformly in ^ as n —)• oo then for
sufficiently large n > N ' > N^ |waJ < a in Q. Hence \qn\ < a on 90, for
n>7V' .
2.6. Since ^n(O) = Vgn(O) = 0 then by Lemma the set 0\Tn contains
at least three components G\^G^^G^ such that 0 e G^, i = 1,2,3. From
(2.5) it follows that for n > N ' and i = 1,2,3
SUp \Qn\ > SUp|gn|.
GiDBp 9Gi
By the maximum principle from the last inequality it follows that any of
the domains €?%, i = 1,2,3 contains a pole of function Qn. Since the function
On has only two poles we get a contradiction which proves the theorem.
BIBLIOGRAPHY
[1] J.L. DOOB, Classical potential theory and its probabilistic counterpart, Springer,
1984.
[2] N.S. NADIRASHVILI, Multiple eigenvalues of the Laplace operator (Russian),
Matem. Sb., 133 (175), 1987; English translation in Math. USSR Sbornik, 61 (1988),
225-238.
Manuscrit recu Ie 13 avril 1994,
revise Ie 21 juillet 1994.
Nikolai S. NADIRASHVILI,
Institute of Earth Physics
Moscow (Russia).


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