# On the Projective Description of Weighted (LF)-Spaces of Continuous Functions

Authors: Catherine V. Komarchuk; Sergej N. Melikhov
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

We solve the problem of the topological or algebraic description of countable inductive limits of weighted Fréchet spaces of continuous functions on a cone. This problem is investigated for two families of weights defined by positively homogeneous functions. Weights of this form play the important role in Fourier analysis.


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## Body

## 1. Introduction

The problem of the projective description of the topology of inductive limits of weighted spaces of holomorphic and continuous functions has attracted the attention of mathematicians after Ehrenpreis [1] proved the fundamental principle. This problem has applications in partial differential equations and convolution equations, (ultra-) distribution theory, in the theory of quasianalytic functionals, and representations of functions by exponential series. The systematic investigation of it was started by Bierstedt and Meise and Bierstedt et al. [2, 3]. The aim of the projective description is to find the conditions under which an inductive limit coincides algebraically with its projective hull or it is its topological subspace. In the case of an inductive limit of weighted Banach spaces of continuous functions the problem of their projective descriptions is very well studied. By now effective conditions are obtained that the topology of a countable inductive weighted limit WC(X) of continuous functions coincides with the one of its projective hull CW¯(X). The algebraic identity WC(X)=CW¯(X) always holds in this case. The case of (LF)-spaces is more complicated. The development in projective descriptions of countable inductive weighted limits of continuous functions had been surveyed in [4, 5]. Abstract conditions for the algebraic and topological description of weighted (LF)-spaces of continuous functions were obtained by Bierstedt and Bonet [6]. Their realizations in concrete situations are also interesting. In this connection we mention the article of Bonet and Meise [7]. In [7] weighted (LF)-spaces of continuous functions defined by weights which arise in the theory of ultradistributions and qusianalytic functionals of Roumieu type were investigated (see Remark after Theorem 9). Bonet et al. [8] have studied the problem of the projective description of (LF)-spaces of continuous functions for weights which are connected with a convex locally closed set in CN.In our article this problem is solved for two families of weights defined by positively homogeneous functions on cones in normed linear spaces. Weights of this form play the important role in Fourier analysis.


## 2. The Problem of the Projective Description and Notations

We recall the necessary notations and definitions and state the problem of the projective description [3, 6].LetX be a locally compact Hausdorff space, and let W=(wn,k)n,k∈N be a double sequence of strictly positive continuous functions on X such that
(1)wn+1,k(x)≤wn,k(x)≤wn,k+1(x),x∈X,
for each n,k∈N. For an upper semicontinuous function w:X→(0,+∞) we introduce the Banach space of continuous functions:
(2)C(w,X)≔f∈CX∣fw≔sup⁡x∈Xfxwx<+∞.
The weighted inductive limit of Fréchet spaces WC(X) of continuous functions on X is defined by
(3)WnC(X)≔projkC(wn,k,X),n∈N;WC(X)≔indnWnC(X).
The space WC(X) is a Hausdorff (LF)-space.The systemW¯ of weights assosiated with W consists of all upper semicontinuous functions w¯:X→[0,+∞) such that for each n there are αn>0 and k=k(n) with w¯≤αnwn,k on X. The projective hull of the inductive limit WC(X) is defined by
(4)CW¯(X)≔sup⁡x∈Xf∈CX∣hhhhhhhhhfw¯≔sup⁡x∈Xfxw¯x<+∞∀w¯∈W¯.
The topology of CW¯(X) is defined by the system of seminorms ·w¯, w¯∈W¯. The space WC(X) is contained in its projective hull CW¯(X) with continuous inclusion.The problem of the projective description for the spacesWC(X) is to determine conditions under which(A)
the spacesWC(X) and CW¯(X) coincide algebraically, or(T)
the spaceWC(X) is a topological subspace of its projective hull CW¯(X).


## 3. Case of Weights Defined by Positively Homogeneous Functions

LetY be a normed (complex or real) linear space with the norm ·. We assume that X is a cone in Y; that is, λx∈X for all λ≥0 and x∈X. The set X is endowed with the induced topology. It is assumed that X is locally compact. Then each set {x∈X∣x≤R}, R>0, is compact in X. Let hn:X→(0,+∞), n∈N, be continuous and positively homogeneous of degree ρ>0 functions such that hn≤hn+1 for each n∈N. Further ω:[0,+∞)→[0,+∞) is a continuous function with
(5)lim⁡t→+∞ωt=+∞,lim⁡t→+∞ω(t)tρ=0.
Put ω(x)≔ω(x), x∈X. We define weight functions by
(6)wn,kx≔exp⁡-hnx+kωx,x∈X,n,k∈N.Theorem 1.
The spaceWC(X) is a topological subspace of CW¯(X).Proof.
Sincelim⁡t→+∞⁡ω(t)=+∞, then for each n the Fréchet space WnC(X) coincides algebraically and topologically with the Fréchet space
(7)WnC0(X)≔f∈CX∣lim⁡x→+∞fxwn,kxhhhhhhhhhhhhhhhhhhhh=0foreachk∈Nlim⁡x→+∞.
By [3, Theorem  1.3.] WC(X) is a topological subspace of CW¯(X).
We investigate now the algebraic identityWC(X)=CW¯(X).Theorem 2.
The following conditions are equivalent.(i)
The spaceWC(X) coincides algebraically with CW¯(X).(ii)
The condition (RD) holds:(8)∀n∃m∀μ∃C:hμ(x)-hm(x)≤Chmx-hnx,x∈X.Proof.
By [6, Proposition 4] algebraic identity WC(X)=CW¯(X) is equivalent to the following condition (wQ) which was introduced by Vogt [9]:
(9)∀n∃m,k∀μ,l∃L,C:wm,l≤Cmax⁡wn,k;wμ,Lon⁡X,
that is,
(10)exp⁡-hm(x)+lω(x)≤Cmax⁡exp⁡-hnx+kωx;exp⁡-hμx+Lωx,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhx∈X.
Since lim⁡t→+∞ω(t)=+∞, the latter is equivalent to
(11)∀n∃m,k∀μ,l∃L,A:∀x≥A-hm(x)+lω(x)≤max⁡-hnx+kωx;-hμx+Lωx.
Thus the algebraic identity WC(X)=CW¯(X) holds if and only if
(12)∀n∃m,k∀μ,l∃L,A:∀x≥A(l-k)ω(x)≤hm(x)-hn(x)
or
(13)hμ(x)-hm(x)≤L-lω(x).
(
ii
)
⇒
(
i
). Suppose that (RD) holds. Fix n. By (RD) ∃m>n∀μ∃C:
(14)hμ(x)-hm(x)≤Chmx-hnx,x∈X.
We choose k≔1 and fix l. If
(15)l-1ωx>hmx-hnx,
then
(16)hμx-hmx≤Chmx-hnx≤Cl-1ωx,x∈X.
Hence the inequality (13) holds with L such that L≥l+C(l-1). Thus the condition (wQ) is valid and consequently the algebraic identity WC(X)=CW¯(X) is fulfilled.
(
i
)
⇒
(
ii
). Suppose that ∀n∃m, k∀μ, l∃L, A:∀x≥A the inequality (12) or (13) holds. Assume that the condition (RD) does not hold. Taking into account the positive homogeneity of functions hn we obtain that
(17)∃n0∀m>n0∃μ∃am,j∈X:am,j=1,hμ(am,j)-hm(am,j)>jhmam,j-hn0am,j,j∈N.
We select m0>n0 and k0 for n0 as in (12) and (13). Set l≔k0+1 and choose μ0=μ0(m0) for m0 as in (17). We take L and A as in (12) and (13). Put Hs≔sup⁡a=1⁡hs(a), s∈N (from the continuity of hs it follows that Hs<+∞). For every j∈N by (17),
(18)0≤hm0am0,j-hn0am0,j<1jhμ0am0,j-hm0am0,j≤Hμ0+Hm0j.
Hence,
(19)lim⁡j→∞hm0am0,j-hn0am0,j=0.
We set D≔sup⁡t≥Aω(t)/tρ. It is clear that D∈(0,+∞). In consequence of (17) and (19) there is j0 such that
(20)hm0am0,j0-hn0am0,j0<min⁡{D;1L-lhμ0(am0,j0)-hm0(am0,j0)}.
Since the function ω(t)/tρ is continuous on [A,+∞) and lim⁡t→+∞ω(t)/tρ=0, there is t0≥A such that
(21)1l-k0hm0am0,j0-hn0am0,j0=hm0am0,j0-hn0am0,j0<ωt0t0ρ<1L-l(hμ0(am0,j0)-hm0(am0,j0)).
We putx0≔t0am0,j0. Then x0∈X and x0≥A and also for x≔x0 both inequalities (12) and (13) do not hold. It is a contradiction.Example 3.
(a) LetKn be convex compact subsets of X=Y≔RN such that Kn is contained in the interior of Kn+1 for each n∈N. We denote by hn(x)≔sup⁡y∈Kn⁡〈x,y〉, x∈RN, 〈x,y〉≔∑j=1Nxjyj, the supporting function of Kn. The functions hn are positively homogeneous of order 1 and convex on RN. We put x≔〈x,x〉1/2.
The sequence(hn)n∈N satisfies the condition (RD) of Theorem 2. Indeed, we can take in the condition (ii) of Theorem 2m≔n+1 for n∈N. The necessary inequality follows from inf⁡a=1⁡(hn+1(a)-hn(a))>0, from the positive homogeneity and continuity of functions hs.
(b) Leth:X→[0,+∞) be a continuous and positively homogeneity of order ρ>0 function and hn≔αnh, n∈N, where (αn)n∈N is a strictly increasing sequence of positive numbers. The sequence (hn)n∈N satisfies the condition (RD).
We will show that the condition (RD) is equivalent to the condition of the regular decrease of the sequence(exp⁡(-hn))n∈N which plays the important role in projective descriptions in the (LB) case [3].Definition 4 (see [3, Definition  2.1]).
The sequence(φn)n∈N of functions φn:X→(0,+∞) such that φn+1≤φn on X∀n∈N is called regularly decreasing if ∀n∃m≥n∀μ>m∀Q⊆X(22)inf⁡y∈Qφm(y)φn(y)>0⟹inf⁡y∈Qφμ(y)φm(y)>0.Theorem 5.
Lethn:X→R be continuous functions which are positively homogeneous of degree ρ>0, hn≤hn+1 on CN, n∈N. The following assertions are equivalent:(i)
The condition (RD) holds.(ii)
The sequence(exp⁡(-hn))n∈N is regularly decreasing.Proof.
(
ii
)
⇒
(
i
). Suppose that the sequence (exp⁡(-hn))n∈N decreases regularly. Then ∀n∃m≥n∀μ>m∀Q⊆X,
(23)sup⁡y∈Q(hm(y)-hn(y))<+∞⟹sup⁡y∈Q(hμ(y)-hm(y))<+∞.
Assume that the condition (RD) does not hold. Then
(24)∃n0∀m>n0∃μ>m∀j∃ym,j∈X:hμym,j-hmym,j>jhmym,j-hn0ym,j.
For n0 we choose m as in the condition (23) and for this m select μ as in the condition (24).
We will prove that from the equalityhm(y)=hn0(y) it follows that hμ(y)=hm(y). Indeed, if hm(y)=hn0(y) for some y∈X, then also for each t>0 the equality hm(ty)=hn0(ty) holds. Therefore
(25)+∞>sup⁡t>0hμ(ty)-hm(ty)=sup⁡t>0tρhμy-hmy.
From this it follows that hμ(y)=hm(y). Hence cj≔hm(ym,j)-hn0(ym,j)>0 for each j∈N. Since for all j∈N(26)1=hm(ym,j)-hn0(ym,j)cj=hmym,jcj1/ρ-hn0ym,jcj1/ρ,
by (24)(27)sup⁡j∈Nhμym,jcj1/ρ-hmym,jcj1/ρ=+∞.
It is a contradiction with (23).
The implication(i)⇒(ii) is obvious.


## 4. Case of Weights Defined by a Composition with Positively Homogeneous Functions

LetΓ be a cone in a normed (real or complex) linear space Z, that is, λx∈Γ for all λ≥0 and x∈Γ. Set for R>0(28)SR(X)≔x∈X∣x=R;B¯RΓ≔x∈Γ∣xZ≤R,
where ·Z is the norm in Z. We endow Γ with the induced topology.Lethn:Γ→R, n∈N, be continuous and positively homogeneous of degree ρ>0 functions such that hn≤hn+1 on Γ for each n∈N. Suppose that ω:[0,+∞)→[0,+∞) is a continuous function such that
(29)lim⁡t→+∞ωt=+∞,lim⁡t→+∞ω(t)tρ=0.
We put ω(x)≔ω(x), x∈X.Further we will use functionsv:X→Γ with a covering property.Definition 6.
A functionv:X→Γ satisfies the condition (SJ) if there is r>0 such that B¯rR(Γ)⊆v(SR(X)) for all R>0.Example 7.
The following functions satisfy the condition (SJ) withr=1:(a)
v(z)≔Im⁡z if X=Y=CN, Γ=Z=RN, N∈N,(b)
v(z)≔(z1,…,zk), z=(z1,…,zN)∈CN, if X=Y=CN, Γ=Z=Ck, k,N∈N, k<N.
We fixα≥0 and define weight functions by
(30)wn,kx≔exp⁡-hnvx-α+1kωx,hhhhhhhhhhhhhhhhhhhhx∈X,n,k∈N,
where v:X→Γ is a continuous function which satisfies the condition (SJ).Theorem 8.
The spaceWC(X) is a topological subspace of CW¯(X).The assertion follows from [3, Theorem  1.3] (see the proof of Theorem 1).We will investigate further when the algebraic identityWC(X)=CW¯(X) holds.Theorem 9.
The following assertions are equivalent:(i)
WC(X)=CW¯(X) holds algebraically.(ii)
There ism∈N such that hμ=hm for each μ>m.Proof.
By [3, Proposition  4] the algebraic identity WC(X)=CW¯(X) is equivalent to the condition (wQ):
(31)∀n∃m,k∀μ,l∃L,C:wm,l≤Cmax⁡wn,k;wμ,Lon⁡X;
that is,
(32)exp⁡-hm(v(x))-α+1lω(x)≤Cmax⁡exp⁡-hn(v(x))-α+1kω(x);hhhhhhhhhhhexp⁡-hμvx-α+1Lωx,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhx∈X.
Since lim⁡t→+∞⁡ω(t)=+∞, this condition is equivalent to
(33)∀n∃m,k∀μ,l∃L,A:∀x≥A(34)-hm(v(x))-α+1lω(x)≤max⁡-hnvx-α+1kωx;hhhhhhhh-hμ(v(x))-α+1Lω(x).
Thus the algebraic equality WC(X)=CW¯(X) holds if and only if the following condition (wQ1) is fulfilled:
(35)∀n∃m,k∀μ,l∃L,A:∀x≥A1k-1lω(x)≤hmvx-hnvx
or
(36)hμvx-hm(v(x))≤1l-1Lω(x).
(
i
)
⇒
(
ii
). Assume that the condition (i) holds but (ii) is not satisfied. We select m and k for n=1 as in (wQ1). Since (ii) does not hold, there are μ>m and a∈Γ such that aZ=1 and hμ(a)>hm(a). Choose l>k so large that
(37)α≔11/k-1/lhma-h1a<l(hμ(a)-hm(a))≕β.
We fix some γ∈(α,β). Define L and A for l as in (wQ1). Since lim⁡t→+∞⁡ω(t)=+∞ and lim⁡t→+∞(ω(t)/tρ)=0, there is R≥A such that
(38)ωR>β,ω(R)Rρ<rργ,
where r>0 is a constant as in condition (SJ).
Lett≔(ω(R)/γ)1/ρ. Then t<rR. By the condition (SJ) there is x0∈X such that x0=R and v(x0)=ta. From
(39)11/k-1/lhma-h1a<ω(R)tρ<β<11/l-1/Lhμa-hma
it follows that
(40)1(1/k)-(1/l)hmvx0-h1vx0<ω(x0)<1(1/l)-(1/L)hμvx0-hmvx0.
There is a contradiction with (wQ1).
The implication(ii)⇒(i) is obvious.
We give a corollary of Theorem9. Let X=Y=CN, Γ=Z=RN; z≔∑j=1N|zj|21/2, z∈CN; xZ≔∑j=1N|xj|21/2, x∈RN; hn:RN→R, n∈N, be continuous and positively homogeneous of degree ρ>0 functions such that hn≤hn+1 for each n∈N; let function ω be such as above. We put v(z)≔Im⁡z, z∈CN. The function v:CN→RN satisfies the condition (SJ).
Put(41)wn,kz≔exp⁡-hnIm⁡z-α+1kωz,hhhhhhhhhhhhhhhhhhhhz∈CN,n,k∈N,
where α∈[0,+∞) is fixed.Corollary 10.
The following conditions are equivalent:(i)
WC(CN)=CW¯(CN) holds algebraically.(ii)
There ism∈N such that hμ=hm for each μ>m.Remark 11.
LetΩ be a convex open subset of RN; let (Kn)n∈N be a fundamental sequence of convex compact subsets of Ω such that Kn⊂int⁡Kn+1 for each n. For each n we denote by hn(x)≔sup⁡y∈Kn⁡〈x,y〉, x∈RN, the supporting functions of Kn. The functions hn are convex and positively homogeneous of order 1. Let ω be a quasianalytic weight function as in Braun et al. [10] (see [7, p. 125] too). In [7, Proposition 7] it was proved that the algebraic identity WC(CN)=CW¯(CN) does not hold for the functions wn,k(z)≔exp⁡(-hn(Im⁡z)-(1/k)ω(z)), z∈CN. This assertion follows from Corollary 10, since hn(x)<hn+1(x) for each n∈N and x∈RN∖{0}.



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