リーマン面で定義された方程式△u=quの解の積分表示について

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(1)Title. リーマン面で定義された方程式△u=quの解の積分表示について. Author(s). 佐藤, 武義. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 24(1) : 7-34. Issue Date. 1973-10. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5967. Rights. Hokkaido University of Education.

(2) • 24 ^ ^ i ^ ^^Mat'^îiB^ (^ II ^ A) P^^B 48 ^ 10 ^. Integral Representation of Solutions of the Equation Au=qu on a Riemann Surface. Takeyoshi SATO Department of Mathematics, Iwamizawa Branch,. Hokkaido University of Education. }) --rVWJE^ht^^ û=qu OÔ^^^^^^l^-C. § 0. Introduction Let R be an open Riemann surface. We consider a partial differential equation. (0,1) An=qu, ^=9219x2+92l9y2, where q{z) is a non-negatlve and locally Holder continuous function on J? of local parameters z=x+iy such that the expression qdxdy is invariant under the change of local parameters z. We always assume that q ^ 0, and the condition \ q{z}dxdy < + oo is assumed in the essential <R. parts of this paper. We shall define a kernel function k{z, w) which has a logarithmic singularity at w and satisfies the condition. (0,2) \ k{z, iv)g{z)dxdy=2Tt -R. for any tv in R. This function is used as a kernel of potential in this paper. And, we shal construct an ideal boundary associated with the equation (0,1) and extend the function k{z, tv) of w to this ideal boudary so that the relation (0,2) remains on a subset of this boundary. The purpose of the present paper is to represent some positive solution of (0,1), which is called a (7-full-solution later and satisfies the condition \ u{z)g{z}dxdy<+ oo, as a potential of the <R. above kernel.. The contents of this paper are quite parallel to those of Kuromachi's theory (C6J) on the ideal boundary of Riemann surfaces. We borrowed ideas, methods and usual tricks from many papers on that theory. To many of them we are indebted [3], [10], and Cl5]. By a q-solution or a solution on a subdomain G of R we mean a twice continuously differentiable function which satisfies the relation (0,1) on G. We call a relatively compact open set or a compact set in R regular if its boundary consists of a finite number of analytic arcs. An exhaustion of R will mean an increasing sequence {Rn} of relatively compact and regular domains such that RnCZRn+i for every n, [j^R,i=R, and no component of R—Rn is compact. And, by a regular closed set in R we shall mean a closed subset of R whose boundary consists of a countable number of analytic curves clustering nowhere in R.. (7).

(3) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. 1. Properties of solutions of the equation (0,1) In this section we state some fundamental definitions and facts for solutions of (0,1) which we shall use later. For proofs of these facts we refer to [II], [12] and [133. The Dirichlet problem on a relatively compact and regular domain V is solvable, that is, for any continuous function /on 9V, there exists a unique continuous function SJr(z) on V such. that lim S}{z) =f{b) on 9V and Sj is a q-solution on V. Furthermere, if /> 0 then S^- > 0 Z—.6. on V. Then, for any disc V there exists a measure w J on 9V with respect to z^V satisfying. 5;(z)=p^< for any continuous function / on 9V, which is called the q-harmonic measure. A function s on an open set G is said to be a q-supersolution on G if s satisfies the following conditions : s{z) > — oo at each z on G and s ^ + °° on any component of G, s is a lower semi-contmuous function on G, and for any disc V such that V d G,. s(z)>^< for all z^=V. For example, any positive constant function is a ^-supersolution on R. Lemma 1,1. a} If Si, $2 are q-supersohttions on G and ai, a^ are positive constants then »iSi+a2S2 and min{s^, 83} are q-snpersohitions on G. b) If {si} is an upper directed family of q-supersohitions {resp. q-solntions} on G, then sup Si is either + co or a q-sttpersohi.tion i. {resp. q-solution) on G.. Lemma 1,2. a} If s is a non-negative q-stipersohdion on G and s{zy) =0 at some point Zy in G, then we have s == 0 on G. b~) If s is a q-super solution on G and for any positive number e there exists a compact subset K of G stick that s{z)>—s on G—K, then me have s > 0 on G. The proofs of the following lemmas are founded in Cll.]. Lemma 1,3. If s is a q-sziper solution on a neighborhood of ZQ, then 1 c'i". lim^_\ s{zo+reie)rd0=s{zo). r_>.0 ^11-ÎQ. Lemma 1,4. If s is a tiuice continuously differentiable fimction on a domain G, then it is a q-supersolution in G if and only if As— qs < 0 on G. The next lemma states an inequality of the Harnack type. . Lemma 1,5. For a compact subset K of G there exists a positive constant c such that. c-l^(^X z<(z'X c%(z) for any non-negative q-sohition zt on G and any two points z and z' in K. Lemma 1,6. A monotone increasing seqzience of q-sohitions on G zuhich is bounded above at a point of G converges to a q-sohition uniformly on any compact set of G. And a bounded sequence of q-sohitions on G contains snbseqnences converging to a q-sohition on G uniformly on any compact subset of G. This fact is called the theorem of Harnack type. Let G{z, zu) be the Green function of the equation (0,1) with pole zu, which always exists unless q == 0 ([II], C13])- If <" is a positive measure on R we see that the function. G{z, <u)= \G{z, zu)dfJi.û}. (8).

(4) ; 24 ^ ^ i -^ »mr^^ (^ ii ^ A) TO 48 4^ 10 H is either = + °o or a q-supersolution on R. If it is not = + °° we call it a potential with the kernel G(z, m}. We can apply Perron-Brelot's method for the Dirichlet problem for the equation (0,1) ([11]). Let G be a domain in R. For the extended real valued function / on 9G U {A}, we define. Sf=={s : a q-stipersohition on G bounded below on G, Urn 5(z)>/(6) z-i-6. for all be8G[J{A} or s is equal to + 0° everyiuhere on G}, where A is Alexandorff's ideal boundary point. And S^={—s ; s^~S^}. We denote S^(z)=inf {s(z) : s^Sf} and Sf{z)=Stip{s{z} : s^Sj}. Then, 5/(resp. ,§/) is either +00 or — oo or a ^-solution on G. For any finite continuous function / on 9G[J {A} it holds that Sf==Sf on G. We denote this common function by 5/. For a point b on 3G lue say that b is a regular point. for G with respect to the equation (0,1), if Jim S/(z)=/(6) for any finite continuous function / Z->.6. on 9G[j{A}. On account of the solvability of the Dirichlet problem on any relatively compact and regular domain, we see that, if 9G is an analytic curve in a neighborhood of b^9G, then b is a regular point with respect to the equation (0,1) by the comparison theorem (ClJ).. § 2. Energy principle For an absolutely continuous function / in the sense of Tonelli defined on a subset G of R, we denote the Dirichlet integral and the energy integral of / taken over G by. ^-^^>dy. and. E^f)=D^f)+ \q{z)f^z}dxdy, Iff respectively. The space E{R) associated with a Riemann surface R and the equation (0,1) is the totality of real functions / on R satisfying the following three conditions : / is quasi-continuous (C3.])> / is an absolutely continuous function in the sense of Tonelli, and the energy integral. En{f) of / over R is finite. For g, gi, g^ in .E{R}, let. (2,1) (A, &).=^(f9yf+^ f)^+ {^(^(^ 9y and. (2.2) \\g\\H={g,g}^ This space E{R) was considered in [11]. The following lemmas with no proof are contained. in [11]. Lemma 2,1. If q ^ 0, Âe% (2,2) is a norm in E{R} and E{R} is a Hilbert space tvith respect to the inner product (2,1) provided that ive identify two fimctions mhich are equal q. p. on R, zuhere " q. p." means "except for a polar set" ([3]). The space SE{R) is the totality of real functions tl on R satisfying the following two conditions : u is contained in E^R), and u is a ^-solution on 7?.. Lemma 2,2. SE{R) is complete with respect to the norm (2,2). Lemma 2,3. If Un^SE(R) for every n and \\tin\\R—>-0 as n—>-+co, then ii,n{z)—fQ locally uniformly on R as n—^+co.. Let C^{R) be the set of all infinitely differentiable functions with compact support on R, and Eo{R) be the closure of C^{R) in E{R) with respect to the norm (2,2).. (9).

(5) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. Lemma 2,4. SE{R} is orthogonal to the space Eo{R), i.e., (^,,0)^=0 for any zi^SE{R} and foÊo{R). For a non-polar compact set K in R, let EK'{R) be the totality of functions / satisfying the following conditions :/ is a function in E{R), and /==0 q. p. on K.. Lemma 2,5. If fn, f are contained in E(R) and \\fn—f\\R->0 as %-^+°°, then there exists a subsequence {fn'} such that fn'->f Q.. p. on R as n'—r +00. Proof. Since \ {fn—f)2qdxdy—>0 as n—>+oo, there exists a subsequence {/»'} such that '.R. {fn'} converges to / almost everywhere on the set {z^R : q{z)>0}. And, since Dn{fn'—f)—>0 as n'-^+oo, by choosing a subsequence again if necessary, we may assume that {/»'} converges to / q. p. on R ([3]). Q. E. D.. Lemma 2,6. The space EK{R} is a closed subspace of E{R}. Proof. This is evident by the previous lemma. Q. E. D.. Theorem 2,7. (Energy principle) Given f in E{R} and a non-polar compact set K there exists a unique fK in E{R) szich that (2,3) fK is a q-solution on R—K;. (2,4) \\fK\\K=min{\\g\\H : gÊ{R) and f-gGEK{R}}. Proof. Let fK be the projection of / on to the orthogonal complement of Esi{R). Then, this theorem can be proved by the same method as that of Satz 15, 1 in ^3_|, applying Lemma 2,4. Q. E. D.. Theorem 2,8. fK has the following properties:. (2,5) // /> 0 on K, then fs>Q on R; (2,6) the mapping f -> fK on E{R) into itself is linear ; (2,7) If KdK,, then {fK)Ki=fK on R-K,; (2,8) If D is a component of R-K, then fK=fw on D. Proof. These properties are evident by [3], also. Q. E. D. In the following, for convenience sake the function which is identically equal to a real number r will be denoted by r also.. Theorem 2,9. If q satisfies the condition \ qdxdy<+^, then lK{z) < 1 on R. IjR. Proof. Since under this condition the function 1 belongs to E{JR), we can define 1 . Let. G be the set {z^R-K : !K{z)>l}. If we write g=min{lK, 1}, then gÊ{R} and l-gÊK[R}. We assume that the set G is not empty. Then it follows that. \\g\\^\\iK^-a+\\ira. < U 1K\\^+\\1K\\2^\\ IK\\H.. This inequality contradicts (2,4) of Theorem 2,7. Therefore, it follows that l-r< 1 on R. Q. E. D. Corollary 2,10. Under the same assumption as Theorem 2,9 zve have. \fK{z)\<max\f\ on R-K 9-S-. Proof. By Theorem 2,8 we can see this corollary easily. Q. E. D. If the relative boundary 9G of a subset G of R consists of a finite number of analytic curves, we generally denote by ds the line element of 9G and by 919n the outer normal derivative on 9G.. Theorem 2,11. Under the condition \ qdxdy<+oo, if a relatively compact and regular <H. subdomain G of R contains a non-polar compact set K, then we have. {10).

(6) ^ 24 ^ it i -^ waw^CT (^ 11 % A) TO 48 ^ 10 ^ (2,7) f <^s^s+ f _fKqdxdy=0 -y>'/ JSG^""" ' Jfl-G' for every f in E{R}. Proof. We take a regular open set U so that Kd U C G. Let g be an infinitely differentiable function on R such that g=0 on U and 1 on R—G. Then 5' is contained in the space Er{R}. By Green's formula we have. {g,nn=Da^{g,fK)+\^ ^qfKgdxdy <B-TT. (^s+C _fKqdxdy.. lac 9n""J ' ^B-O'. Since {g,fs')ff=Q by Theorem 2,7, we established (2,7). Q. E. D.. Lemma 2,12. // /> 0 mid D is a component of R-K, then /8D > S? on D. Proof. Since a positive superharmonic function is a q-supersolution, we can prove this by the same way as that of Satz 15,1 in [,3], Q. E. D. Theorem 2,13. We assume that \ gdxdy<+oo. Let K be a non-polar compact set of I.R. R. If b £ 9K is a regular point of R—K with respect to the Dirichlet problem for the equation (0,1) and if fÊ{R) is continuous on QK, then we have Urn fK{z) =/(&). a->-6. Proof. Let G be a component of R—K such that 9G contains the point b. And, let M= sup \f\. Then, by Theorem 2,8, Theorem 2,9, and Lemma 2,12, we have 9(?. M+f°>{M+f)w>S^f and. 5^<M-/°. Hence, from the regularity of b, it follows that. f{b} = Urn S<M+f{z} - M < Iwi f^{z) z<-6. z-*6. and similarly. f{b)>Um fw{z) z->b. And so, Urn fK{z)=f{b}. Q. E. D. z->.6. 3. Definition of ^-full-supersolutions In the following discussion, we assume that the condition. (3,1) \ g{z)dxdy<+w <R. is satisfied. For a compact set K, let CC°{9K) be the space of restrictions to 9K of every function in C'^{R}. Then, for any function / in C'X°{9K), there exists a function g in C^{R} such that the restriction of g onto 9K is equal to /. Then we can define fr=g on R—K for f^CK'{9K). By Corollary 2,10, fK is well-defined when the condition (3,1) is satisfied. For each z fixed in R—K, the mapping f—> fK{z) is a non-negative linear functional on CCO{9fC) by Theorem 2,8. Hence, for z in R—K there exists a positive measure //f on 9-K' such that \fdp^ =fK{z) for all / in C'°°((?J^). And, for an arbitrary p.K- measurable positive function /, we define. fK{z) by ^fdp.K on 7?-^.. {11).

(7) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. Lemma 3,1. (a) p^-measw 'ability does not depend on the choice of z in R—K and fK is a q-solution on any component of R—K zvhen it is finite at some point of that component. (6) \d/j,f^l for z^R—K. (c) \fK{z) \^ max \f\ for any jjlj '-measurable function f. •QK. Proof, (b) and (c) are evident. For (a) we take an increasing sequence {gi} of upper semi-continuous functions on 9K such that. lim\gidp^=\fd{^, i->00. and a decreasing sequence {hi} of lower semi-continuous functions on 9K such that. lim\hidp^== \fdp^. 1^.00. »1. -. -. ^. -. Since \gid/j,f nd \hid/J,^ are ^-solutions, we can complete this proof by Lemma 1,2 and 1,6. (C15]). Q.E.D. Lemma 3,2. Let K be a non-polar compact set of R and D be any component of R—K. // / is lower semi-contimious on QK and does not assume the value —oo, then lim /-B"(z)> f(b} 2^6. for any regular boundary point b on 9D with respect to the equation (0,1), zuhere the equality holds if f is continuous. Proof. For the case / in Ca°(9K}, it is evident (Theorem 2,13). In the case / in C(9K), i.e., / is a continuous on 9K, taking a sequence {fn} in C°°{'9K) so that fn converges to / uniformly on 9K, we can prove this. If / is lower semi-continuous, we can find a sequence {fn} in C{9K} so that fn converges to / increasingly. Then, the inequality follows. Q. E. D. Lemma 3,3. Let K and K^ be non-polar compact sets of R such as K C K^ and f be a lower semi-continuons function on QK and does not assume the vahie —oo, then we have. (/î(2;)=/^(z) on R-K,. Proof. By Theorem 2,8, in the case f^.Cm{9K) it is evident. For the case f^C{9K), we take fn^CEO{9K) so that fn converges to / uniformly on 9K. Then,. f^z) == Urn f^{z) == Um{f^z) = {U^z) n->.cx> n-»co. in R—K^. If / is lower semi-continuous, there exists fn in C{9K) such that fn converges increasingly to / on 9K, Therefore, we can prove this lemma. Q. E. D. Next, we define a class of q-supersolutions on .R which are very similar to the full-. superharmonic functions due to Kuramochi ([3], [6], [9], [10,], Cl5]). Let s be a positive lower semi-continuous function on R which is not identically equal to -{-co. If s-r<^ s (z^R—K) for any regular compact set K, then s is said to be a q-fidl-supersohttion on R. By Lemma 2,12 we see that a ^-full-supersolution on R is a ^-supersolution on R. For example, if s==l on R, then s is a g'-full-supersolution on R by Theorem 2,9. And, an example of a q-fullsupersolution which is not equal to a real number is given in the next section. The following properties are the immediate consequences of the definition. Lemma 3,4. If Sj and Sg are q-ftdl-supersohitions on R and a'i and a'g ^re positive numbers, then a^s^ + a^s^ is a q-fidl-supersolution on R. And min {Si, $3} is a q-full-super 'solution on R. If {Sn} is an increasing sequence of q-full-super solutions on R and the limiting function s is not identically equal to +00, then s is a q-ftdl-szipersobition on R.. U2).

(8) ? 24 ^ ^ 1 ^ »Mm^!i8^ (^ II ^ A) TO 48 ^ 10 ^ §4. Kernel and potentials In the following, we denote by |[ p. |] the energy of a measure p. defined by. p. I] 2= \ \G{z, zv)dp,{z}d/j.(iv'}. We make some remarks on the potential G{z, /JL) of a measure p. with finite energy. For the next lemma we may use the method in the proofs by Hilfssatz 7, 5 and Satz 7,2 in [3].. Lemma 4,1. The potential G{z, p.} of a measure p. on R belongs to the space E{R} if and only if p. has finite energy. In this case, ive have. (G(.,/<), G{.,^B=2rc^G{.,^d^ for a measure v ivith finite energy.. Lemma 4,2. Any potential G{z, <u) of p. with finite energy belongs to the space Eo{R). Proof. Since any q-solution which is minorant of G{z, /J.') is non-posltive, from the preceding lemma this lemma follows (Lemma 4,6. in [11]). Q. E. D.. Lemma 4,3. For any positive number ^., th function min{G{z, w), ^} of z belongs to the space Eo{R). Proof. From the decomposition theorem of the Riesz type (Theorem 2,2 in [11]) it follows that there existsa measure p.i such that. min{G{z, iv), ^}=G{z, ^x). Since. P.). |[ 2= ^G{z, /J.^dp.^z) <^G'(2, w')dp.^z) < ^, P.), has finite energy. Then Lemma 4,2 implies this lemma. Q. E. D. For a non-empty compact set K in R, we denote by mn- the totality of unit measures supported by K and set. w{K) == inf{ \\{j. [|2 : p.^. ntK}. The quantity l/tu{K) is called the capacity of K, which is denoted by c{K). We stated properties on potentials G{z, /J.) in [17].. Lemma 4,4. If f is a function in C^{R) and 11 is a measure on R luith finite energy, then for any positive number a and. c({ze^:|/(z)|>a})<|?/7T^. fl/i^<ll/IIW^2. Proof. Since, from Green's formula,. (/,G(-,^^-27T^, this lemma can be proved in the same way as the case in [33. Q. E. D.. Theorem 4,5. If f is a function in the space Eo{R} and p. is a measure with finite energy, then we have. {G{.,^,f)s=2^fd^. Proof. We can take a sequence {/„} of C^{R}, such that \\fn-f\\s—^ 0 as %-^+oo. And we may assume that {fn} satisfies \\fn+i—fn\\K<l/2n for every n. Let An== {ze.R: |/n+i(z) -fn{z)\>2-ni2} and B^=[j^Ân. Then, by Lemma 4,4,. (23).

(9) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 n+l—fn\\S^. 1. L^^^ 7t(2~n12'}2 ~~" ~2n'. and so. c(^)<l/2m. We can identify the function / with the series. /1+S (/n+l—/n), n=l. 00. which converges on R— H Bmm=l. We denote by p-m, the restriction of p. onto Bm[J{R—Rm}- Since lim fn=f on ^?—H Bm n-*oo. m=l. and p. (H Bm) =0, we have by Lemma 4,4, m=l. \\f\d^<lzm\\f^\dfJ. n->oo. <^wU/nHM<IHKU/U+D. n->oo. On the other hand, (4,1) Urn W2< Urn ^ ,„„ „ G{', ^==0. m^.co " " ' m^co J Û (^--Km). Therefore, it holds that, for m < n,. \fd^-\fnd/Jl\<^\^ ^ \f-fn\d{J.+\\f\d^m+\\fn\dfJ.m ••-' Rm-Bm'. <72^/1^^(^m-5m)+(ll/ll+l)II^mll+ll/n1111^'111' and that. lim \fdp.-\fndp.. : 2(i|/||+l)||^|!.. Then, by (4,1) we get fdp.=lim \fnd/j,, n-><x>. which implies that (G{',p.\f}s=Um{G{',f^,fn)s 7l<-oo. = Urn 2.-K\ fndfJ.=2Tt\fdfJ.. Q.E.D. n—>oo. In this section we shall construct a kernel k{z, w) having the property K. k{z, w)q(z)dxdy=27t. for any w in R, by the method of Constantinescu and Cornea ([3]). For each tv in R we consider the mapping u -> 2nu{w) on SE(R). Since the condition \\Un\\R—>-0 as n-^+oo implies zin{w}—>-Q as n—^ +00 by Lemma 2,3, this mapping is a continuous linear functional on SE{R). Then, considering SE(R) to be a Hilbert space (Lemma 2,2), we see that there exists Uw in SE{R) such that. (Uw, u}s=2mi{zu) for any u in SE{R). Since Uw{z) == ^.{Uz, Uw)R, we have Ua{w)-=Uw{z) for any <? and w in 7?. Here, we define k{z, zu)=G[z, tv)+Uw{z). For the function k{z, tv) fixed w in R, we write. {14).

(10) •24^ §¥1-^- ^^;gaW^^B^ (^ II ^ A) ^^148^10^ k{', zv}K{z}=^k{-, zu)dfJ.K {z^R-K}, where p^ is the measure defined in § 3. Theorem 4,6. If K is a compact set containing m in its interior, then k{z, iu} = k{ •, w)JB"(z) for all z in R—K. And, k{z, w) is a q-full-supersohdion on R for any tv in R. Proof. Let K be a compac t set containing w in its interior and V/. be a set {z £ R: G{z, w) > A}. We take a sufficiently large num.be.-i ^ so that V^ C 7? and define the function. f^z}=min{G{z, iv), ^,}+Uw{z) on R. Then fi[z} belongs to the space E{R} and there exists a measure p.^ suppoted by the closure of Vi such that. min[G{z, 10), ^} =G{z, ^^). Furthermore /J,^ converges vaguely to 2n8w as ^-^+°°, where Su, is the unit measure at the. point w (see Lemma 4,12 in D-1]). It is enough to show that {fx'}K=:fv on R—K for ^/= sup G{z, -w), i.e., fv is orthogonal zes~. to the space ES'{R). For any g in EE{R}, let g=u+go, u £ <S'£'(7?) and go (= £'o(^)> be the Royden decomposition of g (Lemma 2,4). Then,. (A g}s={G{., ^), ô)^+(^w> ^<)B = 27r\god/ji^ + 2rcn{w), which converges to 2ng{zv')=0 as ^,—f +00. Therefore we have (/^')jr=/^' on R—K, because /,=/,/ on R-Kiox /i>^/. For any regular compact set K we take a sufficiently large regular compact set K^ the interior of which contains K |J {zu}, We define a function on R— K by. it^z)=k{', tv)— \ min {k[', tv), ^}d^.f for every positive number ^. And, let m==inf th{z). Since the function v^z)= \min{k{>, tv), ^}d/J.K SX-i. satisfies [V)}K'i=V). on R—K^ (Lemma 3,3), by the previous property we have. (4,2) ml^{z) < (^)^(z) =k{',tufyWz)-{v^{z)=u,{z) for any z in R—K^.. Les D be any component of R— K. For any boundary point b of D, we have, by Lemma 3,2,. lim y^(z)= min {k{b, tv), /?},. z->b, ZSD. which implies lim Ui{z}û^{b)=0. And, since 11,^ is continuous on QKy, for any b on QK^D 2->-6, z£D. we have, by (4,2),. Um n^z} = U),[b} > ml£'l{b).. z->6, ZGD. Hence, from 1 1<1 and Lemma 1,2, it follows that m>0, because, if m < 0, the non-negative q-supersolution vn—m on R—K takes the value 0 on 9Ki. Then we establised that M^(z)^>0 on R—K, that is,. k{z, w)> \min{k{', w), \}dp.K for z e R—K. Since ^ is any positive number, we have. k{z, w)>^(-, w}K{z), which shows that k{z, w) is a q-full-supersolution on R. Q. E. D.. ( 15 ).

(11) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 Lemma 4,7. For any zv in R, we have Uw = 0 or Uw > 0 on R. And, Uw{z} is continuous on RxR.. Proof. Since Uw{tV')=\\Uw\\sl2n > 0, if Uw{w}=0 we have Uw = 0 on R. In the other case, we can take a compact set K containing w in its interior on which Uw{z) > 0. Then, by Theorem 4,6 and Lemma 2,12,. k{z, w}=k{-, w)K{z). =G(.,w?)+(^?). >SS-K{z}=G{z,tu) for z in R—K, and so, £/u,(z)>0 o% J!?-.K. This implies, by Lemma 1,2, that Uw{z)>0 for all z in R.. Since Lw>0 on R, by the Harnack type inequality (Lemma 1,5) the continuity of Uw{z) on j??xj? can be proved. Q. E. D.. Theorem 4,8. The function k{z, w) defined on RxR satisfies the following conditions : (a) k{z, tv) is contimions on RxR and takes vahies in (0, + °o], (6) k{z, w} is a q-solution on JR— {w} ff%^ /^ffs ê same singzilarity at z==iv zuith the Green function G{z, w). (c) k{z, w} is a q-full-siipersoliition on R for each w in R. And the function k{z,zv) can be characterized. by these properties. Proof. Since these properties can be proved easily, it is sufficient to prove the uniqueness only. We take a neighborhood V of W. Let ^'(-z, iu) be the other function with these properties and. m=inf{k'{z, w}—k{z, w)}. 8F3Z. If we assume that m < 0, then, by Lemma 4,6 and Theorem 2,9, we have k'{z, iu}-k{z, iu)={k{', w)-k{', iu)}^{z). > ml~y{z) > m, for every z in R—V. Since k'{z, w)—k{z, w)>m on V, the function ^'(-z, w~)—{z, tv), which is a q-solution on R, takes the negative minimum m on R. This contradicts Lemma 1,2. Hence, we have m > 0, that is, k'(z, tv}^k{z, zu) for every ^ in ./?. In the same way, we have k(z,w)> k'{z, w). Then k'{z, w}=k{z, zv) on R. Q. E. D.. Lemma 4,9. For every positive mimber X and a point iu in R, the function min {k{z, zu), X} of z belongs to the space E{R). Proof. Let V be the set {z £ R : k{z, iu)>X}. Then,. \\min{{k, w), ^}\\s=\\k{', ZV)\\B-V+^\ qdxdy v. <{||mm{G(., w), /(}||5+H^||5}2+^^ Â^ s. <+°°> where we used Lemma 4,3. Q. E. D.. Theorem 4,10. For every point w in R, zue have. (4,3) \ k{z, iv)q{z)dxdy=2n. B. Proof. Let U be a relatively compact neighborhood of tv. And, we take a positive number X so that 2 ^> sup k{z, zv}. Then, by Theorem 4,6 and 2,9, for z £ R—U zeso-. k{z, tv)=k{-, w?X^l^)<2. Therefore, the set {z^R: k{z, w)>2} is contained in U. Let / be the function win {k{z, zu), 2).. U6).

(12) ^ 24 ^ ^ 1 ^- UMWîlB^ (^ II ^ A) RSft 48 ^ 10 H Then, by Theorem 4,5, we have. f(z)=k{z,tv)=k{-,tunz)=fv(z} for z^R-U. For a relatively compact and regular open set G such that G ~Z> U we get, by Theorem 2,11,. l^/s+f _fqdxdy=0,. ls(?3^"u ' Js-G-"'"'""'-7 "'. for / belongs to E{R) by Lemma 4,9. Then, '-dsz + \ _k{z, iv}g{z)dxdy = 0,. •}9G OHz JS-G. and, by Green's formula,. 9k{^ds^ -2n+ f ^(z, zv}q{z}dxdy.. ISG OTlz J G. Hence, we have (4,3). Q. E. D.. Corollary 4,11. ([13]) If the class SE{R) contains no non-constant function, then we have. (4,4) \ G{z, zu)q{z)dxdy=2Tt for every w <=R. 'B. Proof. Since under the assumption C/;y(z)=0, we have (4,4) from Theorem 4,10. Q.E.D. If p. is a positive measure on R, then function \ k(z, w}dp.(zv) is either = + oo or a qsupersolution on R. When it is not =+cx>, we call it a k-potential or simply a potential with respect to p.. And, this potential is denoted by k{z, /.i). It is easy to see that k{z, p) is a q-supersolution on R and a q-solution on 7?—S(/^), where S(//) is the support of p.. Lemma 4,12. Every potential k{z, <u) is a q-ftdl-sziper solution on R. For any compact. subset K of R such that 5(/-<) is included in the interior of K, we have k{',fJt)K{z)=k{z,i^ on R-K. Proof. This can be proved by Theorem 4,6. Q. E. D. For a circular neighborhood V of tv £ ^?, let -F(^, w) be a fundamental solution of the equation (0,1) on V. Then it is known that. F(2,w)=log^l^+0(l) \Z—l»\ and G{z, tu}—F{z, w) is a q-solution on V. Since K[z, w)=G{z, w')+Uw{z), we can see easily. that k{z, w} = logi^î + zi{z, 10) \Z—tV\. for (z, tv) e 7? x 1?, where %(z, w) is a bounded function on V X V. Therefore, in the same way as in the case of Green's potentials ([3]), we can prove the local maximum principle : Lemma 4, 13. Let p. be a positive measure zvith compact support K. For any point ZQ on K we have. Urn k{z, /^)< Urn k{z, /.<).. z-»zo, ze.K z-'-zo. ze^. Lemma 4,14. If a potential k{z, /J,} of p. supported by a compact set satisfies an inequality k{z, /^XM on S(^) /or some positive number M', then k{z, /^XM on R. Proof. We take a regular compact set K so that the support of 11 is contained in the interior of K. Let m = inf{M— k{z, /^)}. Then we have, by Theorem 2,9 and Lemma 4,12, 9K. (27).

(13) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. (4,5) M-k{z, ^>{M-k(-, /Jt))K{z)>mlK{z) for z in R—K.. For every point b on 9S(,u) we have, by Lemma 4,13, Urn {M-k{z,^~))=M-Um k{z, <«)>0. z->6,z&S(/<) z^-6, zeS(/<). And, for b on 9K,. lim{M—k{z, //))>m. 2-*6. Since the function M.—k[z,ji) is a q-supersolution on the interior of A'-S(^), we have. (4,6) M~-k{z, ^)>min{m, 0} on K-S{/J.}, from Lemma 1,2.. Here, if we assume that m is negative, then the q-supersolution M—k{z, fi) takes a negative minimum at an interior point of 7?—5(^), which is evident by (4,5), (4,6) and 1J5L<1 on R. This is a contradiction, and so, m is non-negative. Then we can complete the proof by (4,5) and (4,6). Q. E. D. By this maximum principle we have the following lemmas, which are consequences of the general theory of potentials with symmetric kernel (for example, see [14] and [3]).. Lemma 4,15. For a positive measure p. zuith compact support the continuity on S(^) of the potential k{z, /J.} at Zy e S(,u) implies the continity on R of k{z, /<) at Zo. Lemma 4,16. (Kishi's lemma) For a potential k{z, fi) there exists an increasing sequence. {Fn} of closed sets ivith the folloiuing properties : Let p.n be the restriction of 11 on Fn. Then k{z, /j.n) is contimions on R and converges to k[z, /^) increasingly. And, the sequence {/j.{R—Fn)} converges to 0 as n—^ +00. We define the notion of capacity of compact sets with respect to our kernel k{z, w). For a measure p. we denote by ||//|| the energy of p., defined by f==\\k{z, tu}dp.{z}d/ji.{iu}. For a non-empty compact set K in R we denote by Mn the totality of unit measures supported by K and set. W{K)=inf{\^:^MK}. The quantity \jW{K} is called the capacity of K, which is denoted by Cap{K}. And, for a Borel set E the capacity of E is defined by Cap{E)=stip{Cap{K) : K is compact set contained in E}. Lemma 4,17. For every compact set K the capacity of zuhich is zero, there exists a measure p. in M.K such that k{z, /<) = + oo for every z in K. Lemma 4,18. Let s be a q-full-super 'solution on R and k{z, ju) be a potential ivhich is. finite on every point of R. If s(z)> k{z, /^) on the support of p. except for a set zuhose capacity is zero, then s(z)~^k{z, /JL) for every point z on R. Proof. We assume that the support of p. is compact. By Lemma 4,12 and 4,17, there exists a q-full-supersolution s' on R which is infinite on the set {z £ S{{X) : s{z) < k{z, p.)} . Then, for any positive number e,. s{z)+£S'{z)>k(z, /l), Z^S[{Ji). By the same way as in the proof of Lemma 4,14, this inequality holds for every z in R. Then. we have s{z)^k{z, ,1} on R. For a general measure /},,. U8).

(14) ^ 24 ^ ^ 1 ^ W»W^CT (^ II ^ A) TO 48 ^ 10 ^ k{z, f^)=lim\_ k(z, '}d/j. < s[z) n-.^.JRn. for every z on R. Q. E. D.. Lemma 4,19. If k{z,/^}=k{z,/j.'} on R except for a set whose capacity is zero, then IJ.= P'.. Proof. Let / be a twice continuously differentiable function with compact support. Since, by Green's formula. 27T/(^)= \ {fq-/lf}k{z zu}dxdy, '-R. we have, by Fubini's theorem,. 2Tt\fd{j.= \ {fq-Af}k{z, n}dxdy. Therefore, we can conclude that p. = p<. Q. E. D.. Theorem 4,20. For a compact set K zvhose capacity is not zero and a q-fti.ll-supersohi.tion s on R, there exists a unique positive measure /j, on K such that k{z, //)< s(z) on R and k{z, /^)= s{z} on K except for a set ivhose capacity is zero. Proof. By Gauss-Frostman's method (C14] and [3]), we can find a measure p. on K so that k{z, /^)<^ s{z) on 5(//) and ^(2,/<)==s(z) on AT except for a set whose capacity is zero. Considering Lemma 4,18 we see this theorem. And the uniqueness follows from Lemma 4,19. Q.E.D.. 5. Compactification of Riemann surfaces In this section we shall define a compactification of R with respect to the equation (0,1) by the analogous argument as in that of the Kuramochi compactification. In the sequel we shall follow [33 in the construction of the compactification.. Let N(R) be the family of all continuous functions / in E{R) for each of which there exists a non-polar compact set K such that f=fK on R—K. Then there exists a unique topological space R* determined by N{R) with the following conditions : R^ is a compact Hausdorff space; R^ contains R as an open and dense subset; Every function in N{R) can be continuously extended to 7?*; If pi and py, are two distinct points in J?*, then there exists a function / in. N{R) such that APiÂPa). The set R*—R is called the ideal boundary of -R and denoted by A. For every point z in R, the function ^(2, tl)} can be extended on R*, since /e(z, tu)=k[', z)v{w),. tveR—V for a neighborhood V of z. We define ^(z, p}=lim k{z, w) for /? in J. w->p. By Lemma 1,5 we can easily see that if k{z', p)>0 for a fixed z' in R, then k{z, p}>0 for any z in R. Hence we set. J*={êJ : k{z, p)>0 on R as a function of z}. Theorem 5,1. For any point p in J, k{z, p~) is a q-sohition on R and a q-fiill-sii.persohition on R.. Proof. At first, by Lemma 1,5, it is easy to see that k{z, p) is continuous on R with respect to z. Let {Zm} be a countable dense subset of R. Since ^(-Zm, tu)—>k{zm,P) as iu—>p, for each m we can take a sequence {Wn} which converges to p as n-->- +00, so that. Um k{z.,n, tVn)=k{Zn,p).. n->ao. By Lemma 1,5 and 1,6, k(z, Wn) converges to a q-solution on R as n—>+oo, which is k{z,p). For any regular compact set K in R,. (19).

(15) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. k{',p)K{z}<lim k{-,tvYi{z) lO-^P. < Urn k{z, zu)=k{z, p), io->p. by Fatou's lemma, which shows that k(z, p) is a q-full-supersolution on R. Q. E. D.. Theorem 5,2. For any p^ and pz (Pimps') on ^, there exists z in R such that k{z, p^\k{z, pz). Proof. By the definition of the compactification I?* of R we can find a function / in. N{R) so that f{Pi)^f{p2), and assume that fK=f for a compact subset K of 7?. Since the second derivatives of a q-solution are locally Holder continuous (IV, § 7' in C4]), there exists a twice continuously different! able function g' on R each second partial derivatives of which are locally Holder continuous in R, such that g'=f outside a compact set K^~Z^K. Let g=Ag'—g'q. Then ^=0 on R—K^ and g is locally Holder continuous on R. We consider the function of tv in R: -[. zi{w}=g'{zu)JT^\ k{zv, z)g{z}dxdy. JR. If g is once differentiable, then the function \ k{z, zv}g{z)dxdy possesses second derivatives and R. ([5]). Aw\ \ k(iv, z) g{z)dxdy\ — q(zu)\ k{w, z}g{z)dxdy= —2ng(iu}. ^R • •- • • - ) ~ • • JK. In the case that g is locally Holder continuous, g can be uniformly approximated by locally Holder continuous functions gi which are differentiable and satisfy a uniform Holder condition. We define functions \ k{tv, z)gi{z)dxdy which have continuous second derivatives and converge IR. to \ k(zu, z)g{z}dxdy uniformly on any closed subset of R (see IV in [4]). Therefore we see lK. Au - qu = {Ag' - g'q} - g= 0 on R, that is, the function u is a q-solution on R. And, by Theorem 2,8 and Lemma 4,12 we have tiF==zi on R—F for any sufficiently large compact set F in R, which implies, by Corollary 2,10 and Lemma 1,2, that %=0 on -R. Hence,. we have. ^. g'w = - ^ \ k{w, z)g{z)dxdy, '7<1. which holds also on the ideal boundary A. Since g'{Pi)^g'(p2)i there exists z in Ky such that. k{z,p^k{z,p,}. Q.E.D. Theorem 5,3. The set J—J* consists of at most one point. Proof. For every / in N{R) we can make a function g' in the same way as in Theorem 5,2 so that f=g' outside a compact subset of R and -[. <g-'(w) = - ^ ^(w, z)g{z)dxdy, where g is a function with compact support. Then for any point p in J—J*, we have. f(P)=-^k{p, 2)g{z)dxdy=Q, that is, the extended function to A of every function / contained in N(R) is zero on J—J*. Since the space N{R) separates J, this theorem follows. Q. E. D. In C7]) IVIaeda showed that an analogous definition of the Kuramochi compactification can be given for a Green space and this compactification is metrizable. Next, we prove that our compactification R* is also metrizable by his method.. {20).

(16) ^24 ^ M 1 -^ W»1?^» (^ II ^ A) P^D 48 ^ 10 ^ Lemma 5,4. If K is a compact set of R and \\fn—f\\s—>-0 as n—>-+co, then there exists a subseqnence {fn'} such that {fn')K converges to fK uniformly on R—G for any relatively compact open set G containing K. Proof. By Theorem 2,8 and (2,4) in Theorem 2,7, we have. \\fK-{fn)K\\S<\\f-fnh, which implies, by our assumption, that \WK-fK\\n-^Q as n-^+^. Hence, by Lemma 2,5, there exists a subsequence {fn'} surh that (/n')JC converges to /JC q. p. on R. Since {fn'}K, fK are continuous on R—K, the convergence is uniform on 9G, Hence, by Theorem 2,8 and Corollary 2,10, we have. {fn'Yi={{MKYG^fKYa=fK as n'-^+oo, uniformly on R—G. Q. E. D.. Theorem 5,5. There exists a coimtable szibfamily of N{R) ivhich separates points of A, and so, R^ is metrizable. Proof. We denote by Cn the family of all infinitely differentiable functions each support of which is contained in Rn+i. Then Cn is a subspace of the space C'(2?n+i) of all continuous functions on Rn+i- Since C{Rn+i) is separable with respect to the uniform convergence topology, Cn is also separable, that is, there exists a countable family Qn which is dense in Cn- We consider a family of functions on R: Q={g«n : g^Qn}, which is a countable subset of N(R). To show that Q separates points of A, it is sufficient to prove that, for any / in N{R} and for any e > 0 there exists n and g in Qn such that \gRn—f\<e on R—Rn+i.. For / in N{R), there is a non-polar compact set K such that f =f on R—K. We take a sufficiently large n so that Rn~2>K and construct an infinitely differentiable function (p on R which equal to 1 on Rn and whose support is contained in Rn+i. For the function (pf, there exists a sequence {f.i} of infinitely differentlable functions wth compact support in Rn+i such that DR{(pf—fi)-^Q and fi—xpf (uniformly) as i—y +00 (see, for example, [16]). Then, it is evident that \\fi—<f>f\\R->0 as i—>- +00, that is, (pf belongs to Eo{R). And, by Lemma 2,5, taking a subsequence we may assume that fi—>^f q.p. on R as i->+oo. Then, by Lemma 5,4, we can. take a subsequence {fv} so that {(/z'pn}i' converges uniformly to ((pf}~Rn=fRn=f on R—Rn+z, that is, there exists g^ in Cn such that l(^P»-/l<e/2 on R-Rn+,. Since Qn is dense in Cn, there exists g in Qn such that |,gr— ^"i [ < e/2, and so, \gsn-{g^n\<el2 on R-Rn, from Lemma 3.1. Therefore, we have \g~Bn—f\<s on R—Rn+i, which completes the proof. Q. E. D.. In the following, for a measure p. on R* we can consider the integral \k{z, p')dp,{p), which. is called the potential of p.. And it is denoted by k{z, p^). § 6. Integral representation of q-full-supersolutions In this section, with k{z, p) we consider an integral representation of q-full-supersolutions satisfying the condition. {21}.

(17) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. (6,1) \ s{z)q{z} dxdy<+ oo. 'K. Lemma 6,1. If s is a q-full-sii.persobi.tion on R and K is a regzdcir compact set in R, then the function v zuhich is equal to SK on R—K and to s on K is represented as the potential. k(z, p} of a measure supported by K. Proof. From Lemma 3,2 and the inequality SK < s on R—K, it follows that v is a ^-supersolution on 7?. Let p. be the measure such that k{z, /j,}<^s{z) on R and k{z, p,)=s{z) on K, where the existence of p. is known from Theorem 4,20 and §2 in [17] by the same way as in the classical potential theory. For a regular compact set K^ containing K in its interior we have, by Lemma 3,3 and 4,12. (y-^(.,^?)=^)-^,^) in R—K^. From this we know, by Lemma 1,2 and 3,1, that the q-solution v{z)—k{z, {JL) on R—K cannot take the positive maximum and negative minimum on 9K^, which shows that v{z)=k{z, fji) on R. Q.E.D. In the following, for every regular compact set K we shall mean by SK the function v in Lemma 6,1, which is a <jr-full-supersolution on R from. Lemma 4,12. Let F be a regular closed subset on jR. Since, if n<m we have spn <; sFm < s by Lemma 4,18 and 6,1, Hm sFn exists, n->.co. where Fn=F^\Rn. Then we define sF(z)==lim s n{z) on R for any q-full-supersolution on R, n-foo. which is also a ^-full-supersolution on R by Lemma 3,4. Lemma 6,2. Let F and F' be regular closed sets in R such that FdF'. For every q-fidl-siiper solution on R, lue have. ^'=sF on R. Proof. By Lemma 3,3 we can prove this easily. Q. E. D.. Theorem 6,3. For any q-ftdl-supersolution s on R satisfying the condition (6,1), there exists a positive measure /j. supported by the clostire of F in R^ such that k{z, //)=s2i'(z) on R. ^. and the total mass of p. is less than -^,\ sqdxdy. l-R. Proof. Let Fn=F^Rn. By Lemma 6,1 there exists a positive measure p.n supported by Fn Such that k{z, /J,n) = SFn(z) on R. We have, by Fatou's lemma and Theorem 4,10,. 2n^n(Fn)=\^, \\ k{z, tu}q{z)dxdy\d{jt,n(tv} lf'n lJji> k{z, p.n}q{z)dxdy < \ s{z)dxdy < + oo,. R ~' '' " R .'. ' -~-}R j^. that is, the total mass of p,n is bounded. Hence, there exists a subsequence of {p.n} converging vaguely to a positive measure p. supported by the closure of F in R*. We denote this subsequence by {/j.n} again. Since k{z, p) is a continuous function of p on 7?*— {z} for any. fixed z in R, it holds that sF(z)=lim sFn{z)=lim k{z, ^n)=k{z, /ji) n-f-oo. n->o°. for any z in R—F.. The restriction of p.n to the interior of Fn is equal to a measure —•^_{^s—qs) as distributtons ([11]). We shall denote it by Vn- And, let v be the restriction of p. to the interior of F. Hence, Vn increases to v as n—> +00, and p.n—v.n converges vaguely to fx—v as n—> +00.. For any z of the interior of F, k{z, /j,)=k{z, v)+k{z, p.-v}. (22).

(18) • 24 ^ ^ i ^- watîe^ (^ n m A) ro 48 ^ 10 n -=-lim k(z, Vn}+lim k{z, p.n—Vn) = Urn k{z, p.ni=s{z}. n->cx). At last, we have to prove that k{z, /^)=s(z) for every z on 9F. Since s-SI is a q-supersolution on R, by Lemma 1,3,. s3!'{z) = lim^S k{z + reie, /j^rdO r-ô ^7U o. =k{z, /jt) for any z on 9F. Q. E. D.. Theorem 6,4. Every q-fzdl-stiper solution s on R satisfying (6,1) can be expressed by the potential k{z, ^) of a measure y. on R*. Conversely, every potential k{z, p.} of a measure /.i on R* is q-full-supersohition on R. Proof. Since s= Um sBn, the first part of this theorem is evident from the same way as n->oo. in the preceding theorem. To prove the converse, if ^ is a positive measure on 7?, it is evident from Lemma 4,12. Therefore we may assume that ^ is supported by J. Then,. k{',^)=^^k{-,pWp^K =^,p)K{zWp}<k{z,^, that is, k{z, fJi} is a q-full-supersolution on R. Q. E. D. Lemma 6,5. Let ^ be a positive measure on R^ and F be a regular closed set in R. Then,. k{.,^=^,pr\wp). Proof. For a regular compact set K in R,. k{-,^=^k{',pWp)}d^ =^k{.,p}K{z}d^p}. Therefore, we have. k{.,^F{z~)=lim k{-,[iYn{z) »;->co. =lim k{',pYn{z)dfi n->oo. =^,pYW^p\ where Fn=F^}Rn. Q. E. D. Lemma 6,6. If s is a q-fidl-snper solution on R contained in E{R), then SF belongs to E{R) and sFn converges to SF in E(R), zuhere Fn=F^\Rn. Proof. If m~>n, we have, by Theorem 2,7, (s^m-s^", sjl'")fl=0,. and hence,. 0 < \\SF^-SF^=\\SF-'n\\s-\\S11'^. Therefore, we have. Us^<Us^<||sb.. From this the existence of lim ||sJ"||.R is inferred and it follows that {sFn} forms a Couchy n<-oo. sequence in the Hilbert space E{R). Since lim sFn=sF on R we can conclude that SF is. (23).

(19) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 contained in E{R) and ||s-s»-si''||^->0 an n-^+oo. Q. E. D. '. Lemma 6,7. Let s and F be as above. For any function f in E{R) tvhich is equal to s on F, ive have {f—sF, SF)R=O. And SF is the unique fimction tvhich has the minimum norm among functions like f. Proof. This is evident by the elementary method in the Hilbert space. Q. E. D. Let A be a closed set of J. Since I?* is a metric space, we can take a sequence {An} of regular closed sets in R such that the closure of A" in 7?* is a neighborhood of A and contained in the l/%-neighborhood of A. For a ^-full-supersolution s on R we consider the sequence {sA } which decreases to a gr-full-supersolution on R as n—> +00. We shall denote lim SA =SA. TO-), co. And, s is a ^-solution on R. Theorem 6,8. Let s be a g-full-supersohi.tion on R satisfying (6,1) and A be a closed. subset of A. Then there exists a positive measure /< supported by A such that sA{z)=k{z, ^) on R. Proof. By Theorem 6,3, there exists a measure /<" supported by the closure of An in R* such that k{z^n}=sAn{z) on R. Since the total mass of ^n is not greater than 1/27TX \ sqdxdy (Theorem 6,3), we can find a 'K. vaguely convergent subsequence of {{in} and a measure ^ on A to which the subsequence converges vaguely. Hence we have. sA{z)=lim k{z, ^n)=k{z, fJt) on R. n->co. Q. E. D.. Theorem 6,9. For a measure [i on R* and a closed set A of A, zve have. k{'^Y{z)=^,pY{z)d^p}. Proof. From the above definition of s^ and Lemma 6,5, it is evident. Q. E. D. Theorem 6,10. For a q-fzdl-szi.persohition s on R contained in E{R}, SA belongs to. E{R) and SA converges to SA in E{R}. Proof. By Lemma 6.7 we can prove this in the same way as in Lemma 6,6. Q. E. D.. Theorem 6,11. For a q-fidl-sttpersohdion s on R contained in E{R}. We have {s-dY=sA on R.. Proof. By Lemma 6,6 and Theorem 2.7, for n<m. \W-{sAmrn\\n<\\s^-sAm^, and || SA — s || R—>O as m—> + 00. Therefore, ||(s ) —SA \\R.—>-Q as m—>-+oo,. that is,. (s^)A"(2)=s^) on R, for every n. Hence, we obtain {sAY=sA on R. Q. E. D.. Corollary 6,12. For a closed subset A of A, {1A}A=1A on R. Proof. This is a special case of Lemma 6,11. Q. E. D. Lemma 6,13. Every q-fzdl-sitper solution s on R is equal to the sum of a q-solution on. R and the potential k{z, v} of a measure v on R. Proof. By Lemma 6,1, sRn is equal to the potential k{z, fZn} of a measure f^n supported by Rn- We denote by Vn the restriction /jtn to Rn, Then we have (.24).

(20) ^24^ ^1-^- A^MBW^^B^ (^ II ^ A) ra48îo^ S{z)=k{z, Vn) +k{2, fJin-Vn} On Rn. Then n—>+oo, k{z, Vn) increases to a potential k{z, v} and k{z, y.n—Vn) converges to a ^-solution on R. Q. E. D.. Lemma 6,14. Let s be a q-f-ull-snper solution on R tuhich is continuous on R. For any closed set A in A with 1^=0 on R, the function S—SA is a q-fzill-super solution on R. Proof. We take a sequence {An} as above, and let K be a regular compact set in 7?.. Then it holds that sK+s^n<s+{sKVn+{s^"}K on K[jAn, and so, by Lemma 6,2 and 4,18, S^-|- SAn= {SK+ S^n)W^n. <,s+{sKVn+{s^n}K on R. Let M= sup s. Then we have {sK)An <M1^", and so, from the assumption, (sji:)4"—)-0 as K. n—>+^. And,. {sA)K{z) = Urn \ s^d^= Um{s^Y<{z). n-».co J • n^.oo. Therefore, making n~>+oo, we have, from (6,2), SK+SA<,S+{SA)K, from which we can see that S—SA is a <3'-full-supersolution on R. Q. E. D.. Theorem 6,15. Let s be a q-fnll-szipersohition on R and A be a closed subset of A ivith 1A==0 on R. Then zue have {sA}A=sA. Proof. Since {sA}A<,sA is evident we have to prove sA>(s4)4 only. By Lemma 6,13 and 6,9 we obtain. sA(z)=M4(z)+ \ k{-, w)A{z)df^{w) <R. and. {sA}A{z)={uA]A{z)+ \ {k{-, wy)A{z}d^w}, <R. •. where tt is a q-solution on R. Since the function min {k{z, w), /t} belongs to JS'(-R) for a positiven number ^ (Theorem 4,9), we see, by Theorem 6,11, that for a large ^. {k{-,tu}A)A{z)={{min{k{-, zv), W)A{z} ={min{k{-, iv), },}}A{z)=k{-, w)A{z). Then we must prove tlA <^ (uA}A for any ^-solution it on R which is also a ^-full-supersolution on R. By Lemma 6,14, we have {t^n-uA)K<,uAn--uA on R-K. for any regular compact set K. Then, by the definition of UA, UA — (tlA}A < UA — u , and so,. {uA}A>uA on R. Q.E.D.. ? 7. Canonical representation To classify points of J* we define the function a{p} for p in J* as follows :. (7,1) a{P) = ^k{.,p} W{z)q{z)dxdy x ^' k{z, p}q[z)dxdy\~ , where {?} is a closed set of J* which contains only one point p.. (25).

(21) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 Theorem 7,1. For p £ J*, we /zaye a(^)==0 or 1. Proof. Since. (7,2) \ k(z, p}q{z)dxdy < lim \ k{z, tu}q{z)dxdy 'B. w^p^S. •. =2?r. from Fatou's lemma and Theorem 4,10,we have. k{-,p)W{z)=a{p)k{z,p}, applying Theorem 6,8. If 1 (p) •=. 0, then. k{',p)w{z)={k{.,pmw(z) =a^p}k{z,p} by Thenrem 6,15, which implies a(^)=l or 0. In the case 1{P]>0, we have 1(p} (2) = ck{z, p) (c>0) on account of Theorem 6,8 and (3,1). Hence, from Corollary 6,12 we obtain. ck{ -, p) w (^) = (1 (Pl) (Pl (^) = c^(^, ^) . Therefore, (7,1) implies a{p)=l. Q. E. D. Theurem 7,2. According as a{p}=0 or 1, ive have. k{',p)^{z)=0 or k{z,p). Proof. Since q{z) is non-negative and not constantly zero, it is evident by (7,1) and. k{z,p}>k{',p)W{z). Q.E.D. We shall denote by A^ and Jf respectively the sets of points of J* for which a{p) has the value 0 or the value 1. Theorem 7,3. For every point p in Jj", we have 'R. k{z, p)q{z)dxdy = 2rc.. Proof. Let Am be a regular closed set such that its closure Am in ^?* is a neighborhood of p and the sequence {Am} converges to p. And, we set A^=Am(~}Rn. Then, there exists a measure fJtm on Am such that. k{',p)^m{z)=k{z,^m). Also, there exists a measure ^ on A^ such that. k{',p}A'^z}=k{z,^}. Here, we may assume that ^ converges vaguely to f^m as n—>- +00 and fzm converges vaguely to a point mass c8p as m —> + °°.. Since the sequence {k{-, p)Am{z)} converges to ^(*, Jf?)(p) (2), the Lebesque convergence theorem. implies that. (7,3) \k{ •, p~) ^ {z)q{z)dxdy = Hm \ k{., pym{z)q{z)dxdy, 'S. ~. ~. m->x')R. by k{', p)Am{z)^k{z, p) and (7,2). Then, for any positive number e we have, applying Fubini's theorem and (4,3),. k{ •, p) w {z)q{z)dxdy + e > \k{', pym{z)q{z)dxdy. fl. ". '. •. •. -. J^. 4?B,. > \'B k{', PTn {z)q{z)dxdy = \ _k{z, ^}q{z)dxdy ' -IR. = ^[^k{z, zu}q{z)dxdyY^{iu} = 27;^^, where m is a sufficiently large integer. Therefore letting %—>• +00 and m—>-\-co,. (26).

(22) :24 ^ ^ i -^ AMaitîie^ (^ n ^ A) TO 48 ^ 10 ^. 'fl. k{ •, p) tpl {z)q{z)dxdy + s > 2nc,. and so,. Ifl. k{ •, p}[p} [z}q{z)dxdy > 2^c.. Since. k(.,p)W{z)==ck{z,p), by Theorem 7,2 it is evident that c=l. On the other hand,. 2TT>\ k{z, p}q{z)dxdy IR. '. ~'~''. ~. >\_k{',p)^{z)q(z)dxdy. <R. Then we got the equality. Q. E. D. Theorem 7,4. The set J? //s an F^-set. Proof. For a positive integer m, we denote by 8m the set of all points p of J* having the following property; for every regular closed set F in R the closure in R^~ containing a neighborhood of p and is contained in the 1/w-neighborhood of p in R*, it holds that. _k{',pY{z)q{z)dxdy<^.. lfi. Since if p £ J? then lim\ k{ -, pYm{z)q{z)dxdy = 0. m—>.ooa/. by Theorem 7,2 and (7,3), we have Jor== U^=i^m. Therefore, it is sufficient to prove only that 8m, is closed in jR*.. Let F be any closed set in R defined as above and let Fn be F(~]Rn. By the fact that {k{-, p) n{z}} converges increasingly to k{',p) (z) on R, for any positive number e we can find a sufficiently large number n so that. (7,4) f k{., p}F{z)q{z)dxdy -e<\ k{., pYn{z)q{z)dxdy. Ifi. •-•••-•. -. J^. If a sequence {pi} in 8m converges to p, we have, by (3,4), that. k{', piY^z)qWxdy- \ k{-, p)Fn{z)q{z)dxdy \. IR. '. ~. .... ^^. sup I k{z, pi) — k{z, p} I x \ q{z)dxdy -> 0 as i—> + oo, Fn'. •. '. '. •. ~. '. •. -^R-. that is,. k{-, p}Fn{z)q{z}dxdy = lim \ k{-, pi)Fn{z}q{z)dxdy.. ]R. ~. '. '. -. i->coJR. Hence, from (7,4) it follows that, for a sufficiently largen n,. 7i > /?m ^ ^( •, piY{z)q[z)dxdy i-r^^R. > Urn \ k{ -, pi)Fn\z)q{z)dxdy >\ k{ •, pY{z)q{z)dxdy - e. Z->.ooJ.R. •. -. .. -. .. -. ^^. Since e is any positive number, p is contained in 8m.. Thus 8m is a closed set in R*. Q. E. D. Lemma 7,5. Let s be any q-fzill-supersolution on R. If {An} is a sequence of closed sets of A increasing to a closed set A such that sAn == 0 for each n, then zve have SA = 0. Proof. ([15]) By the fact stated in § 6, for a fixed Zo in ^ and e> 0 we can take a closed. (27).

(23) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 regular ste Fn so that its closure in R^ is a closed neighborhood of An and contained in the suf&ciently small neighborhood of An and s TO(zo)<s/2't. By the compactness of A, we can choose a finite number of {Fn} which can be denoted by F^, F^, ... Fp, so that [3^Fn is a neighborhood of A. Let Lm be a closed regular set such that its closure Lm in R*' is a closed neighborhood of A and contained in the 1/w-neighborhood of A. Since Lmd\J^Fn for large m, we have sLmî^s^^lFn^Ri< S SF^Ri n==l. p. for each i, which implies that s4<SS • Therefore, it follows that sA{zo)<s, that is, sA{zo)=0. n=i. Hence, Lemma 1,2 implies SA = 0 on -R. Q. E. D.. For a positive integer m, we denote by Tfm the set of all points p of J* satisfying the following property: For every regular closed set F in R the closure in 7?* containing a. neighborhood of p and which is contained in the 1/m-neighborhood of p in R*, it holds that. ' k{', pY{z)q{z)dxdy <^-_ (* k{z, p}q{z}dxdy. ' - ' ^TI^B.. IR. '-, denoting ^{p)=sttp \ k{', p')ll'{z)q{z)dxdy,where {F} is the family of all sets with the (-Ff ->R. above property, we have. r^={^£j* .- W <^k{z,p}q{z)dxdy}. For any real number r, it is easy to prove that the set. {p^^:W<r} is closed, by the same way as in the proof of Theorem 7,4. Since the function \ k{z, p}q{z)dxdy Ij?. is lower semi-continuous on R by Fatou's lemm.a, fm, is a Borel set for every m. And, it is. clear that J^= U 7'mm==l. Lemma 7,6. If s is a q-fzdl-super 'solution on R zvith the property (6,1), then sa=:0 on R for any closed subset E of r»i. Proof. At first, we consider a q-full-supersolution s such that {sB)B=sB on JR for any closed subset E of 7-m. Let E be a closed set of fn the diameter of which is less than l/2m and H be a regular closed neighborhood of E such that its closure in R^ is contained in the If 2mneighborhood of E. Since H is contained in the 1/m-neighborhood of any p in E,. (7,5) f k{ -, p~)s{z)q{z)dxdy < ^ f k{z p)q{z)dxdy IR ' ' ~ ' • •- • • - ^ Jffl. for any p in E. Since s is represented as the potential k{z, [ji) of a measure ^ supported by E by the assumption and Theorem 6,8, we have. sE{z)q{z)dxdy= \ k{', [ji)B{z)q{z)dxdy. 'fi. •. -. JQ. <\ k{', ^}s{z)q{z)dxdy <R. = Hf^(-> P)s{{z)QWxdy~^d^p} ^. <-^\ k{-, {ji}q{z)dxdy ';;. _sB{z}q{z)dxdy,. >R. (2<S).

(24) •24^ ^1-^- ^b^Mat^Ê^ (^ II ^ A) PS^P48^10^ from (7,5) and Lemma 6,5. Then. •'R.. sE{z)q{z)dxdy=0,. which implies that ss{z) =0 on R. Since any closed set E can be divided into a finite number of closed sets with a diameter less than l/2m, we can prove ss=0 for any closed subset E of fm by Lemma 7,5. Since {1E)S=1S (Corollary 6,12) for any closed set E on A, we have 1s =0 on R for a closed set E of Tm by the above proof. Therefore, by Theorem 6,15, we can assume {sB)B=ss for any s. Hence, in general we obtain SE=0 for ECZTm- Q. E. D.. For the following discussion we prepare some notations. Let F be a regular closed set in R and A be a closed subset of A. We can take a sequence {Am} of regular closed sets in R so that each closure Am in J?* is a neighborhood of A and Am converges to A as m—>-+°°. Let A^=Am^\Rn. Then, by theorems in §6, there exist measures •^m and ^ supported by Am and A^ respectively such that. sAm{z)=k{z, //71), sA^(z)=^^). And, we may assume that ^ converges vaguely to //m as %—>-+oo for each m. Let v^ be. the restriction of ^ to F. Then we can assume that v"^ converges vaguely to a measure vm as n-^+oo, which is supported by Amf~}F and that vm converges vaguely to a measure v as m—>-+oo, -which is supported by A^~}F. Lemma 7,7. Let F and F' be regular closed sets on R such that FC.F' and F does not contain any boundary point of F'. Then we have. k{z,v)<sFt{z) on R. Proof. It is evident that. ^,^)<^,^)=s^(z), and. k{-,vy{z)=k{z,v^ by Lemma 4,12. Therefore, we have. k{z, ^)<(sA»y(z)<sy'(z), W\ ^- t Am\Vt / \ ^ Vf. and so, by letting n —r + oo and then m —^ + oo, we obtain. k{z,^<sFr{z). Q.E,D,. Theorem 7,8. Let s be any q-fidl-stiper solution on R satisfying (6,1) and A be a closed subset of A. Then SA can be represented as an integral. ^k{z,p)d^p). '^1. Proof. ([15]) In this proof, we use notations defined as above. It was seen that sA{z)= k{z, [i), where {j. is supported by A. Here, we must show that ^(J^)=0. Let e be any positive number and ZQ be a fixed point in R. Since ss=0 for any closed subset of rm by Theorem 7,6, there exists a regular closed set F' in ^? such that its closure in R* is a neighborhood of E and sF'{zo)<s. We take a similar set 77 as F' so that FC.F' and 77 does not contain any boundary point of F'. We can find a continuous function / on 7?* so that 0</<1, /=1 on E, its support is. contained in F, and \fdfJ, and \fdv are sufficiently closed to ^(£1) and ^(£) respectively. Then we have. {29).

(25) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973. ^fdfjim =[iw^fd^ =lim\fdv^=\fdvm, n--».co.. and so, letting m—> +00, \fd^=\fdv. That is, ^{E)=v{E). It holds from Lemma 7,7 that. _k{z,,pWp}<sF'(z,)<s.. 3. Therefore we have \ k{z, p)v(p')=0 on R, which implies v{E}=Q. Then it follows that <E. fJi{rm)=SUp fJt{E)=SUp V{E)==0 Ec.rm sc.rm. for each m. Since J^= U fm we can complete this proof. Q. E. D. m=l. Corollary 7,9. Any q-solution u on R which is also a q-full-super solution on R, satisfying (6,1), is represented as \ *k{z, p)dfjt{p). ^1. Proof. This is evident from u=lim uR~~s"n-=uA. Q. E. D. n->co. Theorem 7,10. For any q-full-supersohition s on R satsfying (6,1) and closed subsets A and A' of A such that A d A', it holds that (sA)4'=s4 on R And, if F' is a regular closed set in R and its closure in 7?* is a dosed neighborhood of A, then ive have {sAY'=sA on R.. Proof. By Theorem 7,8 we may assume that there exists a measure p. on J? HA such that sA{z)=k{z, [i). On the other hand, applying Theorem 7,2, we have. k{z,p)=k{',p)W(z} <k{',p)A'(z)<k(z,p), that is, k{z, p) == k{', p}A> \z) for pÂ^\A^. Therefore, from Theorem 6,9, it follows that. {sA^'{z}=\_.k{',p)A'{z}d^p) ^.n-î. =sA{z}. For the proof of the other equality we can prove similarly by Lemma 6,5. Q. E. D. In the following, a measure ^ on A will be called canonical if ^(J^)=0. A representation of the form such as k{z, [i) is canonical representation if the measure occurring in its integral is canonical.. ? 8. Uniqueness of canonical representation If a q-full-supersolution on R is a ^-solution on J?, it is called a q-full-solution on i? simply. A (7-full-solution on R satisfying >K. u{z)q{z)dxdy < + o°. will be called q-minimal if v=cu on R whenever v and n—v are q-full-solutions and 'R. v{z)q{z)dxdy < + oo.. (30).

(26) ^24 ^ ^ i -^- WMair^^ (m n ^ A) ro 43 ^ 10 ^ In this section we shall establish the uniqueness of the canonical representation obtained in the preceding section. Although the method is almost due to [3J and Cl5], some modifications may be necessary.. Lemma 8,1. Suppose that n is q-minimal. Let A be any Borel subset of ^. If a relation of the form. u{z}=\k{z,p)d^p} A. holds for all z in JR, then {i is a point measure supported by p' £ A, so that, it{z)=ck{z, p'} 1^. and c=^_\ u[z)q{z)dxdy'. JR. Proof. In the inductive way it is possible to construct a decreasing sequence {An} of closed subsets of A the diameters of which approach zero and each of which has a positive /y-measure. Let p' be the point common to all the An. Now, since the function. u{z)- ^ k{z,p)d^p-)=\ ^ ^ k[z,p}d^p} An ••*••-• ,/ A—4n. is a g'-full-solution by Theorem 6,4, there exists Cn such that. tl{z)=Cn\ 'An , k{z,p}d;jt{p~). Therefore, u can be represented as k{z, y.n), where p.n is a measure on An. Here, from. 27rxthe total mass of ^n=\^ \ \ k{z,p}q{z}dxdy\d{j,n{p') J^n ^<^B. ''R. 'B. k{z, p.n)q{z)dxdy n{z}q{z)dxdy < + °°,. there exists a measure ô which is a vague limit of a subsequence of {^n}. It is easy to see that [J.Q is a point measure at ^)/ e A, and u{z)=k{z, fjio)=ck{z, p'). Considering Theorems 5,2 and 7,3, by the above reasoning we can see that // is a point measure at p'. Q. E. D.. Theorem 8,2. Every q-full-solution n zuhich is q-minimal is a positive multiple of some k{z, p), ivhese p is in A!. Proof. By Corollary 7,6 we can assume that ti{z)==k{z, jjt), where p. is a canonical measure. Then, from Lemma 8,1 it follows that tt(z) == ck{z, p), where p is contained in 4^. Q. E. D.. Theorem 8,3. If p is a point of A^, the k{z, p) is q-minimal. Proof. If v and iu{z)=k{z, p}—v{z) are (7-full-solutions and <R. v{z}q{z}dxdy < + oo,. by Theorem 7,2 we have. v^{z)+wW{z')=k{',p}^{z) =k{z,p)=v{z)+w{z), which implies v[p[=v. Therefore, it follows from Theorem 6,8 that. v{z)=ck{z, p} on R, that is, k{z, p) is q-minimal. Q. E. D.. For any /in C^, 'K. G{z, .w}{AJ-qf}dxdy = - 2nf{w}, ( 31'}.

(27) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 from Green's formula. Hence, there exists a measure a with compact support on R such that f(w}=k(zv, <r). By Lemma 6,1, for a regular compact set K of R and a point p in J* there exists a measure /^p,K supported by K such that. k{',pW=k{2,^). Now, we shall show that \ fdp.p,K is a continuous function of p for any continuous function / on K. At first, since, for / belonging to C^,. f{z}=k{z,a')-k{z,ff"}, -where a' and a" are both non-negative, it holds that. ^fd^, K= ^( •, p}K{w}dff'{zv} - ^(., p}K{zu}da"{w}. Then, the continuity of \fdfjip,K is inferred from the inequality. sup I k{', pi)K(,w) — k{ •, ?2)K{w} I < max \ k{w, p^) — k(w, ps) \. R3w. IBw. For any continuous function / -with compact support in R, approximating / by a sequence of functions of C^° the supports of which are contained in a fixed compact set in R, the continuity of \fdfJt.p,K follows. In the following, for simplicity we shall denote ^ for /^p,^n. We have, by Theorem 4,10 and Fubini's theorem,. 2^d^= ^ k{z, ^)g{z)dxdy < \ k{z, p}q{z)dxdy < 2n, IB. which shows that the total mass of [iv is bounded. Let ^v be the vague limit of a subsequence. W of {^}. Lemma 8,4. fJ,p is a canonical measzire.. Proof. Let ZQ be a fixed point in R. Then, by Theorem 7,6, for any closed subset E of fm and any positive number s there exist regular closed sets F and F' which are similar sets as those in Theorem 7,8, such that, k{-, p}s" {zy) < s and ZQ ^ F'. Let v^ be the restriction of ^ to F. Then we may assume that ^ converges vaguely to the measure on A and v^ converges vaguely to a measure vv supported by F H A. And, let / be a continuous function on 7?* supported by F such that 0 <^ / ^ 1 on R^, f = 1 on E and iv and \fdvv are sufficiently closed to ^P{E) and ^p(£'), respectively.. Then, since \fd^=\fd^ for every n, \fd;jp=\fdvp, that is, pv{E)=vp{E~). On the other hand, from. k(z,^)<k{z,^} =k{-,p)Kn{z)<k{z,p}, it follows that k{z,^==k{-,^r{z}<k{',pr'{z) and, letting n-^+oo, k{Zo, vp)^k{', p)J!"{Zo)<s. Therefore, we have v{E}=Q, and so, ^p(£)=0. Hence, we can see that p,p is a canonical measure by the same way as in Theorem 7,8. Q. E. D.. Lemma 8,5. For p £ A^, ^v is the unit umeasure at p.. ( 32}.

(28) ^ 24 ^ M l -^ UMat^W^ (^ II ^ A) TO 48 ^ 10 fi Proof. Let {^,} be a subsequence of {^} converging vaguely to //p. Since. k{z,p)= Urn k{z, //^,) rt'-ôo. =k{z,^ and k{z, p) is q-minimal (Theorem 8,3), p.p is the unit measure at p by Lemma 8,1. Q. E. D.. Theorem 8,6. Canonical representation of any q-full-solution satisfying u{z)q{z)dxdy < + oo. 'R. is unique.. Proof. We assume k{z, ^)=k{z, v} on R, where y. and v are canonical measures. We consider measures jji.n and Vn defined as follows ;. ^n=^fd^dfz{p) and. p^»=J ^fd^dv{p}, respectively. Then, by Lemma 6,5, k{z, {jtn)=k{-, f.t)Rn(z} and k{z,Vn}=k(-,^n{z). Hence, k{z, fjin)=:k{z, Vn}- Since any function in C^ is represented as k{z, a), ^(•, <J)d[in= \k{', p.n}da = \k{', Vn)d<J = \k{', a}dvn, which implies that ^n=^n. For any continuous function / on R*, by Lemma 8,5, we have. Hm \f d iin -= \ (lim \fdyQd^ =. n->cxi«/ */ n->ooi. and similarly Um\fdVn:= \fdv.. 7i->oo<. Therefore, we can show that p.=v. Q. E. D.. References [1] Brelot, M. (1960), Lectures on potential theory. Tata Inst. of J. R., Bombay, p. 170. F2] Brelot, M. (1965), elements de la throne classique du potentiel. C. D. U., Paris. p. 209. [3] Constantinescu, C. and Cornea, A. (1963), Ideals Rander Riemannscher Flachen. Springer, Berlin. p. 244. [4] Courant, R. and Hilbert, D. (1962), Mlethods of mathematical physics. Vol. II, Interscience, New York. p. 830. [5] Duff, G. F. (1956), Partial differential equations. Univ. of Toronto press, Toronto, p. 248. [6] Kuramochi, Z. (1962), Potentials on Riemann surfaces. J. Fac. Sci. Hokkaido Univ. Ser-I. 16, p. 5-79. [7] Maeda, F-Y. (1964), Notes on Green lines and Kuramochi boundary of a Green space. J. Sci. Hiroshima Univ. Ser-A-I. 28, p. 59-66. [8] Maeda, F.M. (1966), Axiomatic treatment of full-superharmonic functions. J. Sci. Hiroshima Univ. Ser-A-I. 30, p. 197-215.. (33).

(29) Vol. 24, No. 1 Journal of Hokkaido University of Education (Section II A) October 1973 [9] Maeda, F-Y. (1968), Introduction to the Kuramochi boundary. Lecture notes in math. 58, Springer, Berlin, p. 1-9. [10] Maeda, F-Y. (1968), On full-superharmonic functions. Lecture notes in math. 58, Springer, Berlin. p. 10-29.. [11] Maeda, F-Y. (1968), Boundary value problems for the equation Au—qu=Q with respect to an ideal boundary. J. Sci. Hiroshima Univ. Ser-A-I. 32, p. 85-146.. [12] Myrberg, L. (1954), Uber die Integration der Differential Gleichung Au=c{p')u auf ofEenen Riemannscher Flachen. Math. Scand. 2, p. 142-153.. [13] Myrberg, L. (1954), Uber die Existenz der Greencher Function des Gleichung du=-c{p~)u auf Riemannscher Flachen. Ann. Acad. Sci. Fenn. A-I. 170.. [14] Ninomiya, N. (1969), Potential theory. Kyoritsu, Tokyo, p. 190. [15] Ohtsuka, M. (1964), An elementary introduction of Kuramochi boundary. J. Sci. Hiroshima Univ. Ser-A-I. 28, p. 271-299. [16] Sario, L. and Nakai, M. (1970), Classification theory of Riemann surfaces. Springer, Berlin, p, 446. [17] Sato, T. (1972), Evans' solution of the equation Au=qu on Riemann surfaces (II). J. Hokkaido Univ. of Education. Sec-II-A. 23, p. 13-19.. (34).

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