3 ) P(p, q) is a continuous function of (p, q) whenever p q.

Transcription

1 No. 1.] 7 3. Positive Definite Integral Quadratic Forms and Generalized Potentials. By Syunzi KAMETANI. Tokyo Zyosi Koto Sihan-Gakko, Koisikawa, Tokyo. (Comm. by S. KAKEYA; M.I.A., Jan. 12, 1944.) I. Positive definite integral quadratic forms. Let Q be a separable, metric space with metric p(p, q) (p e A? and q E A). Suppose that the function 'P(p, q) defined for all points p e A and q E 9 satisfies the following conditions 1 )-5 ) 1 ) 1)(p, q)== P(q, p) ~ 0, 'P(p, p) = + 00, 2 ) lim (1)(p, q) = +0, p(p, q)50 3 ) P(p, q) is a continuous function of (p, q) whenever p q. Before the condition 4 ) is mentioned, it seems convenient to begin with some preliminary remarks. Given a bounded Borel set E in 9, let c be any completely additive function of Borel sets on E. Then by Jordan's decomposition theorems), we may write c = ~+ - o ~, where c+ and Qr are the positive and negative variations of respectively, each of which is itself a non-negative, completely additive set-function defined for all Borel sets contained in E. Now, consider the following integral : (D(p, q)dc(p)dr(q) =1im PN(p, g)dc*(p)dr (q)+lim ~N(p, q)di (p)dr"(q) -lim ~N(p, q)dcr(p)dr~(q) -lim ~N(p, q)dc "(p)dr~(q), N-boo N-~bo where PN(p, q)=min {N, P(p, q)}. f is used for f throughout this Note, so that ff for E E E If all the four terms involved are finite, then the integral to be absolutely convergent. Thus the 4th condition is : is said 4 ) + 0 ~ P(p, q)dc(p)d6(q) > 0 except when the integral is meaningless, 5 ) if P(p, e)do<p)de(q)=0, then we have c(e)=0 " for any Borel set e c E. 1) S. Saks : Theory of the Integral, (1937), Chap. I.

2 8 S. KAMETANI. [voḷ 20, In case when 2 is a three dimensional Euclidean space, the function P(p, q)= 1 (2> a > 0) satisfies the required properties stated [P(p, q)]a above. Let ' = ~E be the set of all the completely additive function a of Borel sets on E such that the integral (P(p, q)da(p)da(q) is absolutely convergent. Then is a linear space, namely, if a E C~ and z E C~, then as + ~'z E c for any real constants a and i. It is evident from the definition that if a E l, then a+ E C~ and a- E C~, so that if a E and z E C~, then (P(p, q)da(p)dz(q) is also absolutely convergent, which we shall denote by (a, z) and say the positive definite integral quadratic form. Evidently the symbol (a, z) defined above for any pair of a E C~ and z E c possesses the following properties of inner product : I1) (a, z)=(z, a) (Fubini-Tonelli's Theorem), 12) (aa, -)= a(a, z) for any real constant a, I3)' (a, ri + T2) - (a, Ti) + (a, Z2) 14) (a, a) > 0, where equal sign holds when and only when a = 0. Thus, writing II a II = V(a, a), we have introduced a "norm into the linear space considered : II a II >_ 0 for any a E C~, II a 11=0 implies a = 0, II as II = I a I II a II, a being any real constant, and Notice also for any a E and z E C~ I (a, z) I <_ II a II II z II (Schwarz's inequality). II a + z II < II a II + II z 11. The purpose of the present Note is to show how the general principle, introduced by 0. Frostman3', of sweeping out process in the theory of Potential may be reduced to some simple considerations of the normed space ( just stated. II. Positive mass-distributions on the compact set E. Given a Borel set E in the space 2, let p be a completely additive, non-negative function of Borel sets on E. Then, p is called a positive mass-distribution on E, of total mass p(e). We have p - 0, when and only when µ(e) = 0. It is well known that if {p,j is a sequence of positive mass-distributions on the compact set E, of total masses bounded : 2 0. Frostman : Potentiel d'equilibre et Capacite des Ensembles, etc. These, (1935), Chap. II. 3) 0. Frostman : Loc, cit.

3 No. 20.] Positive Definite Integral Quadratic Forms and Generalized Potentials. ) pn(e) ~ M(< + co) (n=1,2,...), then there exists a sub-sequence {p.} such that, for every continuous function f (p) defined on E, we have always (1) lim f (p)dpnj(p)= f(p)dp(p), which is often expressed as "p,. converging weakly to p "4'. This fact is also true for the "product" distribution µn x pn in the product space 2 x 2, namely, for any continuous function g{p, q) ((p, q) e E x E), we have always (2) lim JJ g(p, q)dpn.(p)dpn.(q) = g(p, q)df4p)dµ(q) If g(p, q) is replaced by the discontinuous function GO(p, q), then such relations as above will not hold in general. But we have by (2) for the fixed N(> 0) llm PN(p, q)dpnj(p)dpnj(q)= PN(p, q)dp(p)dp(q) and, from we have also o(p, q)dpn.(p)dpnj(q) > N(P, q)dpnj(p)dpn/q), l m (P, q)dpnj(p)dpnj(q)? ~N(P, q)dp(p)dp(q), whence, making N> co, (3) lim 11 pn.ij2 > II p I(2. If f (p) is lower semi-continuous on the compact set E, then there exists a monotone increasing sequence {fm(p)} of continuous functions on E such that fm(p) T f (p) (m -~ oo ). Similar argument as above shows also (4) lim f (p)dp.(p) > f (p)dp(p) j~oo III. Generalized Potentials and Capacity. Given a bounded Borel set E in 2, let p be a positive massdistribution on E. We shall consider now the following function: (5) u(p) = P(p, q)dp(q) =Nim ON(P, q)dp(q), which is well defined at every point of 2. We call this function the (generalized) potential of the distribution p with respect to P(p, q). 4) N. Kryloff and N. Bogoliouboff : La theorie generale de la mesure Bans son application a 1'etude des systemes de la mecanique non lineaire. Ann. of Math. Val. 38 (1937).

4 u. 10 S. KAMETANI. [Vol. 20, If there exists a neighbourhood U of p E 9 such that µ(u. E) = 0, then the potential u(p) is continuous at that point. Being a limit of a monotone increasing sequence of continuous functions as seen by the right-hand side of (5), u(p) is in general lower semi-continuous a t p 9, if, for every neighbourhood U o f p, p(u E) > 0. The set Eo of points p E 2 such that, for every neighbourhood U of p, µ(u E) 0 is called the kernel of p. It is evident that Eo is closed and contained in Eb, the closure of E. It is also evident that u(p) is continuous outside E0. Let p be a positive mass-distribution on E, of total mass 1, and consider the upper bound of the potential : Writing V m(e) = sup (P(p, q)dp(q) V m(e) = inf V (E), we shall define pet CA(E) _ [V m(e)]'1, which is called the (P-capacity of E. We notice here that C0(E) = 0 if and only if V ~(E) _ + oc. A little consideration shows that the necessary and sufficient condition for the existence of a positive mass-distribution p on E, of total mass positive, with bounded potential is that the (P-capacity of E should be positives'. Let E be a compact set of positive capacity. Then there exists a positive mass-distribution p on E, of total mass 1, such that for every p E Q u(p)= J (P(p, q)dp(q) <M(<+ 00), from which we have, integrating both sides with p, whence µ EVE, so that we have (P(p, q)dp(p)dp(q) <M, inf 11/1112= W(E) < 00, the lower bound being taken over all p E `~ E with p(e) =1. Then, there exists a sequence of positive distributions {p,j, of total mass 1, such that lim if p,112= W(E). From the sequence, we can select a sub-sequence {p n.} converging weakly to the positive distribution Po of total mass 1. By (3), we have W(E) = lim II pn. Ill _> II ~o 112 > W(E), 5) S. Kametani : On some Properties of Hausdorff's Measure and the Concept of Capacity in Generalized Potentials, Proc. 18 (1942), 617.

5 No. 1.] Positive Definite Integral Quadratic Forms and Generalized Potentials. 11 from which we find W(E) = II fro 112>0, since otherwise, we would find µo = 0. Hence we have at last (6) 0 < W(E) < + oo. IV. Gauss' Variation. Let E be a compact set of positive capacity. Given a upper semicontinuous function f(p) > 0 on E, let us write G(p)=I1 µ f (p)dµ(p), where p e is any positive mass-distribution on E. C0(E) being positive, there exists a positive distribution p e C, of total mass positive. Let v = m-1 p, where m = p(e) (> 0), then v E C~ and v(e) =1. Hence G(p) = m211 v 12_ 2m f(p)d(p) y (7) > m2w(e)-2mm=m2 W(E)-2M m-1) where M= sup 1(p) < + cc by the upper semi-continuity of f(p), and pee I'V(E) > 0 by (6). Since the right-hand side of (7) is positive for m > 2M/ W(E)= K, G(µ) > 0 holds for any.positive distribution p on E, of total mass p(e) > K. As G(µ) = 0 for p = 0, we have g=inf G(p) <O, a from which and also from the consideration made above we have inf G(p) = inf G(p). u p(e) Let {µn} be a sequence of positive mass-distributions such that lira G(pn) = g and p(e) _< K. Then there exists a sub-sequence {p.} which converges weakly to a positive mass-distribution Po with the following properties : Jim II Pn.II > II Poll (by(3)) and Hence lim (--f(p) dpn.(p) > -,f(p))dpo(p) (by(4)). we have g=lim G(µ,.) >_ lim IIan lim (-f (p))dp.(p) >_ II fro (--f(p))dpo(p) = G(p0) 11

6 . 12 S. KAME'~AN!. Vol. 20, from which follows G(p0) =9. Thus we have proved the following : Theorem 1. Let E be a compact set of positive capacity. Then there exists a positive mass-distribution fro E which minimizes the value of G(µ) for all the positive distributions p on E. (Gauss' Variation). Let v e be any positive distribution on E. Then No+ev e C~ for e > 0. By Theorem 1, we have for any e > 0 0 < G(po + ey)--g(po) = II Po+ e.' Sf(p)dpo(p) -- 2e J f (p)dy(p) II fro f(p)dpo(p)} 24(P o, y) - f (p)dy(p) + e211 p j2 Noticing II v 112 < oo and e(> 0) arbitrary, we must have the following inequality (Po, y) - f (p)dy(p) 0, which is valid for any positive mass-distribution v e C~. If II µo l1 0 on the one-hand, G(tpo)= t211 po 112_2t f (p)dpo(p) attains its minimum at t= f (p)dpo(p)/ 11 Po 112, while G(tpo) must be minimized at t= 1, from which we have (8) II µo 112_ f(p)dp0(p) = 0. If 11 poll= 0 on the other hand, then we have No = 0, from which we have also (8). Thus we have proved the following fundamental : Theorem 2. Let E be a compact set with G0(E) > 0. Then, for any positive mass-distribution u on E such th >t ' E we have (No, y) ' f (p)dy(p)} 0. In particular, 'if = Po, then we have II µo 112_ f (p)dpo(p) = 0.,i,' Theorem 3. The positive mass-distribution p on E which minimizes G(p) is uniquely determined. Proof. Let fro and v be two distributions minimizing G(p). Then by the preceding theorem, we have (Po, "o) f(p)d.'0(p) = II X0112,

7 No. 20.] Positive Definite Integral Quadratic Forms and Generalized Potentials. 13 Hence (vo, fro) > f (p)d'tio(p) = II loo Ill. ll po - vo 112 = ll p112-2(p,) + ll yo ll -< 0, which shows 11 fro- "oil = 0, that is, Po = v~. IV. Properties of the potential of the distribution Po. Throughout this section, we shall denote by u(p) the potential (P(p, q)dµo(q) of the minimizing distribution µo on the compact set E of positive capacity. Theorem 4. The potential u(p) is not less than f (p) everywhere on E with the possible exception of (-capacity 0, namely C~{p; p E E, u(p) < f (p)} =0. Proof. Denoting the set {p; p E E, u(p) < f (p) } by eo, we find this Borelian, since u(p) -f(p) is lower semi-continuous. Now if C~(eo) > 0, then there would be a positive distribution v on eo, of total mass 1, such that eo eo (P(p, q)dv(p)dv(q) < +00. Hence, regarding as the distribution on E, v belongs to C. Then, by Theorem 2, we must have 0 < (po, v) - f (p)dv(p) -= (u(p)-f (p))dv(p) _ (it(p)'-f(p))clz.(p) <0, which is absurd. Lemma. The set e0= {p; p e E, u(p) < f (p)} contains no p0-mass, namely : fro{p ; p E E, u(p) <f (p)} =0. Proof. Supposing the contrary, let po(eo) > 0. Then we can define a positive mass-distribution v on E~ of total mass po(eo) (> 0), writing for every Borel set e, v(e) = po(eo. e). Since o(p, q)dv(p)dv(q) = P(p, q)dpo(p)dpo(q) eu eo eo we have v E c ii X0112 By Theorem 2, it would follow 0 < (fro, v) - f (p)dv(p) (u(p)-f (p))dv(p)= eo ~u(p) " f (p))dpo(p) <0, whence a contradiction. Theorem 5. Let E~ be the kernel of i. Then. at every rooint

8 14 S. EAMETANI. [Vol. 20, Proof. Supposing the contrary; let po be a point of Eo such that u(p0) --.f (po) > 0. The function u(p) -f(p) being lower semi-continuous on E, there would exist a neighbourhood U of po such that (9) u(p) -f(p) > 0 for every p e U E. Since Po is a point of the kernel of Po, we have po(u E) > 0. By the preceding lemma, eo= {p; p E E, u(p) < f (p)} contains no p0-mass, whence we must have by Theorem 2 0= u(p) -f(p))dpo(p)= L.eo(u(p) -r.f (p))dpo(p) (i(p) --f(p))dpo(p) E. U-e0 _ L.U(u(p) --f(p))dpo(p) > 0 (by(9)), which is a contradiction. Remark 1. If f(p) is a positive constant, then the distribution µo gives the potential which corresponds to the equilibrium potential. If f (p)= P(p, s), where s E, or more generally, if f (p) = P(p, s)di (s), where p(e, F) > 0, then the potential obtained corresponds to the one given by sweeping out the mass charging s or F. Remark 2. Having not assumed on P(p, q) none of subharmonicity, etc., the results obtained seem general enough, though there might be rooms for more precise and complete results under additional conditions. The author wishes to express his gratitude to all the members of the Mathematical Institute of Nagoya Imperial University for their close attention and kind remarks given concerning his lecture during his stay with them,