NETUKA, IVAN

Netuka, Ivan

Osaka University Knowledge Archive

The Ninomiya operators and the generalized Dirichlet problem in potential theory

Osaka City University Repository

Netuka, I.
Osaka J. Math.
23 (1986), 741-750
THE NINOMIYA OPERATORS AND THE GENERALIZED
DIRICHLET PROBLEM IN POTENTIAL THEORY
IVAN NETUKA
(Received July 9, 1985)
Introduction
Investigating various aspects of Keldys theorem [11], N. Ninomiya in-
troduced and studied special operators related to the generalized Dirichlet
problem in classical potential theory.
To recall his interesting result from [18], let us introduce the following
notation.
Let {Bj} be the sequence of balls in the Euclidean space Rm of dimension
m>2 having a rational center and rational radius. Denote by λ, the nor-
malized surface measure on dBj and fix positive numbers Uj such that the
Newtonian potential q of the measure Σ αy λ; is continuous on R
m
. (The
potential q will be called the Cartan potential here.)
Suppose that UdRm is a bounded open set and denote by C(QU) the set of
all continuous functions on QU. Let £?(£/) stand for the set of all real-valued
functions defined on U. As usual, Hu denotes the operator of the Perron-
Wiener-Brelot solution of the generalized Dirichlet problem on U.
To state the Ninomiya uniqueness result, suppose that A: C(9t/)-»£F(i7)
is an operator having the following properties:
(i) A is linear and positive;
(ii) sup Af( U) ^  sup f(Q U) whenever /e C (9 C7)
(iii) A(p\
w
)=p\u whenever^ is a continuous Newtonian potential of a measure
supported by the complement CU of the set C7;
(iv) A(q\
w
) is harmonic (or subharmonic) on U for the Cartan potential q.
(Obviously, the operator A=HU enjoys (i)-(iv), thus no existence problem
arises.)
N. Ninomiya [18] was able to prove that such an operator A is uniquely
determined by conditions (i)-(iv). This remarkable result improves the state-
ment of Keldys uniqueness theorem for the generalized Dirichlet problem (see
Theorem 1 below; note that conditions (4) and (6) stated there are automatically
satisfied in classical potential theory; cf. also [6] and [16]).
The proof of uniqueness given by Ninomiya makes use of potentials of
742 I. NETUKA
finite energy and of Lebesgue measure. Hence it does not admit a straight-
forward modification to a more general situation as in the case of other partial
differential equations or of abstract potential theory.
In this paper, in the context of harmonic spaces, necessary and sufficient
conditions for uniqueness of the Ninomiya type operators are established and
relations to previous investigations of Keldys operators are shown. The validity
of the theorems given below requires that the relevant function spaces are rich
enough, as illustrated by a counterexample. Also, a density result for harmonic
functions is proved. This result does not seem to be known even in classical
potential theory where the corresponding statement can be formulated as fol-
lows: Every function continuous on U and harmonic on U is a uniform limit
of continuous potentials of signed measures supported by CU.
1. The Ninomiya and Keldys operators
In what follows, let X be a ^-harmonic space with countable base in the
sense of axiomatics of Constantinescu and Cornea [7]. (All notions not recalled
here are to be found in this monograph.) The corresponding harmonic sheaf
is denoted by M and the cone of continuous potentials on X is denoted by 3?.
For a potential p, the symbol C(p) stands for its superharmonic carrier.
Suppose that UdX is a nonempty, relatively compact open set. Denote
by U
r
 and t/t the set of regular and irregular points of C7, respectively. A Borel
set MddU is said to be negligible, if M has harmonic measure zero at every
point of C7, i.e. 8%U(M)=Q whenever #et7.
As above, ΞF(U) stands for the space of real-valued functions on U9 <S(U)
is the cone of (not necessarily continuous) superharmonic functions on U. Of
course, &(U)=<S(U)Π(-<S(U)).
Let us introduce the following notations:
H(U)= {/*eC(f7); A l£,€ΞeΛ(E/)}, H(QU) =
Q(U) = P(U)-P(U).
Clearly, Q(U), the space of differences of functions of P(U), is a subspace
ofH(U).
The following definition is a slight modification of that proposed by N.
Ninomiya. (It should be noticed that the above mentioned condition (ii) is a
consequence of conditions (i) and (iii) and so does not appear in our definition.)
An operator A: C(dU)-*ζF(U) is said to be a Ninomiya operator on C7, if
(1) A is linear and positive;
(2) A(p
m
)=p}u whenever />eP(C7);
(3) there is a strict potential #e£P such that
If (1) and (2) hold and (3) is replaced by
THE NINOMIYA OPERATORS AND THE DIRICHLET PROBLEM 743
(3*) there is a strict potential q^S such that A(q\
w
) e — <S(Ϊ7),
then A will be termed a weak Ninomiya operator.
Recall that an operator A: C(θ £7)->£?(£/) is said to be a Keldys operator,
provided (1) holds and the following conditions are fulfilled:
(2') A(h
m
)=h\u whenever Aeff(E7);
(3') A(C(QU))dJl(U).
Clearly, every Keldys operator is a Ninomiya operator and every Ninomiya
operator is a weak Ninomiya operator. Of course, the operator Hu of the Perron-
Wiener-Brelot solution of the generalized Dirichlet problem is a Keldys operator.
The main question studied in this paper is to investigate under what cir-
cumstances a Ninomiya operator is uniquely determined.
The uniqueness problem for Keldys operators has been extensively studied;
for references see [15], [16], [17]. Recall the following result; for a proof see
[12] or [21] cf. also [15].
Theorem 1. Suppose that
(4) the space H( U) linearly separates the points of 0 and contains a strictly posi-
tive function.
Then the following conditions are equivalent:
(5) there is exactly one Keldys operator on U;
(6) the set [7, is negligible.
The main result of this paper reads as follows:
Theorem 2. Suppose that
(7) the space Q( U} linearly separates the points of U and contains a strictly posi-
tive function.
Then the following conditions are equivalent:
(8) there is exactly one weak Ninomiya operator on U;
(9) there is exactly one Ninomiya operator on U;
(10) the set t/t is negligible.
The proof of Theorem 2 will be postponed until after having proved several
auxiliary results.
2. Simpliciality and density of a space of harmonic functions
First recall that, for x^U, the set M
x
 of representing measures (with respect
to Q(U)) is the set of all positive Radon measures μ on U such that μ(q)=q(x)
whenever qEίQ(U). The Choquet boundary of U with respect to Q(U) is defined
by
CAβαo £7 ={*€=#; M. = {6,}}.
Note that Ch0(U) 0 is a Gs set. If (7) holds, then by the Choquet theorem (cf.
744 I. NETUKA
e.g. [19], p. 43), for every x^U, there is a μ
x
^M
x
 carried by the Choquet
boundary. This means that μ
x
(U\CfiQ(U)U)=0.
Let us denote by F the essential base of OE7; see [4], p. 94.
Theorem 3. Suppose that condition (7) holds. Then ChQ(U) U=FΓί U and
the space Q(U)is simplicial (i.e. for every x€ΞU there is a unique μ
x
€ΞM
x
 carried
by ChQ(U)U).
Proof. Since Q(U)c:_H(U), we have BFX<=MX for every *<Ξ U by [4], p. 103.
If x€ΞChQ(u)U, then x<=U_ and £f = 6X. Thus x<ΞFΓi U by [4], p. 102. Con-
sequently, ChQ(u) t/cFn U.
Then X(p) is increasingly filtered by [10], p. 500 and by [4], p. 98,
RFP = sup X(p) .
Let x^FΠ U and v
x
^M
x
 be a measure carried by Chg^U. We are going
to prove that v
x
=£
x
, which, in turn, shows that x€ΞChQ(u)U.
To this end, fix p e 3* and put P
x
 = -C(p) \ a - Then P1 is increasingly filtered,
zndp=sup Pl on Ffl U because Rp=p on F. Since ^Λ is carried by
y we have
dv
x
= \ sup P1dvx = sup {I A d ^
-sup {A^ AePJ =£(*).
Consequently, v
x
(p)=p(x) whenever p^3? and v
x
=ε
x
 by [7], p. 45. It follows
Let Λ:G £7 and p.,, T
Λ
 be elements of M
x
 carried by ChQ(u)U and, as above, let
be a fixed potential. Then
^P*= J supP^p^^sup {J A r f p
Λ
;
= sup {A(*); AePJ = Rζ(x)
and, similarly,
Consequently, ρ
x
(p)=r
x
(p) for every ^e^ and p
x
=τ
x
. Hence Q(U) is
simplicial.
THE NINOMIYA OPERATORS AND THE DIRICHLET PROBLEM 745
Theorem 4. Suppose that (7) holds. Then Q(U) is uniformly dense in
H(U).
Proof. Clearly, Q(U)c:H(U). By Theorem 2, we have Choiu)U=ChQ(u)U
=FΠ U and it follows from [4], pp. 102, 103, that Chff(u)0=FΓ( 0 and H(U) is
simplicial. Thus Q(U) and H(U) have the same annihilating measures; cf. [9],
p. 20 or [15], p. 240. Consequently, Q(U)=H(U).
REMARK. One also could use the Stone-Weierstrass type theorem esta-
blished in [8]. For classical potential theory in Λ3, the result of Theorem 4
for regular sets is mentioned in [1].
3. Uniqueness of Ninomiya operators
In this section, we are going to prove Theorem 2. To this end, some
auxiliary results are needed.
Lemma 1. For x&U and /<Ξ C(dU), put Duf(x)=eζ(f). Let an operator
A : C(dU)-+3(U) satisfy (1) and (2). Then
whenever p& 3>.
Proof. For#et7,
, /€ΞC(8t7),
is a positive Radon measure on 3 U. Let p e ίP and q £_£"(/>).
we have at(q)= q(x). Thus, for any
= sup X(p) (*) = Λf (*) = eζ(p) = DP(pι
w
) (*) .
REMARK. For a similar result, cf. [13], p. 175.
Lemma 2. Let (7) hold and let an operator A: C(3 [/)->£?( U) satisfy (1)
and (2) . // z e ChQ(U) U, then
P(x) f°r x-*z,
whenever p^ S.
Proof. By (7), there is pQ^P(U) with inf pQ(U)^l. If ax has the same
meaning as in the proof of Lemma 1, then, for every x^U,
= ρQ(x)^sup p0(U)<oo .
746 I. NETUKA
So, for a given sreCΛρ^t/, it is sufficient to prove the following statement:
If #
n
e [/, x
n
 -> % and aXn -> α weakly, then a = βz .
To show this, ύxp^P(U). Then ccXn(p)=p(xn) and, consequently, α(s)=
s(x) for every se<2(t/). Hence a^M2 and α=£2, since
Lemma 3. Suppose (6) αrad (7). Let A : C(dU)->3ί(U) be an operator
satisfying (1) and (2). If q^S and A(q\
w
) e — <S( C7),
Proof. Let q satisfy the hypothesis. By (7), there is p^P(U) such that
and the function s=Hu(q\
w
)—A(q\
w
) is superharmonic and lower bounded on U.
Condition (6) and Theorem 3 imply that Du(q\
w
)=Hu(q\
w
} and U
r
=ChQ(u}U\
see [4], p. 106. By Lemma 2,
lim s(x) — 0
x-*z
for every z^U
r
. Since the set C7, is negligible by (6), the minimum principle
(cf. [2], p. 145) yields s^O on U. But ίfgO by Lemma 1. Consequently,
Proof of Theorem 2. Obviously, (8) implies (9). Since (4) follows from
(7), (9) implies (10) by Theorem 1. It remains to prove that (10) implies (8).
Let A be a weak Ninomiya operator. By definition, there is a strict po-
tential jeίP such that A(q\
w
)G—<S(U).
Fix *<E U. By (10) and [4], p. 106, ec
x
u
=BF
x
 , thus
a
x
(p):>εc
x
u(p},
by Lemma 1. We have
by Lemma 3. Since g is a strict potential, a
x
=8χU by [6], pp. 166, 43.
We conclude that A=HU and (8) is verified.
REMARK. The implications (5)=Φ>(6), and also (8)=Φ(9)=^(10) are true
without hypotheses (4) and (7); cf. [15], pp. 239, 244. The following theorem
shows, however, that the implication (6)=>(5), and, consequently, the implica-
tion (10)==>(8) do not hold in general.
THE NINOMIYA OPERATORS AND THE DIRICHLET PROBLEM 747
4. Non-uniqueness of Keldys operators and the negligible set of
irregular points
We shall construct a harmonic space Y and a relatively compact open set
Ud Y for which the Keldys theorem fails despite of the fact that U{ is negligible.
It is also shown that the uniqueness of a Keldys operator implies that the space
H(dU) is in a sense rich enough.
Theorem 5. There is a ^-harmonic Bauer space Y with countable base
and a nonempty, relatively compact open set t/C Y such that [7t is negligible and
there exist two distinct Keldys operators on U.
Proof. If Fc R is an interval (possibly degenerated), denote by -C(V) and
JC(V) the set of all affine and constant functions on V9 respectively.
Define F— [0, 1[ (endowed with the relative topology from R) and Λf={l/
(fl+1); n^N} . Every open set VdY not containing 0 is a union of a disjoint
system of intervals which are open in R.
Suppose that Fc Y is a nonempty open connected set. If Oe F, put M(V)
=JC(V). If OφF and MnF-0, put M(V)=£(V). If, finally, OφF and
Λf Π FΦ0, define a
v
=mf (MΠ V), V^\ inf V, a
v
[9 V2=] avy sup F[. Notice
that Fι=0, if and only if inf F=0. A function h^C(V) is said to belong to
M(V), if h\
v
^-C(Vύ and h\
v
^<K(V^. If F'c Y is an open set, then h is said
to belong to M(V')y provided h\v^M(V) for every component V of the set V.
Then M is a harmonic sheaf possessing the Doob convergence property and con-
taining constant functions. It is easily seen that ]a,b[ and [0,tf[ are regular sets,
if 0<α<6<l. Hence there is a strong base of regular sets in Y. Notice that
]0,tf[ is not regular.
One easily verifies that x\-*\— x is a superharmonic function on F, thus Y is
a Sβ-harmonic space by [7], p. 44.
Put E7=]0, — [. Then c#(E7)=JC(E7) and also H(U)=JC(U). We have
Huf=fί—\ on U for f<=C(QU). Obviously, — is a regular point, OeZ/. and
\ 2 / 2 /
 1 v
the set {0} is negligible, since Huf= 0, provided /(0) = l,/ί— J=0. (Note that
for any a^M and β^R, the function Λ;H->(I— a~1x)++β is superharmonic.)
For/eC(9ί7), define
Then 1^ is a Keldys operator on C7 and Af=Huf if and only if f(Q)=f(— V
REMARK. The above example is a modification of a construction given in
[7], p. 71; cf. also [20], [21]. A similar example for a Brelot space is shown in
748 I. NETUKA
[6] but U is not relatively compact there.
In the example described above, H(U) is the space of constant functions.
Thus H(U) does not separate points of 0.
The following theorem shows that, for nontrivial H(U), the space H(dU)
contains a strictly positive function if Keldys uniqueness theorem holds. The
proof uses an idea from [2].
Theorem 6. Suppose that H( U) Φ {0} . If there is a unique Keldys operator
on U, then H(dU) contains a strictly positive function.
Proof. Let j denote the function which is equal to 1 on 9 U. Suppose
that there is a unique Keldys operator on U and there is no strictly positive
function in H(dU). We deduce from these hypotheses that H(U)= {0} .
Notice that for fly f2^H(dU), the function /i—/2 is constant, if and only if
/!=/»•
Define
Then -ί/i(9f7) is a majorizing subspace of C(QU). If λe#i(9C7), then there is
exactly one g^H(U) and exactly one c^R such that h=g\
w
-\-c j. If
then £^0, since otherwise g\
w
 would be strictly positive.
The mapping
Ak:h>->g}u+c H
u(k j ) , k=l,2,
is obviously linear on ί/i(9f/) and
for every g^H(U). Suppose now that λ^O and z^U
r
. Then £^0 and
lim (gπ(x)+c H°(k j) (x)) = g
X-+Z
Notice that Huj, being majorized by a continuous potential, is bounded and
Uf is negligible since there is a unique Keldys operator on U\ cf. the remark
following the proof of Theorem 2. Thus Akh^0 by the minimum principle;
see [2], p. 145. One can extend Ak to a Keldys operator; cf. [15], p. 253. Thus
uniqueness gives, in particular, A1j=A2j or Huj=Hu(2 j). Consequently,
Huj=0 on U. It follows that H(U)= {0} .
Theorem 7. Let VdX be an open relatively compact set, SdX be a closed
semipolar set such that S Π V is not polar. Then there are at least two distinct
Keldys operators on U= V\S.
Proof. Denote S1=Vf}S. Then dU=QV\jSl9 because S is nowhere
THE NINOMIYA OPERATORS AND THE DIRICHLET PROBLEM 749
dense by [7], pp. 118, 153. It is easily seen that there is a non-polar compact
set KdS^ see [7], p. 144. For/eC(9f7), define
Then A19 A2 are clearly positive linear operators from C(dU) into <3l(U). Let
ht=H(U). Then h^H(V) by [14], p. 121, and Al(hlw)=Aί(hlw)=hlu. Thus
A19 A2 are Keldys operators. Let /e C(9 C7), /= 0 on 8 V and / ^  1 on K. Then
A2f=0 but A/is not identically zero on C7, since J£ is not polar; see [7], p. 147.
Thus AU A2 are distinct Keldys operators on U.
Corollary. If there is exactly one Keldys operator on every nonempty open
relatively compact subset of a harmonic space Ύ
 y then the axiom of polarity holds
in Y.
Proof. Suppose that the axiom of polarity does not hold in Y. Then there
is a nonempty open subset X of Y such that X is a Sβ-harmonic space in which
the axiom of polarity does not hold; see [7], pp. 225, 48. By [7], p. 219, there is
a compact non-polar totally thin subset of X. The rest follows from Theorem 7.
REMARKS. The result stated in the corollary is known; see [12], [15] for
another proof. Theorem 7 gives a method to construct sets on which the
Keldys uniqueness theorem does not hold. The special case for the harmonic
space associated to the heat equation was investigated in [12],
The main results of this paper were presented in a talk at the Conference
on Potential Theory, Oberwolfach, July 1984.
During the Conference, Professor W. Hansen communicated to me another
proof of Theorem 4 using continuous dilations studied in [5].
In September 1984, Professor L.I. Hedberg informed me that a similar
density result for classical potential theory was proved in his manuscript con-
cerning approximation by harmonic functions. Instead of Q(U)y however,
potentials of measures of finite energy are considered.
References
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Faculty of Mathematics and Physics
Charles University
Sokolovska 83
186 00 Praha 8
Czechoslovakia


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The Ninomiya operators and the generalized Dirichlet problem in potential theory

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