OF A RESISTANCE FORM

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OF A RESISTANCE FORM

HABIB BENFRIKA, ILEANA BUCUR, ANTONIO NUIC ˘A and SPERANT¸ A VL ˘ADOIU

Given a resistance form (ε,F, X) on a spaceX, we present a sufficient condition forX to be semisaturated in terms of resistive distance associated with (ε,F, X).

AMS 2000 Subject Classification: 31D05, 60J45, 60J35, 60J40.

Key words: resistance form, energy, fine topology, natural topology, semisaturated set.

1. PRELIMINARIES

We recall (see [2]) that a resistance form is a system (ε,F, X) whereXis an arbitrary set whose cardinal is greater than 2, F is a pointwise real vector lattice of real functions on X which contains the constant functions onX and separates the points of X,ε :F × F →R is a symmetric bilinear form such that the conditions below are fulfilled:

(1) (ε(f, f)≥0,∀f ∈ F

and ε(f, f) = 0⇔f is a constant function);

(2)ε(f, g)≤0 iff, g ∈ F and f∧g:= inf(f, g) = 0;

(3) there exists a pointx₀ ∈X such that the linear vector spaceF_x₀ defined by F_x₀ ={f ∈ F |f(x0) = 0} endowed with the scalar product hf, gi= ε(f, g)∀f, g∈ F_x₀ is a Hilbert space;

(4) the convex coneF_x⁺

0 of all positive functions ofF_x₀ is a closed subset of the Hilbert space (F_x₀,h·,·i).

It is known (see [2]) that any linear functional ϕ : F_x₀ → R which is increasing with the pointwise order relation is continuous on F_x₀, i.e., there exists “the potential g^ϕ” of the functionalϕ,g^ϕ∈ F_x₀ such that

ϕ(f) =ε(g^ϕ, f) =ε(f, g^ϕ), ∀f ∈ F_x₀.

In particular, for any point x∈X there existsg^x∈ F_x₀ such that f(x) =ε(g^x, f) =ε(f, g^x), ∀f ∈ F_x₀.

Sometimes, the function g^x is called the potential onX with pole at x.

REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 407–415

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Since F_x₀ is a pointwise vector lattice of real functions, we have the relations

inf(ε(f, g^x), ε(h, g^x)) =ε(f∧h, g^x), ∀f, h∈ F_x₀, sup(ε(f, g^x), ε(h, g^x)) =ε(f ∨h, g^x), ∀f, h∈ F_x₀.

In the sequel we always assume that the space X endowed with the metric d given by d(x, y) =kg^x−g^yk, x, y∈X, is complete.

Remark. If there exist a filterF onX and a pointy∈Xsuch thaty6∈A for some element A of this filter, and for anyf ∈ F we have lim

F f(x) =f(y), then the system (ε⁰,F⁰, X⁰) given by X⁰ = X \ {y}, F⁰ = {f|_X⁰;f ∈ F }, ε⁰ : F⁰ × F⁰ → R, ε⁰(f|_X⁰, g|_X⁰) := ε(f, g) also is a resistance form on X⁰ since any element of F is entirely determined by its restriction to X⁰. So, it is natural to assume the following type of completion of X: if there exists a filter F onX such that for anyf ∈ F the filterf(F)onR converges to a real number ϕ(f), then we add to the spaceX the pointϕand extend the functions f ∈ F to the space X∪ {ϕ} as

f(ϕ) := lim

F f(x).

If we consider the functionf on X∪ {ϕ} defined by f(y) =

(f(x) ifx∈X, limF f(x) ify=ϕ,

and define a bilinear form εeon the linear space F = {f | f ∈ F } by the equality

ε(f , g) =e ε(f, g), ∀f, g∈ F,

then the system (ε,eF, X∪{ϕ}) is again a resistance form which does not differ from the starting resistance form (ε,F, X). In the sequel we denote

S={s∈ F_x₀ |ε(s, h)≥0,∀h∈ F_x⁺

0}.

Obviously, S is a closed, convex subcone of F_x₀ such that the linear space S−S is dense in the Hilbert spaceF_x₀. We recall (see [2]) that ifs∈S then s≥0 onX. Indeed, by the properties of a resistance form we have

ε(s−, s−) =ε(s−,−s+s+) =−ε(s₋, s) +ε(s−, s+)≤ −ε(s₋, s)≤0, s≥0.

We remark that for any x ∈ X the Green potential g^x belongs to S and, moreover, g^x lies on an extreme ray of S, i.e., ifs1, s2 are two elements of S such that s₁ +s₂ = g^x, then there exists a real number α, 0 ≤ α ≤1, such that s1=αg^x, s2 = (1−α)g^x.For more details see [2].

It is known (see [2], [3]) thatS has the following properties:

a)s1, s2 ∈S⇒s1∧s2∈S;

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b) for any decreasing family (si)i∈I ofS the function∧_i∈Isi defined by (∧i∈Is_i)(x) := inf

i∈Is_i(x), x∈X, belongs to S and lim

i,I ks_i − ∧_i∈Isik = 0 following the section filter of the family (si)i∈I;

c) for any increasing family (si)i∈I of S which is pointwisely dominated by an element f ∈ F_x₀, the function∨_i∈Is_i defined by

(∨_i∈Is_i)(x) := sup

i∈I

s_i(x), x∈X, belongs to S and lim

i,I ks_i − ∨_i∈Isik = 0 following the section filter of the family (s_i)i∈I;

d) for anys, s1, s2 inS such thats≤s1+s2 there exists⁰₁, s⁰₂ inS such that s=s⁰₁+s⁰₂,s⁰₁≤s₁, s⁰₂ ≤s₂;

e) inf(α, s)∈Sfor anys∈S and any positive constant functionαonX0. Hence S is an H-cone in the sense of [3]. Moreover, the restriction to S×S of the “energyε” has the properties:

(1) for any s∈ S the mapt → ε(s, t) defined onS is additive, increasing, positively homogeneous and continuous in order from below, i.e., for any increasing and dominated family (si)i∈I in S, the family (ε(s, si))i increases toε(s,∨_i∈Is_i);

(2) for any decreasing family (si)i∈I inS, the family (ε(s, si))i decreases to (ε(s,∧_i∈Is_i));

(3) ifϕ:S →R+ is an additive, positively homogeneous and increasing map, then there exists (and is uniquely determined) sϕ∈S such that ϕ(s) = ε(s_ϕ, s) ∀s∈S.

2. EXTENSION OF THE ENERGY FORM

We shall further denote bySthe set of all functionst:X0:=X\ {x₀} → [0,∞] such that for anys∈S the function t∧sdefined by

t∧s(x) = inf{t(x), s(x)}, ∀x∈X₀,

belongs to S,t(x0) = 0 and the set [s <∞] ={x∈X|s(x)<∞}is dense in X with respect to the “weak topology” onX, i.e., the coarsest topology onX making continuous any function f ∈ F_x₀. Since the set S−S is dense in the Hilbert space F_x₀, for anyf ∈ F_x₀ we may consider a sequence (d_n)_n in the linear space S−S such that lim

n→∞kf −d_nk= 0. Using the continuity of the reduite operators (see [3]) we have

n→∞lim kR(f−d_n)k= 0, d_n−R(f −d_n)≤f, ∀n∈N.

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Hence the function f is the limit in the Hilbert space F_x₀ of a sequence (s⁰_n− s⁰⁰_n)n such thats⁰_n−s⁰⁰_n≤f,∀n∈N. We consequently have

limn ε(s⁰_n−s⁰⁰_n, s) =ε(f, s) = sup

n

ε(s⁰_n−s⁰⁰_n, s), ε(s⁰_n−s⁰⁰_n, s)≤ε(f, s), ∀s∈S.

Let us consider on S the “fine topology”, i.e., the coarsest topology on S for which for anyu∈Sthe function defined onS bys−→^b^u ε(u, s) is made continuous. From the previous consideration we deduce that for anyf ∈ F_x₀ the functions−→^f^b ε(f, s) onSis lower semicontinuous with respect to the fine topology.

Proposition1. The fine topology onS coincides with the trace onS of the weak topology of the Hilbert space F_x₀.

Proof. The assertion follows from the previous considerations, since for any f ∈ F_x₀ the functions fband −fb are both lower semicontinuous with respect to the fine topology.

We recall that thefine topology on X is the coarsest topology on X for which any function s∈S is continuous onX.

Corollary 2. The fine and the weak topology on X coincide.

Proof. The assertion follows from definitions and Proposition 1.

We know (see [2]) that for any subsetA of X the map onS given by s→B^As:=∧{s⁰ ∈S |s⁰≥sonA}

is a balayage on S, i.e., it is additive, positively homogeneous, increasing, idempotent, contractive, i.e., B^As ≤sfor all s∈ S, and continuous in order from below. Moreover, we have

BÂs=son A, ∀s∈S, ε(BÂs, s⁰) =ε(s, BÂs⁰), ∀s, s⁰ ∈S.

Therefore, for any x∈X and any A⊂X, with x∈A we have

ε(BÂg^x, s) =ε(g^x, BÂs) =BÂs(x) =s(x) =ε(g^x, s), ∀s∈S,

i.e., B^Ag^x=g^x. In particular, B^{x}g^x=g^x,∀x∈X andg^x(y)≤g^x(x) for all y ∈X. The last assertion follows from the fact that for anys∈Sthe function s∧1X belongs to S.

Proposition 3. Any elementt∈S is finite on X.

Proof. If we suppose that y ∈ X is such that t(y) = +∞, then by the previous considerations we have

1 nt

∧g^y ∈S,

1 nt

∧g^y

(y) =g^y(y), 1

nt

∧g^y ≥B^{y}g^y =g^y

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for anyn∈N,n6= 0. Sincetis finite on a fine dense subset ofX, we deduce that g^y = 0 on a fine dense subset ofX, i.e.,g^y ≡0. But in this case for anyf ∈ F_x₀ we have f(y) = 0, i.e.,y=x0, which contradicts the fact that t(y)>0.

Remark. On account of Proposition 3, we may consider onS a topology called the “natural topology”, namely, the coarsest topology on S for which the functions on S defined by S 3t → t(y), ∀y ∈X, is made continuous. If we identify any element x of X with the element g^x of S (S ⊂S), we obtain a new topology on X called again “natural topology”. Obviously, it is coarser than the fine topology on X. However, the following assertion holds.

Proposition 4. If t : X → R+ is such that t∧s ∈ S for any s ∈ S, t(x₀) = 0and the set[t <∞]is dense inXwith respect to the natural topology, then t∈S.

Proof. Assuming the contrary we deduce, as in Proposition 3, the exis- tence of an element y ∈ X, y 6= x₀, such that the element g^y is zero on a naturally dense subset ofX. Butg^y(y)>0 and the set [g^y >0] is a nonempty naturally open subset of X.

Definition 5. For anyu, v∈S we denote byε(u, v) the element fromR+

defined as ε(u, v) = sup{ε(s, t)|s, t∈S,s≤u,t≤v}.

It is not difficult to show that

a)ε(u, v) =ε(u, v) if u, v∈S;ε(u, v) =ε(v, u), ∀u, v∈S;

b) ε is increasing in each variable u and v and u⁰ ≤ u⁰⁰ ⇔ ε(u⁰, v) ≤ ε(u⁰⁰, v),∀v∈S;

c)ε(u, g^x) =u(x), ∀x∈X,∀u∈S;

d) εis additive and positively homogeneous in each variable.

The following assertions can be easily verified (see also [2]).

a)S is a convex cone of positive real functions on X such that for any family (s_i)i∈I on S there exists the greatest lower bound on S denoted by

∧_i∈Isi and we have (∧_i∈Isi)(x) := inf

i∈Isi(x),∀x∈X.

b) For any increasing family (si)i in S such that sup

i∈I

si(x) <∞ for any x ∈ X (or only for any x ∈ X⁰, X⁰ dense in X with respect to the natural topology) there exists the smaller upper bound in S denoted by ∨_i∈Is_i and (∨i∈Isi)(x) := sup

i∈I

si(x),∀x∈X.

c) For any s, t1, t2 in S, s ≤ t1 +t2 there exists s1, s2 ∈ S such that s=s₁+s₂,s₁≤t₁,s₂≤t₂.HenceS is anH-cone (of functions onX\ {x₀}).

d) ε is continuous in order form below in each variable, i.e., for any increasing family (u_i)i∈I inS such that sup

i∈I

u_i(x)<∞for any x∈X we have ε(u, v) = sup_iε(ui, v),∀v∈S,where u=∨_i∈Iui.

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e) For any map θ:S →R+ which is additive, increasing, continuous in order form below such that θ(g^x)<∞ for eachx∈X, there exists (and it is uniquely determined) v∈S such thatθ(u) =ε(u, v) ∀u∈S.

From now on we suppose that there exists a countable subsetCinX such that C is dense inX with respect to the natural topology.

We remark that the countable subsetQ₊(C) ofS defined as Q₊(C) =

( X

v∈F

r_vg^v|F ⊂C, F finite, r_v ∈Q+

)

is dense in order from below in S (see [2]).

Indeed, for any elements∈S and any finite subset F of X we have B^Fs_S X

v∈S

B^{v}s=X

v∈F

s(v)

g^v(v)g^v=X

v∈F

α_vg^v, α_v ∈R+,

whereu_S vmeans that there existst∈S such thatv=u+t. SinceS is an H-cone and for each v ∈ X, the function g^v lies on an extreme ray of S, we have B^Fs= P

v∈S

βvg^v.Therefore B^Fs= sup

( X

v∈F

r_vg^v |F ⊂C, r_v ∈Q+, 0≤r_v ≤β_v,∀v∈F )

.

Further, we obviously haves= sup{B^Fs|F finite, F ⊂X} and, therefore, s= sup{u|u∈Q+(C),u≤s}.

Hence S is a standard H-cone because any element of Q₊(C) is univer- sally continuous (see [2], [3]).

Considering now the unitu₁ ofS defined by u₁(x) =

(1 ifx6=x₀, 0 ifx=x0,

we denote by Ku1 the set of all elements v from S for which ε(u1, v) ≤ 1.

It is known that the set Ku1is a compact convex subset of S with respect to the natural topology while the set X1 of all extreme elements of this compact convex set is a G_δ set ofK_u₁ with respect to the natural topology.

We have

e∈X₁ ⇒ε(u₁, e) = 1 if e6=g^x⁰ and ε(u₁, g^x⁰) = 0.

In particular, for anyx∈X we haveg^x∈X₁, i.e., if any elementxofX is identified with the element g^x of S, then we haveX ⊂X₁. Moreover, since K_u₁ is a cap of the convex coneS and the linear vector space of function onX generated byS is a vector lattice with respect to the specific order given byS, namely,f _S g⇔g=f+s for somes∈S, we deduce thatKu1 is a simplex.

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Since any element u of S is the supremum of all its minorants from Q+(C), the function bu : S → R+ defined as u(v) =b ε(u, v), v ∈ S, is lower semicontinuous with respect to the natural topology on S. Hence any such function buis a positive lower semicontinuous affine function on K_u₁.

Now, by [3], we have

ε(u⁰∧u⁰⁰, v) = inf{ε(u⁰, v⁰) +ε(u⁰⁰, v⁰⁰)|v⁰, v⁰⁰∈S, v⁰+v⁰⁰ =v}

and, therefore, we have

ε(u⁰∧u⁰⁰, e) = inf{ε(u⁰, e), ε(u⁰⁰, e)}= inf{ub⁰(e),cu⁰⁰(e)}, e∈X₁. Hence any elementu∈Smay be uniquely extended as a function onX1by

u(e) =u(e) =b ε(u, e), bu(g^x) =ε(u, g^x) =u(x), ∀x∈X.

Therefore, S becomes a standardH-cone of functions on X1. Moreover, since the order relation between these functions is given by the order relation between their restrictions toX, we deduce thatX is dense inX1 with respect to the fine topology on X₁ associated with the cone S of functions on X₁. The setX₁is the so calledsaturation ofX with respect to theH-coneSorS. The set X is called semisaturatedif the set X₁\X is polar.

Any balayageB on Sor, equally, on Smay be represented as a balayage operation on a basic set Aof X₁, i.e.,

Bu=∧_S{v ∈S |v≥_Au}=:B^Au, ∀u∈S, A={x∈X₁ |Bu=u, ∀u∈S}.

More details on the above consideration may be found in [3].

Proposition 6. Let u be an element of S. We have u ∈S if and only if ε(u, u)<∞.

Proof. Ifu ∈S obviously we have ε(u, u) = ε(u, u)<∞. Suppose now that ε(u, u)<∞. Consider the upper directed family (s_i)i∈I of all minorants fromSofu. By the properties of the extended energyε, the family (ε(s_i, s_i))i∈I

of positive numbers is bounded, upper directed and sup

i∈I

ε(s_i, s_i) =ε(u, u)<∞.

Let i, j∈I such that si ≤sj. We have

ε(sj−si, sj −si) =ε(sj, sj) +ε(si, si)−2ε(sj, si)≤

≤ε(sj, sj) +ε(si, si)−2ε(si, si) =ε(sj, sj)−ε(si, si).

From this inequality we deduce that for any ε > 0 there exists i_ε ∈ I such that ks_j−s_ik ≤ε∀i, j∈I,i, j≥i_ε.

Therefore, the family (s_i)i∈I from S is convergent to an element s∈ S following the section filter on I. Since the family (si)i∈I is upper directed we have s(x) = sup

i∈I

si(x) =u(x),∀x∈X, i.e.,u=s,u∈S.

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Theorem 7. If U is a filter in S convergent to the element u∈S with respect to the natural topology on S, then ε(u, u)≤lim inf

v,U ε(v, v).

Proof. For anyF ∈ U let us denote v_F =∧_v∈Fv.

Obviously, the family (vF)F∈U of elements from S is upper directed. By [3, Theorem 4.5.2], uis the supremum (in S) of this upper directed family.

From the properties of the extended energy fromεwe get ε(u, u) = sup

F∈U

ε(vF, vF)≤ sup

F∈U

v∈Finf ε(v, v)

= lim inf

v,U ε(v, v).

Theorem8. If there existss0∈S such thatg^x(x)≤s0(x)for anyx∈X, thenX is semisaturated with respect to theH-cone of functions fromS (orS).

Proof. Suppose that there exists s0 ∈S such thatg^x(x)≤s0(x) for any x∈X. We shall prove that the setX₁\X is polar with respect to theH-cone S of functions on X₁.

Letebe an arbitrary element of this set. If ε(e, e)<∞, by Proposition 6, elies on an extreme ray of the coneS,ε(u₁, e) = 1 and, therefore, by the initial assumptions onX we havee=g^y for some pointy ∈X. This contradicts the fact that e∈X1\X. Therefore, we haveε(e, e) = +∞. Since the set X is fine dense in X1, there exists a family (g^xⁱ)i∈I with xi ∈X which is convergent toe with respect to the fine topology onX1. Hence for anys∈S we have

s(e) :=ε(s, e) = lim

i,I ε(s, g^xⁱ) = lim

i,I s(x_i) following the section filter of I. In particular, we have

s0(e) = lim

i,I s0(xi)≥lim

i,I g^xⁱ(xi).

Since the family (g^xⁱ)i ∈ I also is convergent to e with respect to the natural topology on X1, by Theorem 7 we have

+∞=ε(e, e)≤lim inf

i,I ε(g^xⁱ, g^xⁱ)≤s₀(e), s₀(e) =∞.

Hence s₀ = +∞ on the set X₁\X, i.e., this set is polar with respect to the H-coneS of functions on X₁.

Example 9. Let us consider on the interval [0,1] of the real lineRthe set F of all absolutely continuous functions such that their derivatives belong to L²(λ), where λ is Lebesgue measure on [0,1]. We consider the bilinear form ε(f, g) =R1

0 f⁰g⁰dλonF × F.

It is not difficult to show that the system (ε,F,[0,1]) is a resistance form on [0,1]. One can show that the resistance distance between two points x, y from [0,1] is |x−y| and g^x(x) = x for all x ∈ [0,1]. But the function s0 : [0,1] → R+ defined by s0(x) = x belongs to S and we have g^x(x) ≤

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s0(x). In fact, we have g^x(x) = s0(x) for all x∈[0,1]. Hence the set [0,1] is semisaturated (in fact it is saturated). For more details see [3].

If we change the space X in the above example taking X = [0,∞), the function s0 : X → R+ defined by s0(x) = x, x ∈ [0,∞), belongs to S and s₀(x) =g^x(x) for allx∈[0,∞). The spaceX₁in this example is the set [0,∞]

and is semisaturated in this case, too.

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Received 10 April 2009 Romanian Academy

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Technical University of Civil Engineering Departament of Mathematics and Computer Science

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University of Pite¸sti

Faculty of Mathematics and Computer Sciences Str. Tˆargul din Vale nr. 1, 110040 Pite¸sti, Romania

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University of Bucharest

Faculty of Mathematics and Computer Sciences Str. Academiei nr. 14, 010014 Bucharest, Romania

svladoiu@fmi.unibuc.ro