SIMULTANEOUS SOLUTIONS OF THE WEAK DIRICHLET PROBLEM Jan Kolar and Jaroslav Lukes Abstract. The main aim of this paper is a geometrical approach to simultaneous solutions of the abstract weak Dirichlet problem. We answer partially a question from the paper [2] where a similar problem was discussed from a potential-theoretical point of view for the case of function spaces consisting of harmonic functions. 1. Introduction Let A be a function space on a compact Hausdor space K. By this we mean a closed linear subspace of C(K) (the space of all real-valued continuous function on K equipped with the sup-norm jj:jj K ) containing the constant functions and separating the points of K. Let M 1 (K) denote the set of all probability Radon measures on K and " x the Dirac measure at x 2 K. Let further M x (A) be the set of all A-representing measures for x 2 K, i.e. M x (A) = f 2 M 1 (K) : f = f(x) for any f 2 Ag: The Choquet boundary Ch A (K) is dened as the set fx 2 K : M x (A) = f" x gg. We say that A is simplicial if for any closed set H Ch A (K) and any f 2 C(H) there is a function F f 2 A such that F f = f on H and kfk H = kf f k K. A function F f is called a solution of the weak Dirichlet problem for f on H. In general, it is by no means unique. Let H be subset of a topological space K and A H a subspace of C(H). By an extender L we understand any mapping L : A H! C(K) such that L(f)jH = f for any f 2 A H. We introduce main examples of function spaces. Examples. 1. Continuous functions. For A = C(K) we have Ch C(K) (K) = K and C(K) is simplicial. The existence of the solution of the weak Dirichlet problem is a consequence of the Tietze-Urysohn extension theorem. 2. Ane functions. Let K be a metrizable convex compact subset of a Hausdor locally convex space E and A the linear space A(K) of all continuous ane functions on K. In this case the Choquet boundary Ch A(K) (K) coincides with the set ext K of all extreme points of K and it is always a Borel subset of K. We call K a Choquet simplex if the function space A(K) is simplicial. 1991 Mathematics Subject Classication. 46 A 55, 31 D 05, 54 C 65. Key words and phrases. weak Dirichlet problem, function space, linear extender for Choquet simplexes, Lazar selection theorem, Borsuk-Kakutani and Borsuk-Dugundji theorems. Research supported by the grant GAUK 165/99 and CEZ:J13/98:113200007. Typeset by AMS-TEX 1
2 JAN KOL AR AND JAROSLAV LUKES 3. Harmonic functions. Let be a bounded open subset of the Euclidean space R n, K the closure of and A the linear space H() of all continuous functions on which are harmonic on. In this case H() is simplicial and the Choquet boundary of H() consists of all regular points of. More generally, let be a relatively compact open subset of an abstract harmonic space and H() again a linear subspace of C() of functions which are "harmonic" on. J. Bliedtner and W. Hansen showed in [1] that H() is simplicial and characterized the Choquet boundary of H() (which is in this case contained in the set of all regular points of and the inclusion can be proper). In a subsequent paper [2] they proved that for any closed set H Ch H() () there is even a linear isometric extender L : C(H)! H() and posed a question whether a linear isometric extender exists in more general situations. We will answer their question positively for the case of simplicial function spaces. 2. Results In what follows, 1 A will denote the constant function having the value 1 on a set A, co B and co B the convex hull and the closed convex hull of a set B, respectively. Theorem A. Let K be a Choquet simplex and H a closed metrizable subset of ext K. Then there exists a positive linear isometric extender L : C(H)! A(K) such that L(1 H ) = 1 K. Proof. Considering C(H) equipped with the w -topology, the set M 1 (H) of all probability measures on H is a convex w -compact subset of C(H). Since H is assumed to be metrizable, C(H) is separable and there exists an invariant metric on C(H) which is on bounded sets (hence also on M 1 (H) ) compatible with the w -topology. Although is not itself complete, the completion Y of (C(H) ; ) is a Frechet space and M 1 (H) is a compact subset of Y. Let the multi-valued map : K! M 1 (H) be dened as (x) = f" x g M 1 (H) for x 2 H and (x) = M 1 (H) for x 2 K n H. Since H ext K, it follows that is ane: For any 2 (0; 1) and x; y 2 K we have (x) + (1? )(y) M 1 (H) = (x + (1? )y). Since H is closed and the mapping " : x 7! " x 2 M 1 (H) is continuous, the multi-valued mapping is lower semi-continuous on K. Indeed, for any open set G, G\M 1 (H) 6= ;, the set?1 (G) = (K nh)["?1 (G) is open, otherwise it is empty. The Lazar selection theorem ([4], Theorem 3.1) tells us that has a continuous ane selection : K! (M 1 (H); w ). Let x := (fxg) for x 2 K. Then for any f 2 C(H), the function L(f) : x 7! x (f) is continuous and ane on K. Since x = " x for any x 2 H, L(f) is an extension of f. As x is a probability measure on H, we have L(1 H ) = 1 K and L(f) 0 whenever f 0. Hence L : f 7! L(f) is a positive linear isometry from C(H) to A(K). Corollary. Let K be a metrizable compact convex subset of a Hausdor locally convex space. Then the following conditions are equivalent: (i) K is a Choquet simplex, (ii) for any closed set H ext K and any function f 2 C(H) there exists F 2 A(K) such that Fj H = f and kfk H = kfk K,
SIMULTANEOUS SOLUTIONS OF THE WEAK DIRICHLET PROBLEM 3 (iii) for any closed set H ext K there exists a positive linear isometric extender L : C(H)! A(K). Proof. The implication (i) ) (iii) follows from the Theorem A, (iii) ) (ii) is trivial and (ii), (i) is a well known characterization of Choquet simplexes. Theorem B. Let K be a compact Hausdor topological space, A a closed simplicial subspace of C(K) and H a compact metrizable subset of the Choquet boundary of A. Then there exists a positive linear isometric extender L : C(H)! A such that L(1 H ) = 1 K. Proof. It is well known that K can be embedded in the state space S := f 2 A : kk = 1 = (1 K )g in such a way that functions from A are in a one-to-one correspondence with (restrictions of) ane continuous functions A(S) on S. In this setting H Ch A (K) is to be understood as a subset of ext S. From the Theorem A it follows that there is a positive linear isometric extender L e : C(H)! A(S). Let L(f) be the function from A corresponding to L(f) e (i.e. restriction of L(f) e to K). Then L : C(H)! A is an extender we looked for. Corollary. Let K be a metrizable compact Hausdor space and A a function subspace of C(K). Then the following conditions are equivalent: (i) A is simplicial, (ii) for any closed subset H of the Choquet boundary Ch A (K) and for any function f 2 C(H) there is F 2 A such that Fj H = f and kfk H = kfk K, (iii) for any closed subset H of the Choquet boundary of A there exists a positive linear isometric extender L : C(H)! A. 3. Final remarks and open problems Remarks. 1. Notice that our Theorem B is a generalization of the theorem II.4.14 of [5]. The last mentioned theorem from [5] bears usually the Borsuk-Kakutani name and concerns linear extenders in the case when K is a compact Hausdor topological space, H a metrizable subset of K and A = C(K) the function space of all continuous functions on K (our Example 1). The Borsuk-Kakutani theorem fails to be valid if the metrizability condition of H is omitted. For example, if K is the Stone-Cech compactication N of N and H = NnN, then there is no continuous linear extender from C(H) to C(N). It seems to be an open problem to characterize those compact Hausdor spaces where the analogy of the Borsuk- Kakutani theorem holds. 2. As mentioned in the Introduction, the case of linear extenders for the spaces of harmonic functions is treated in [2]. The authors use pure potential-theoretical arguments. Even we answered their question on the existence of linear extenders for simplicial spaces it would be interesting to examine more general case of function cones of lower semi-continuous functions on locally compact Hausdor spaces. 3. The proof of the Theorem A could be done also using [4], Corollary 3.7 under the assumption that the set co H is metrizable. But the metrizability of the set H in consideration implies the metrizability of co H (a personal communication of V. Fonf). Having
4 JAN KOL AR AND JAROSLAV LUKES this in mind, we can argue, for example, as follows. Let K be a Choquet simplex and H a metrizable closed subset of ext K. Any continuous function f on H can be uniquely extended to co H resulting in a continuous ane function. Since K is a Choquet simplex, co H is a metrizable closed face of K and we can apply another Lazar's theorem from [4] (Corollary 3.7) to get a requisite linear extender. A version of the result can be also obtained using an abstract theory of M-ideals (see [8]). 4. When we left the assumption of compactness of the space, the question of linear extenders is more delicate. The classical Borsuk-Dugundji theorem says that the linear extender L from C(H) to C(K) such that L(f)(K) co(f(h)) for any f 2 C(H) exists provided K is a metric space and H is a closed subset of K. (A topological space X is said to have the Dugundji extension property if for every nonempty closed subset H of X there exists such an extender. Thus metrizable spaces have the Dugundji extension property as do have stratiable spaces.) 5. Consider now the density topology on the real line R and a Lebesgue null set H R. It is known that any function on H which is the restriction of a Baire-one function from R can be extended to a density-continuous function on R (cf. [6], Theorem 8.1.b). Nevertheless the existence of linear extenders is not assured in general. The rst named author of this paper proved in [3] the following theorem: Let H R be a Lebesgue null set, A the space of all bounded density-continuous functions on R, B 1 the space of all bounded Baire-one functions on R and A H = B 1 H. Then there is a continuous (or positive) linear extender from A H to A if and only if the set H is scattered. 6. In a paper of E. Michael and A. Pe lczynski [7] there are other theorems on linear extenders for function spaces. They are based mainly on the so-called bounded extension property. Since the simplicial function spaces have this property rarely it would be interesting to derive some more general results. References 1. Bliedtner J. and Hansen W.: Simplicial cones in potential theory, Inventiones Math. 29 (1975), 83{110. 2. Bliedtner J. and Hansen W.: The weak Dirichlet problem, J. Reine Angew. Math 348 (1984), 34{39. 3. Kolar J.: Simultaneous extension operators for the density topology (to appear). 4. Lazar A.: Spaces of ane continuous functions on simplexes, Trans. Amer. Math. Soc. 134 (1968), 503{525. 5. Lindenstrauss J. and Tzafriri L.: Classical Banach Spaces, Lecture Notes in Math. 338, Springer- Verlag, 1973. 6. Lukes J., Maly J. and Zajcek L.: Fine topology methods in real analysis and potential theory, Lecture Notes in Math. 1189, Springer-Verlag, 1986. 7. Michael E. and Pe lczynski A.: A linear extension theorem, Illinois J. Math. 11 (1967), 563-579. 8. Werner D.: Some lifting theorems for bounded linear operators, Functional analysis (Essen, 1991) (1994), Lecture Notes in Pure and Appl. Math., 150, 279{291. Jan Kolar, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 18675 Praha 8, Czech Republic
SIMULTANEOUS SOLUTIONS OF THE WEAK DIRICHLET PROBLEM 5 E-mail address: kolar@karlin.m.cuni.cz Jaroslav Lukes, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 18675 Praha 8, Czech Republic E-mail address: lukes@karlin.m.cuni.cz