TitleOn the kernel of positive definite. Citation Osaka Journal of Mathematics. 17(1)

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1 TitleOn the kernel of positive definite Author(s Ninomiya, Nobuyuki Citation Osaka Journal of Mathematics. 17(1 Issue 1980 Date Text Version publisher URL DOI Rights Osaka University

2 Ninomiya, N. Osaka J. Math. 17 (1980, ON THE KERNEL OF POSITIVE DEFINITE TYPE NOBUYUKI NINOMIYA (Received March 31, 1979 In a locally compact Hausdorff space, let k(p, Q be a real-valued function, continuous for any points P O, may be oo for P=Q always finite for PΦζ, n(p, Q a real-valued function finite continuous for any points P Q. A complex-valued function K{P,Q = k(p,q+in(p,q is said to be of positive definite type if the double integral (called energy integral \\κ{p,qd σ {Qde{P of any complex-valued measure σ supported by a relatively compact Borelian set, whenever it is finitely determined, is non-negative. As is well-known, any function K(P, Q of positive definite type is symmetric: K(P,Q = K{Q,P i.e. k(p,q = k(q,p n(p,q =-n(q,p, ^0 I K(P,Q ^ sup K(P y P for any points P Q. In the real function theory, we see some results which characterize functions of positive definite type. In the present paper, we shall try to characterize functions of positive definite type on the point of view of the potential theory. We shall advance the argument adopting an idea a method in the previous paper [2]. For any measure a β (real-valued or complex-valued supported by a relatively compact Borelian set, consider the potential taken with respect to a kernel K(P, Q K(P, a = J K{P y Qda{Q y K(a, P = \ K(Q } Pda(Q the double integral (called mutual energy integral K(a,β=\da(P\κ(P,Qdβ(Q.

3 188 N. NINOMIYA Similarly, we shall consider k(p, a, k(a, P, k(a, /?, n(p, α, n(a y P n(a, β with respect to kernels k(p, Q n(p, Q. We are going to prove following theorems. Theorem 1. Suppose that a kernel K(P, Q is symmetric. A necessary sufficient condition that the kernel K(P, Q is of positive definite type is that the following property is satisfied: [PJ Let E ly F 1 be compact sets, E x E 2 being disjoint, being disjoint. Let μ ly, v λ be positive measures supported by E ly E 2, F 1 respectively with total mass a ly b x b 2 (= 1 respectively. For certain constants A B, if there hold inequalities k{p, μi^k(p, μ^- n (v x -i P+A on the support of μ λ k(p, v x ^k(p, -n{p y μi - +B on the support of v ly then there holds the inequality n(9 μi +a 1 A+b 1 B. Furthermore, consider the kernel of positive definite type in a stronger form. DEFINITION. A kernel K(P, Q of positive definite type is said to satisfy the energy principle, if the double integral (called energy integral of any complex-valued measure σ supported by a relatively compact Borelian set, whenever it is finitely determined, is non-negative vanishes only when σ = 0. Then, we have: Theorem 2. Suppose that a kernel K(P, Q is of positive definite type. A necessary sufficient condition that the kernel satisfies the energy principle is that the following property is satisfied: [P 2 ] Let E ly E 2, F 1 be relatively compact Borelian sets, E λ E 2 being disjoint, being disjoint. Let μ ly y v x be positive measures supported by E ly respectively with total mass a ly b λ b 2 (= 1 respectively. For certain constants A B y if there hold inequalities k(p, μi ^k(p, μ^-n^-v^ P+A

4 KERNEL OF POSITIVE DEFINITE TYPE 189 almost everywhere with respect to μ λ k{p, v x ^k{p } -n(p y μi - +B almost everywhere with respect to v ly then there holds the inequality n(v 1 y n(v μ μ +a A+b B It is the following lemma that is important to prove the theorems. For any complex-valued measure supported by a relatively compact Borelian set we have σ = μ+iv, K(σ, a = j j {k(p, Q+in{P, Q} {d β {Q+idv{Q {dμ{p-idv{p}. K{P, Q being symmetric, i.e. we have further k(p,q = k(q,p n(p, Q = -n(q, P, K{σ y σ = k{μ, μ+k(v, v+2n{v y μ. When μ is a real-valued measure of variable sign, it's positive part μ + negative part μ" in Harm's decomposition are positive measures supported by disjoint relatively compact Borelian sets respectively. This is similar with respect to v y too. Hence, for relatively compact Borelian sets E u F 2, E 1 E 2 being disjoin 4 :, F 1 being disjoint, each of E x E 2 possibly intersecting with F 1, each of possibly intersecting with E 1 for positive measures μ l9 f v x supported by E ly respectively, consider the quantity 2n{v x y μ x = k(μ u μi 2k(μ 1} +k(, +k(v ly v λ 2k(v ly +k(y +2n(v 1 y μi~ We are going to study the minimum of this quantity. We should like to suppose that each of E ly is of β-transfinite diameter positive 1 *. Lemma. Let E ly be relatively compact Borelian sets as stated 1 For a symmetric kernel k(p, Q, a compact set is said to be of fc-transfinite diameter positive if it supports a positive measure with total mass 1 of ^-energy integral finite. A Borelian set is said to be of /ί-transfinte diameter positive when it contains a compact set of k- transfinite diameter positive. Concerning the relation between the transfinite diameter energy integrals, see for instance [1] (p. 45.

5 190 N. NINOMIYA above, μ/ y y v x ' positive measures supported by E ly ί\ respectively with total mass a ly b x b 2 (=1 respectively. Suppose that, among all the pairs (μι,, v x ' y of such measures, a pair (μ ly y v ly makes minimum of I{μι, \ v x ' y \ Then, we have (En ; v ly V 2 ^l{μ x f ', '\ i//, v{. Ai^Le x (F on E x except for a set of k-capacity zero 2 \ where e x {P = k(p, μi-k(p, μj+n(v λ -y P (E 12 e 1 (P^A 1 on E 1 almost everywhere with respect to μ x. When E λ E 2 are disjoint, the exceptional set of k-capacity zero might be replaced by of k- transfinite diameter zero. Similary, putting e 2 (P = k{p, -k{p, μ,-n{ Vl -, a,a 2 = j e 2 {Pd {P, P, UP = k{p, v 1 -k(p, +n(p, μi-, we have UP = k(p, -k{p, Vl-n(P, μi-, b 2 B 2 = B 2 =\f 2 {Pd {P, (E 21 A 2^e 2 (P on E 2 except for a set of k-capacity zero, (E 22 e 2 (PtίA 2 on E 2 almost everywhere with respect to y (F u B^f^P on except for a set of k-capacity zero, (Fi 2 f 1 (P^B 1 on Fi almost everywhere with respect to v ly (F 21 B 2^f 2 (P on F 2 except for a set of k-capacity zero, (F 22 f 2 (P^B 2 on F 2 almost everywhere with respect to. When E x E 2 are disjoint disjoint, the exceptional sets of k-capacity zero might be replaced by of k-transfinite diameter zero. We are going to prove (E π (E 12 only. Take any positive measure 2 A compact set is said to be of ^-capacity positive if it supports a positive measure with total mass 1 whose Λ-potential is bounded from above on any compact set. A Borelian set is said to be of ^-capacity positive when it contains a compact set of A-capacity positive. Evidently, if a set is of Λ-capacity positive, it is of A-transfinite diameter positive.

6 KERNEL OF POSITIVE DEFINITE TYPE 191 cίi supported by E x with total mass^ such that k{a u tfi< k(a u Then, we have for any positive number 6 smaller than 1 an inequality which, in consideration of the symmetricity of k{p, Q y induces an inequality (2 Sk(μ ly μι Making ->0, we have ly μι -2k{au hence almost everywhere with respect to any positive measures CL\ supported by E x with total mass a x such that k(cd, (Xi< o k(a u < ' The inequality naturally holds for θίi=μ\ there is an evident equality = J so we have almost everywhere with respect to μ x. When E λ E 2 are disjoint, the inequality (E n holds with respect to any positive measure a.\ of ^-energy integral finite supported by E x with total mass a x. Then, the exceptional set might be considered as of &-transfinite diameter zero. REMARK. of«(p,0. The proof of Lemma never takes need of the anti-symmetricity Proof of Theorem 1. First, we shall prove that if a kernel K(P, Q is of positive definite type, it satisfies the property [PJ. Let E ly be compact sets in the property [PJ μ u, v λ positive measures in the same manner. For certain constants A B, suppose that there hold inequalities on the support of μ x k(p, μ^kip μ^-n^-v^p+a Λ(P, v x ^k{p, -n{p, μi - +B on the support of v v We have naturally inequalities

7 192 N. NINOMIYA Unless we have the result of the property [PJ? we have the inequality a ι A+b ι B<k{y μi+k(, v^ k{, k(y 2y +n(v 1 P 2j +n(9 μ λ Then, we have from those three inequalities which is a contradiction. Next, we shall prove that if a kernel K{P, Q satisfies the property [PJ, it is of positive definite type. Let E ly be relatively compact Borelian sets, E 1 E 2 being disjoint, being disjoint. We suppose that each of E ly F 1 contains a compact set of ^-transfinite diameter positive. We are going to prove whenever it is finitely determined, for any pair (μ/ y y v{ y of positive measures supported by E ly respectively with total mass a ly b x b 2 (=l respectively. Consider the case when E u Έ 2y all are compact sets. Then, we have a pair that gives minimum of I(μ/, ; v/, ' among all the pairs of four positive measures stated above. Since, putting /= inf/(μ/, '\ I is finite there exists a sequence of pairs of positive measures μ lny ny v u n supported by E ly respectively with total mass a ly b x b 2 (=l respectively such that We may suppose that the sequence {μ ίn } y {n }, {^i«} {n } all are vaguely convergent by means of taking suitable sub-sequences out of them. Let μ ly y v γ be their limiting measures respectively. They are positive measures supported by E ly respectively with total mass a ly b γ b 2 (= 1 respectively. The ^-potential of positive measures with compact supports being lower semi-continuous in the whole space, finite continuous outside of their supports the w-potential being finite continuous everywhere, we have inequalities μi^ lim k(μin> μu > n

8 KERNEL OF POSITIVE DEFINITE TYPE 193 those similar with respect to v x further an equality So, we have «(*Ί *2> ^1 ^2 = limn^ ny μ ln n Then, by Lemma we have inequalities on the support of μ λ k(p, μi^k(p, -n{v 1 -, P+A ι k(p y v x ^MP, -n(p } μ ι - +B ι on the support of v v Therefore, we have an inequality similarly, an inequality +n(p 2, μι ^ n(v ly μ x ^ Thus, we have the inequality looking for: 0 2(a ι A ι +b ι B ι +a 2 A 2 +b 2 B 2 = 2I(μ u, v u. Finally, suppose that some of E u are not compacts. Let μ u y v λ be positive measures supported by E ly E 2, respectively with total mass a u a 2, b x δ 2 ( 1 respectively. Then, taking sequences of compact sets {E ln }, {E 2n }, {F ln } {F 2n } such that u c 12 C...c lλ C...c 1, μi(e ln f a,, 2n^. t *i, taking restrictions μ lny ny v u n of μ ly y v x to E lny E 2ny F u n respectively, we have making n-*oo

9 194 N. NINOMIYA Proof of Theorem 2. First, we shall prove that if a kernel K(P, Q is of positive definite type satisfies the energy principle, it satisfies the property [P 2 ]. Let E ly be relatively compact sets in the property [P 2 ] μi> μ2> v ι an d V 2 positive measures in the same manner. For certain constants A B y suppose that there hold inequalities k(p y μi ^k(p, -n(v ι -> P+A almost everywhere with respect to μ x k(p y v^kip, -n(p y μi - +B almost everywhere with respect to v v We have naturally inequalities Unless we have the result of the property [P 2 ], we have the inequality ^A+biB^tyμ^ μi+k(y Vi k(, μ^+nfa v^ ~k(y 2, +n(, μ x. Then, we have from those three inequalities which is a contradiction. Next, we shall prove that, if a kernel K(P, Q is of positive definite type satisfies the property [P 2 ], it satisfies the energy integral. Let E u E 2, F 1 be relatively compact sets, E λ E 2 being disjoint, F 1 being disjoint. Let μ lf y v λ be positive measures supported by F l9 E 2J F 1 respectively with total mass a ly b x b 2 (=l respectively. We are going to prove whenever it is finitely determined. If the pair (μ ly y v ly is a minimal pair in Lemma there hold inequalities k{p y μι Sk(P y μ^-niy^v^ F+A ι almost everywhere with respect to μ λ k(p y v x ^k{p, -n{p y μ 1 - +B 1

10 KERNEL OF POSITIVE DEFINITE TYPE 195 almost everywhere with respect to v x. inequality Then, by the property [P 2 ] we have an Accordingly, we have (, Vi<k(, n(v 1 y +k(y n{> μi μ^+a^+b^. which is a contradiction. Let us consider the kernel of positive definite type in a weaker form. DEFINITION. A complex-valued function = k(p,q+in(p,q is said to be of positive definite type in restricted sense, if the double integral (called energy integral \\κ(p,qd σ (Qdά(P of any complex-valued measure σ supported by a relatively compact Borelian set with total mass 0, whenever it is finitely determined, is non-negative. DEFINITION. A kernel K(P, Q of positive definite type in restricted sense is said to satisfy the energy principle, if the double integral (called energy integral "κ(p,qd σ (Qdσ(P of any complex-valued measure σ supported by a relatively compact Borelian set with total mass 0, whenever it is finitely determined, is non-negative vanishes only when σ=0. For a kernel of positive definite type in restricted sense, Theorems 1 2 are expressed in the following styles. Theorem Γ. Suppose that a kernel K(P, Q is symmetric. A necessary sufficient condition that the kernel K(P, Q is of positive definite type in restricted sense is that the following property is satisfied: [Pi] Let E lf be compact sets, E x E 2 being disjoint, being disjoint. Let μ λ be positive measures supported by E x E 2 with total mass a (>0 respectively v λ positive measures supported by with total mass 1 respectively. For certain constants A B, if there hold inequalities k(p, μi ^k(p, -n(y x -,

11 196 N. NINOMIYA on the support of μ λ k(p, v λ ^k(p, -n(p, μι - +B on the support of v u then there holds the inequality #( > /*i+%2, v λ ^k{y n(v λ v u +k(y n(y μ ι +aa+b. Theorem 2' Suppose that a kernal K(P, Q is of positive definite type in restricted sense. A necessary sufficient condition that the kernel K(P, Q satisfies the energy principle is that the following property is satisfied: [P ] Let E ly be relatively compact Borelίan sets, E x E 2 being disjoint, being disjoint. Let μ λ be positive measures supported by E x E 2 with total mass a (>0 respectively v x positive measures supported by F 1 with total mass 1 respectively. For certain constants A B, if there hold inequalities k(p, μi ^k(p, -n( Vl - V2, P+A almost everywhere with respect to μj k{p, Vl ^k{p, -n{p, μi - +B almost everywhere with respect to v l9 then there holds the inequality ^l<k/^2y ^ n^ V^ (y n(, μ x +aa+b. Corollary. A kernel K(P, Q, which is symmetric, is of positive definite type in restricted sense if the following property is satisfied: [P*] Let E u E 2, Fι be relatively compact Borelian sets, E x E 2 being disjoint, F 1 being disjoint. Let μ x be positive measures supported by E 1 E 2 with total mass a (>0 respectively v x positive measures supported by F 1 with total mass 1 respectively. For certain constants A B, if there hold inequalities k(p, μi^k(p, - n (v x -y P+A on the support of μ λ k{p, v 1 ^k(p, -n(p, μi- +B on the support of v ly then these two inequalities hold at the same time in the whole space. Using this corollary, we should like to terminate the paper presenting a simple example of a kernel satisfying the above property [P*], therefore of positive definite type in restricted sense.

12 KERNEL OF POSITIVE DEFINITE TYPE 197 EXAMPLE. On the plane i? 2, let [Xi, x 2, \J yyii y 2, n(p, Q = c 1 (x 1 y 1 +c 2 (x2 y 2 > c x c 2 being any real constants. A complex-valued function, which is symmetric, K(P,O = k(p,q+in(p,q is of positive definite type in restricted sense. In fact, let E lf E 2, be compact sets, E x E 2 being disjoint, being disjoint. Let μ x be positive measures with total mass β(>0 whose supports are E x E 2 respectively, v λ be positive measures with total mass 1 whose supports are respectively. For certain constants A B y suppose that there hold inequalities on E 1 k(p, μι^k{p y μ2 -n(v γ -y P+A k(p, v x ^k{p, -n{p > μ ι - +B on E 2. We are going to prove that the former inequality holds everywhere. Then, we shall see that the latter one holds everywhere, too. E x being a compact set of logarithmic capacity positive, let λ be the equilibrium measure on E x. That is, λ is a positive measure supported by E x with total mass 1 such that ik(q is equal to a constant V on Έ x except for a set of logarithmic capacity zero, ^ V everywhere. The logarithmic potential of a positive measures with compact support is superharmonic in the whole plane harmonic in each component outside of the support, the w-potential is harmonic in the whole plane. Then, being any positive number, consider the function g{p = k(p, μi + k(p, λ-λ(p, +n(v 1 -p a9 P-(A+SV. This is subharmonic in each component outside of E l9 we have at each boundary point M of E λ

13 198 N. NINOMIYA P being outside of E x P' being in E x. We should like to prove lim (P=-oo. First, we have limk(p, λ= oo. Next, we have, 0 denoting the origin, lim {k(p> μλ MP, = lim {I log dμλq \ log d (Q \ = 0 p-> p->«u PO J PO Finally, let us notice that n{v ι y P is bounded. Since, taking the Dirac measure at the origin 0, we have In(y x -, P I = In(P, ^-n(p, P 2 \ \n(p, v x -n(p y + n(p, ^2»n(P, 6 So, we have in each component outside of Eχ Making -»(, we have k(p, ^k(p, μ -n(v -p, P+A μi everywhere. QUESTION. A kernel K{P, Q which is of positive definite type in restricted sense but is not symmetric, does it exist? References [1] O. Frostman: Potentiel d'equilibre et capacite des ensembles avec quelques applications a la theorie des functions, Medd. Lunds Univ. Mat. Sem. 3 (1935, [2] N. Ninomiya: On the potential taken with respect to complex-valued symmetric kernels, Osaka J. Math. 9 (1972, 1-9. Department of Mathematics, Osaka City University Sugimoto-cho, Sumiyoshi-ku, Osaka 558, Japan 3 For a kernel k(p, Q logarithmic on R 2, Newtonian in R 3 or generally satisfying the maximum principle of Frostman, the potential k(p, β of a positive measure β with compact support satisfies at each boundary point M of the support of β an inequality P being outside of the support of β P' being on the support of β (cf.: [1], p. 69.