数理解析研究所講究録 1293 巻 2002 年 31-41

31Regularity of function-kernels
and the vanishing property of potentials
in the neighborhood of the point at infinity
Isao HIGUCHI ( 樋口 功)
Science Division, Aichi Institute of Technology

1. Introduction
Let $X$ be alocally compact but non-compact Hausdorff space satisfying the second
countability axiom. Afunction $G=G(x, y)$ on $X\cross X$ is called acontinuousfunction-kernel on $X$ if $G(x, y)$ is continuous in the extended sense and satisfies,$0<G(x, x)\leq+\infty$for$0\leq G(x, y)<+\infty$

for $\forall(x, y)\in X\cross X$ with$\forall x\in X$$x\neq y$.

is called acontinuous kerneltoIn general, apositive linear mapping fromon $X$ .
satisfies the continuity principle, we
When acontinuous function-kernel
can verify, under the additional condition that every non-empty open set in $X$ is of$X$
positive -inner capacity, that there exists apositive mesure 4everywhere dence in$C$$C_{K}$$G$$G$

satisfying

is locally 404-summable,(1)$G(x, y)$(2)

$V_{G}^{\xi}(f)(x)= \int G(x, y)f(y)d\xi(y)$is continuous on $X$ for$\forall f\in C_{K}$.

as acontinuous kernel on $X$ .
Then we can considerof continuous kernels on
be acontinuous kernel on $X$ . The familyLet$X$
is colled aresolvent family associated with $V$ , if it satisfies the following
equalities:$V_{G}^{\xi}$$(V_{p})_{p>0}$

$V$

(3)

$V_{p}-V_{q}=(q-p)V_{p}\cdot V_{q}=(q-p)V_{q}\cdot V_{p}$(4)$\lim_{parrow 0}V_{p}=V$.

,$\forall p$, $\forall q>0$ ,32.Hunt[9] verified that, when acontinuous kernel
satisfies the complete maximum
principle, we can associate aresolvent familywith, under the assumptionthat $V(C_{K})\subset C_{0}$ and $V(C_{K})$ is dense in.
The existence of aresolvent family may be developped to the theory of asemi-group.
So we can consider acontinuous kernel as the elementary solution of the infinitesimal
generator of the semi-group and hence we can enter analytically into the arguement of
the generalized Poisson and Dirichlet problems.
Subsequently, G.Lion[18] obtained the same result without the condition that $V(C_{K})$is dense in.
On the other hand, P.A.Meyer[19], J.C.Taylor[20] and F.Hirsch[8] constructed the resolvent family replacing the condition that $V(C_{K})\subset C_{0}$ with the weaker conditions on
the vanishing properties of potentials at infinity.
Now, let us recall here the arranged results in the theory of convolution kernels.
Aconvolution kernel $N$ on alocally compact abelian group $X$ is said to be aHunt
kernel when there exists avaguely continuous semi-groupof positive measures$X$on satisfying$V$$\mathrm{G}.\mathrm{A}$$(V_{p})_{p>0}$$V$$C_{0}$$C_{0}$$(\alpha_{t})_{\mathrm{t}>0}$$N= \int_{0}^{\infty}\alpha_{t}dt$( i.e.,$\int fdN=\int_{0}^{\infty}\{\int fd\alpha_{t}\}d\mathrm{t}$for$\forall f\in C_{K}$).

Concerning the characterization of aHunt kernel, the following rerults are well known.
Anon-periodic convolution kernel $N$ becomes a Hunt kernel if and only if $N$ satisfies
one of the following conditions:(A) $N$ is balayable, that is, there exists abalayaged mesure on every open set notnecessarily relatively compact (cf. G.Choquet-J.Deny[l]).

(B) There exists aresolvent family associated with N (cf. M.It\^o[10]).
(C) $N$ satisfies the domination principle and$\mathrm{N}$is regular (cf. M.It\^o[10]).

(D) $N$ satisfies the domination principle and has the dominated convergent
property ( , M.It\^o[10] and M.Kishi[17]).
$\mathrm{c}\mathrm{f}$Remark1. The author has investigated the relations $(A)\sim(D)$ with respect tothe continuous function-kernel and verified already the equivalence of (C) and (D) andobtained the relations $(C)arrow(A)$ and $(C)arrow(B)$ (cf. I.Higuchi[4], [5], [6]). Bu the
inverse relations $(A)arrow(C)$ and $(B)arrow(C)$ fail to hold in general (cf. I.Higuchi[6]and M.It\^o[12] . These facts suggest that the treatments of the function-kernels are more
complicated than that of of the convolution kernels.
The regularity of function kernel is concerned deeply with the vanishing property ofpotentials in the neighborhood of the point at infinity.
The purpose of this paper is to characterize the regularity of anon-symmetriccontinuous function-kernel $G=G(x, y)$ satisfying the complete maximum principle andto prove that at least one of and converges to 0quasi-everywhere at infinity.
$\mathrm{s}$$\mathrm{t}$

$)$

$G$

$\check{G}$33
2. preliminaries
The$\mathrm{G}$-potential $G\mu(x)$ of aRadon mesure$\mu$on X is defined by$G \mu(x)=\int G(x, y)d\mu(x)$ .

Put
$M=$

{

$\mu$

: positive Radon mesure on $X$ },$E=E(G)= \{\mu\in M ; \int G\mu(x)d\mu(x)<+\infty\}$ ,is finite continuous on $X$ },$F=F(G)=$ { $\mu\in M$ :$G\mu(x)$$D=D(G)=$ { $\mu\in M$ :$G\mu(x)<+\infty$

G-n.e. on $X$ }.

And we write their sub-families consisting of the measures with compact support by,respectively.
,. The notations
the totality of -potentials of the measures in
We denote byof the families of various class of potensiala are also denoted similarly.
$M_{0}$$E_{0}$$F_{0}$$P_{M_{0}}(G)$$M_{0}$$\mathrm{G}$.
is said to be -negligible if
ABorel measurable set$X$
We say that aproperty holds -nearly everywhere on asubset of and write simply
that holds G-n.e. on , when it holds on except for a -negligible set.
$B$$\mathrm{P}$$\mathrm{G}$$A$$\mathrm{G}$

$A$

$A$

$\mathrm{P}$$\mu(B)=0\mathrm{f}\mathrm{o}\mathrm{r}.\forall\mu\in E_{0}(G)$$\mathrm{G}$Alower semi-continuous function on $X$ is said to be -Superharmonic when$0\leq u(x)<+\infty$ G-n.e. on $X$ and for any $\mu\in E_{0}(G)$ , the inequality $G\mu(x)\leq u(x)$ G-n.e.
implies the same inequality on the whole space $X$ .
on
We denote by $S(G)$ the totality of -superharmonic functions on $X$ .
$\mathrm{G}$$u$$S\mu$$\mathrm{G}$supported by
For afunction $u\in S(G)$ and aclosed set $F\subset X$ , apositive measuresatisfying the following conditions is called abalayaged mesure of on $F$ , if it$\mu’$$F$$u$exists,Gfi(x) $=u(x)$ G-n.e. on$G\mu’(x)\leq u(x)$$F$,

on $X$ .

We denote by $S_{bal}(F, G)$ the totality of -superharmonic functions for which the balayaged mesure on $F$ exists and write simply $S_{bal}(G)$ instead of $S_{bal}(X, G)$ .
$\mathrm{G}$Potential theoretic principles are stated as follows:(i) We say that G satisfies the domination principle and write simply G
when $P_{M_{0}}(G)\subset S(G)$ .

$\prec G$(ii) We say that G satisfies the complete maximum principle and write simply.
G $\prec G+1$ when we have $P_{M_{0}}(G)\cup\{c\}\subset S(G)$ for$\forall c\geq 0$34(iii) We say that$G$

satisfies the balayage principle if we have

$P_{M_{0}}(G)\subset\cap k_{j}\mathrm{c}ompact\subset X$

Sbal $(K, G)$ .

(iv) We say that $G$ is balayable when we have

$P_{M_{0}}(G)\subset S_{bal}(G)$.

(v) We say that satisfies the continuity principle if, for any, the finite
continuity of the restriction of $G\mu(x)$ toimplies the finite continuity of $G\mu(x)$ on
the whole space $X$ .
$G$$\mu\in M\mathrm{o}$$S\mu$For anon-negative Borel function and aclosed set $F$ , the -reduced functionof u on F and the -reduced function of u on F at infinity , aredefined respectively by$\mathrm{G}$$u$$\mathrm{G}$$R_{G}^{F}(u)(x)= \inf$$\delta${ $v(x)$ ; $v\in S(G)$ , $v(x)\geq u(x)G-n.e$ . on

$R_{G}^{F,\delta}(u)(x)= \inf_{\omega\in\Omega_{0}}R_{G}^{F\cap C\omega}(u)(x)$

where$\Omega_{0}$

},

$F$

,

denotes the totality of all relatively compact open sets in $X$ .

And we write simply

instead of$R_{G}^{\delta}(u)(x)$$R_{G}^{X,\delta}(u)(x)$.

Put, for aclosed set $F$ ,$S_{0}(F, G)=$

{ $u\in S(G)$ ;$R_{G}^{F,\delta}(u)(x)=0$

G-n.e. on $X$ }.

And write simply $S_{0}(G)$ instead of $S_{0}(X, G)$ .

Remark 2. When$G$

satisfies the domination principle, the following (1) and (2)hold:(1) We have

$R_{G}^{F}(u)\in S(G)$(2) We haveregularization of

for any closed set $F$ and for any$\hat{R}_{G}^{\delta}(u)\in S(G)$$R_{G}^{\delta}(u)(x)$for any $u\in S(G)$ , where

and write simply
The kernel$G${ $u\in S(G)$ ;$S_{0}(G)$$\hat{R}_{G}^{\delta}(u)(x)$.

denotes the lower.

Further we put, for aclosed set$S_{0}(F, G)=$$u\in \mathrm{S}(\mathrm{G})$$F$,$R_{G}^{F,\delta}(u)(x)=0$

G-n.e. on $X$ },instead of $S_{0}(X, G)$ .

is said to be regular when we have $P_{M_{0}}(G)\subset S_{0}(G)$ .

Remark 3. Suppose thatsatisfies the complete maximum principle and that,, $G\mu(x)$ converges uniformly to 0at infinity , that is, forforand forexists an
satisfying $G\mu(x)<\epsilon$ on. Then becomes regular.
Therefore, regularity means akind of vanishing property of potentials at infinity .
$\forall\mu\in M_{0}$$\forall\mu\in M_{0},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$G$$\forall\epsilon>0$$\delta$$\omega$$\in\Omega_{0}$$C\omega$$G$$\delta$

35
Remark 4 (cf. I.Higuchi[6] and M.It\^o[13]). Whwn G satisfies the domination principle, the following statements are equivalent:
(1)

$G$

is regular.

(2) $P_{E_{0}}(G)\subset S_{0}(G)$ .
(3) $P_{F_{0}}(G)\subset S_{0}(G)$ .

(4) $P_{D_{0}}(G)\subset S_{0}(G)$ .
(5)$\check{G}$is regular.

Therefore, it suffices to obtain the weakest condition (3) when we show the regulariry$G$
of $G$ and we may use the strongest condition (4) when we apPly the regularity of .
And the duality of regularity follows from the equivalence of (1) and (5).

Remark 5(cf. Theorem 1 and Corollary 1).
(1) When$G$satisfies the domination principle, we have $S_{0}(G)\subset S_{bal}(G)$ .

(2) If $G$ satisfies the domination principle and is regular, then$G$is balayable.

The inverse of (2) in Remark 5does not necessarily hold in general.
In fact, there exists an example of continuous function-kernel such that is balayabble
but not regular (cf. I.Higuchi[6]).

Remark 6.

$G$

$G$

Three notions of the thinness of aclosed set $F$ are defined as follows:(i) $F$ is said to be $G$ -thin at infinity(ii) $F$ is said to be $G-1$ -thin at

(iii)

$F$

is said to be

Remark 7.
the implications$G-\mathrm{c}\mathrm{a}\mathrm{p}$$\delta$, if-thin at$\delta$$\delta$, if $P_{M_{0}}(G)\subset S_{0}(F, G)$ holds.

$1\in S_{0}(F, G)$, ifholds.

$\inf_{\omega\in\Omega_{0}}cap_{G}^{i}(F\cap C\omega)$$=0$

holds.

When both G and adjoint satisfy the complete maximum principle,
can be shown (cf. Theorem 3and Theorem 4).
(iii)$\check{G}$$\Rightarrow(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$In the rest of this paper, we always assume that every non-empty open set

in X is of positive$\mathrm{G}$-inner capacity

36
3. Thinness and balayability
By using the equivalence of the relative domination principle and the relative balayage
principle obtained by M.Kishi[15], we can prove the following
Theorem 1.

If

$G$

satisfies the domination principle, we have
$S_{0}(F, G)\subset S_{bal}(F, G)$.

Remark 8. The inverse inclusion relation of Theorem 1does not necessarily holdin general. But if we suppose thatsatisfies the domination principle and thatisregular, then we have $S_{0}(G)=S_{bal}(G)$ Further, the following (1) $\sim(4)$ are equivalent:
$G$

(1)

$u\in S_{0}(G)$(2)$u\in S_{bal}(G)$(3)$\hat{R}_{G}^{\delta}(u)\in S_{0}(G)$(4)$\hat{R}_{G}^{\delta}(u)\in S_{bal}(G)$

$G$

.

.
.
.

Corollary 1 (cf. I.Higuchi[6]). Suppose that $G\prec G$ . Then$G$

is balayable if

$G$

is

regular.

Remark 9. The inverse of Corollary 1does not correct in general.
In fact, We denote by $N=N(x, y)$ the Newton kernel on $R^{n}(n\geq 3)$ and by 4apositive measure such that $N\xi(x)$ is finite continuous on $X$ and that $\int d\xi<+\infty$ . The
the continuous function-kernel defined by$G(x, y)=N(x, y)+N\xi(x)$satisfies the domination principle. And we can prove is balayable but not regular (cf.
I.Higuchi[6] . Therefore, the regularity is astronger property than the balayability.
$G$

$)$

Remark 10. Similarly we can prove the following propositions concerning thethinness of aclosed set at infinity$\delta$:(1) Letsatisfy the domination principle and $F$ be -thin at
potential $G\mu(x)$ of any $\mu\in M_{0}(G)$ , there exists abalayaged mesure of
set contained in $F$ .
$G$

$G$

$\delta$$G\mu$

Then, for aGon every closed(2) Let satisfy the complete maximum principle and $F$ be G-l-thin atexists the equilibrium measure of every closed set contained in $F$ .
$G$(3) Letsatisfy the maximum principle and $F$ be G-cap-thin at$cap_{G}^{i}(F)<+\infty$ .
$G$$\delta$$\delta$. Then there. Then we have37
4. Thinness and strong balayability
be acontinuous function kernel on $X$ and $F$ be aclosed set in $X$ . We say thatLet
is strongly balayable on $F$ , if, for every -superharmonic function dominatedby some -potential in $P_{M0}(G)$ , there exists abalayaged measure on every closed set
contained in $F$ .
$G$

$G$

$G$

$G$

Acontinuous function-kernel $G=G(x, y)$ is said to be non-degenerate when for
any , $y_{2}\in X$ , $G(x, y_{1})$ and $G(x, y_{2})$ are not proportional each other.
$y_{1}$

The following theorem is acharacterization of -thinness of aclosed set at infinity
and is an answer to the inverse problem of (1) in Remark 10.
$G$$\delta$

Theorem 2. Suppose that satisfies the domination principle and that is nondegenerate. Then, for any compact set , the following statements are equivalent:(1) F is-thin at infinity .
$G$

$G$

$F$

$G$

$\delta$(2) The following (a) and (b) hold:strongly balayable on $F$ .
(b)is balayable on $F$ .
and are strongly balayable on $F$ .
(3) Both
(a)

$G$

is

$\check{G}$$G$$\check{G}$The next result is acharacterization of G-l-thinness of $F$ at and is an answer to theinverse problem of (2) in Remark 10. This theorem asserts that the G-l-thinness at is
astronger property than the -thinness at .
$\delta$$\delta$$G$$\delta$

Theorem 3. Suppose thatsatisfy the complete maximum principle.
andThen, for any closed set $F$ , the following statements are equivalent each other:(1) F is$G$

G-l-thin at infinity$\check{G}$$\delta$.

(2) Next (a) and (b) hold:-thin at .
(b) The equilibrium measure(a) F is$G$

And further, if boththe following (3).

$\delta$$G$

and$\check{G}$are$\gamma F$

of F exists.
, then (1) or (2) is equivalent to$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$(3) Next (c) and (d) hold:(c) For any bounded -superharmonic function
measure of on every closed set contained in $F$ .
$G$

$u$

, there exists aG-balayaged$u$(d) For any bounded -superharmonic function
measure of on every closed set contained in $F$ .
$\check{G}$

$)$

$u$

$u$

, there exists a$G\vee$balayaged

38
The following theorem is acharacterization of G-cap-thinness at infinity of aclosed
set and is an answer to the inverse problem of (3) in Remark 10. And, by this result, we
can assert that the G-cap-thinness is astronger property than the G-l-thinness.
$\delta$

Theorem 4. Suppose that bothandThen the Slowing (1) and (2) are equivalent:(1) F is$G$

satisfy the complete maximum principle.

$\check{G}$G-cap-thin at infinity .
$\delta$(2) The following 3statement hold:(a)$cap_{G}^{i}(F)<+\infty$.

(b) F isG-l-thin at infinity C5.

(c) F isG-l-thin at infinity

And further, if we suppose that bothor (2) is equivalent the following (3):

$G$

and

$\delta$.

$\check{G}$are, the above (1)$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$(3) The above (a) and both of the following (c) and (d) hold:(c) For any bounded -superharmonic function
measure of on every closed set contained in $F$ .
$G$

$u$

, there exists aG-balayaged$u$, there exists a$u$(d) For any bounded -superharmonic function
measure of on every closed set contained in $F$ .
$\check{G}$$G\vee$-balayaged$u$

5. Thinness and the vanishing property of potentials
Remark 11 (cf. I.Higuchi[7]).

In the case thatis symmetric and thatsatisfies the complete maximum principle, we have already verified that the following (1)and (2) are equivalent:
$G$

$G$

(1) G is regular .

(2) For $\forall c>0$ and for $\forall\mu\in F_{0}(G)$ , we have$\inf_{\omega\in 0}cap_{G}^{i}\{x\in C\omega ; G\mu(x)\geq c\}=0$.

Therefore, the regularity is an extension of the property that0in the neighborhood of the point at infinity.

$G$-potential converges to, the circumstance is more complicated. In
When
is non-symmetric, that is,
the rest of this section, We consider the relations between the regularity and the vanishingproperty of potentials at infinity , camparing the three notions of thinness at , when
the kernel is non-symmetric$G\neq\check{G}$$G$$\delta$$G$$\delta$

39
satisfy the complete maximum principle.
Then, for any closed set F, the following three statements are equivalent:

Theorem 5. Suppose that both G and(1) F is(2) For$G$-thin at infinity$\forall c>0$

, and$\forall\mu\in M_{0}$$\delta$$\check{G}$.

, the set

$F\cap\{x\in X ; G\mu(x)\geq c\}$

is

$G-1$

-thin at$\delta$.

(3) For $\forall c>0$ ,$\forall d>0$

and$\forall\mu$,$\forall\nu\in M_{0}$, the set

$F\cap\{x\in X ; G\mu(x)\geq c\}\cap\{x\in X ; \check{G}\nu(x)\geq d\}$is

G-cap-thin at$\delta$.

Putting $F=X$ , we have following main theorem.

and satisfy the complete maximum principle.
Then the following (1), (2) and (3) are equivqalent one anathor:

Theorem 6. Suppose that both(1) G is$G$$\check{G}$regular .

(2) For $\forall c>0$ , and for$\forall\mu\in M_{0}$, the set$\{x\in X ; G\mu(x)\geq c\}$is

G-l-thin at(3) For.

$\delta$$\forall c>0$,$\forall d>0$

and$\forall\mu$,$\forall\nu\in M_{0}$, the set$\{x\in X ; G\mu(x)\geq c\}\cap\{x\in X ; \check{G}\nu(x)\geq d\}$is

G-cap-thin at .
$\delta$areandAnd further, if we suppose that bothabove (1) $\sim(3)$ are equivalent to the following (4):$G$(4) Both (a) and (b) hold:(a) G is(b)$\check{G}$strongly balayable.

is strongly balayable.

$\check{G}$$\mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$, then the40
Remark 12.

Roughly speaking, Theorem 6asserts that, when G andsatisfy
the complete maximum principle, G is regular if at most one of the -potential and-potential converges to 0at infinity .
$\check{G}$$G$$\check{G}$$\delta$

Example.

let $Y=Y(x, y)$ be afunction-kernel on$Y(x, y)=\{$ $e^{-(x-y)}1,$,$R^{2}$$x\geq yx<y,$defined by.

This kernel is called the Yukawa kernel. It is not so easyt to prove that yhe
Yukawa kernel is regular. We can prove bothandsatisfy the complete maximum
principle and that the -potentials converge to 0at infinity but the -potentials do notconverge to 0at .
By our Theorem 6, we can assert that both and are regular.
$Y$$\check{Y}$$Y$$\delta$$\check{Y}$$\delta$$Y$$\check{Y}$参考文献[1] G.Choquet-J.Deny, Noyaux de convolution et balayage sur tout ouvert, Lecture Notes
in Math.,Springer-Verlag, No. 404(1973), 60-112.
[2] J.Deny, Noyaux de convolution de Hunt et noyaux associees \‘a une famille fondamontale, Ann.Inst Fourier, 12(1962), 643-667.
[3] I.Higuchi, Duality of domination principle for non-symmetric lower semi-continuousfunction-kernels, Hiroshima Math.J., $5(1975)$ , 551-559.

[4] –, R\’egularit\’e et propri\’et\’e de convergence domin\’ee des potentiels d’un noyaufonction non-sym\’etrique, S\’eminaire de Th\’eorie du Potentiel, Paris, No. 6, Lecture(1982), 158-202, Springer-Verlag.
Notes in$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}.,906$[5] –, Existence de r\’esolvantes associ\’ees \‘a un noyau v\’erifiant le principe de dominathat, ibid. No. $6(1982)$ , 203-224.
[6] –, La balayabilite au sens fort des noyau-fonctions continues du potentiel, Nagoya
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[7] –, On the thinness of aclosed set in the neighborhood of the point at infinity, 京
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[8] F.Hirsch, Conditions n\’ecesaires et suffisantes d’existence de r\’esolvantes, Z.
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[11] –, Positive eigen elements for an infinitesimal generator of adiffusion semi-groupand their integral representations, Potential Theory Copenhagen 1979, Lecture Notes
in Math., 787(1979), 164-184.
[12] –, On weakly regular Hunt diffusion kernels, Hokkaido Math. J., 23(1981),Sp.,

303-335.
[13] –, Remarks on the regulariry for continuous function-kernels satisfying the domination principle, Nagoya Univ. preprint series No. $4(1987)$ .
[14] M.Kishi, Inferior limit of asequence of potentials, Proc. Japan Acad., 33(1957),$\mathrm{S}\mathrm{p}$,303-335.
[15] –, Unicity principles in the potential theory, Osaka Math. J., 13(1961), 41-74.
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[17] –, Some remarks on the existence of aresolvent, Ann. Inst. Fourier, 25, fasc,
3-4(1975), 345-352.
[18] G.Lion, Families d’operateur et frontieres en theorie du potentiel, Ann. Inst. Fourier,16, Fasc. $2(1966)$ , 389-453.

[19].Meyer, Caract\’erisation des noyaux potentiels des semi-groupes discret, Ann. Inst.
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$\mathrm{P}.\mathrm{A}$[20] J.C.Taylor, On the existence of sub-markovian resolvent, Invent. Math. 17(1972),

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