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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 187952, 10 pages
http://dx.doi.org/10.5402/2012/187952
Research Article

Some Dense Linear Subspaces of Extended Little Lipschitz Algebras

Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran

Received 18 November 2011; Accepted 3 January 2012

Academic Editors: S.Β Anita and H.Β Hedenmalm

Copyright Β© 2012 Davood Alimohammadi and Sirous Moradi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let (𝑋,𝑑) be a compact metric space. In 1987, Bade, Curtis, and Dales obtained a sufficient condition for density of a subspace 𝑃 of little Lipschitz algebra lip(𝑋,𝛼) in this algebra and in particular showed that Lip(𝑋,1) is dense in lip(𝑋,𝛼), whenever 0<𝛼<1. Let 𝐾 be a compact subset of 𝑋. We define new classes of Lipchitz algebras Lip(𝑋,𝐾,𝛼) for π›Όβˆˆ(0,1] and lip(𝑋,𝐾,𝛼) for π›Όβˆˆ(0,1), consisting of those continuous complex-valued functions 𝑓 on 𝑋 such that 𝑓|𝐾∈Lip(𝐾,𝛼) and 𝑓|𝐾∈lip(𝐾,𝛼), respectively. In this paper we obtain a sufficient condition for density of a linear subspace 𝑃 of extended little Lipschitz algebra lip(𝑋,𝐾,𝛼) in this algebra and in particular show that Lip(𝑋,𝐾,1) is dense in lip(𝑋,𝐾,𝛼), whenever 0<𝛼<1.

1. Introduction

Let Ξ© be a locally compact Hausdorff space. The linear space of all continuous (bounded continuous) complex-valued functions on Ξ© is denoted by 𝐢(Ξ©) (𝐢𝑏(Ξ©)). It is known that 𝐢𝑏(Ξ©) under the uniform norm on Ξ©, that is,β€–β„Žβ€–Ξ©ξ€½||||=supβ„Ž(𝑀)βˆΆπ‘€βˆˆΞ©ξ€Ύξ€·β„ŽβˆˆπΆπ‘ξ€Έ,(Ξ©)(1.1) is a commutative Banach algebra. The set of all 𝑓 in 𝐢(Ξ©), which vanish at infinity, is denoted by 𝐢0(Ξ©), which is a closed linear subspace of (𝐢𝑏(Ξ©),β€–β‹…β€–Ξ©). Clearly, 𝐢0(Ξ©)=𝐢𝑏(Ξ©)=𝐢(Ξ©), whenever Ξ© is compact. The linear space of all complex regular Borel measures on Ξ© is denoted by 𝑀(Ξ©). It is known that 𝑀(Ξ©) under the norm β€–πœ‡β€–=|πœ‡|(Ξ©) (πœ‡βˆˆπ‘€(Ξ©)) is a Banach space, where |πœ‡| is the total variation of πœ‡βˆˆπ‘€(Ξ©).

The Riesz representation theorem asserts that there exists a linear isometry from (𝐢0(Ξ©),β€–β‹…β€–Ξ©)βˆ—, the dual space (𝐢0(Ξ©),β€–β‹…β€–Ξ©) onto (𝑀(Ξ©),β€–β‹…β€–). In fact, for each Ξ›βˆˆ(𝐢0(Ξ©),β€–β‹…β€–Ξ©)βˆ—, there exists a unique measure πœ‡βˆˆπ‘€(Ξ©) with β€–πœ‡β€–=β€–Ξ›β€– such thatξ€œΞ›(𝑓)=Ξ©ξ€·π‘“π‘‘πœ‡π‘“βˆˆπΆ0(ξ€Έ.Ξ©)(1.2) Let (𝑋,𝑑) be a compact metric space and 𝛼>0. The Lipschitz algebra Lip(𝑋,𝛼) is defined as the set of all complex-valued functions 𝑓 on 𝑋 such that𝑝𝛼||||(𝑓)=sup𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)∢π‘₯,π‘¦βˆˆπ‘‹,π‘₯≠𝑦<∞.(1.3) Then lip(𝑋,𝛼) is a subalgebra of 𝐢(𝑋). The subalgebra Lip(𝑋,𝛼) of Lip(𝑋,𝛼) is the set of all those complex-valued functions 𝑓 on 𝑋 for which lim|𝑓(π‘₯)βˆ’π‘“(𝑦)|/𝑑𝛼(π‘₯,𝑦)=0 as 𝑑(π‘₯,𝑦)β†’0 and is called little Lipschitz algebra of order 𝛼.

We know that Lip(𝑋,1) separates the points of 𝑋, 1∈Lip(𝑋,1) and Lip(𝑋,𝛽)βŠ†lip(𝑋,𝛼)βŠ†Lip(𝑋,𝛼), where 0<𝛼<𝛽⩽1. Also, if 𝑋 is infinite and 0<𝛼<1, then lip(𝑋,𝛼)β‰ Lip(𝑋,𝛼). The algebras Lip(𝑋,𝛼) for 𝛼⩽1 and lip(𝑋,𝛼) for 𝛼<1 are Banach function algebras on 𝑋 under the norm ‖𝑓‖𝛼=‖𝑓‖𝑋+𝑝𝛼(𝑓). Since these algebras are self-adjoint, they are uniformly dense in 𝐢(𝑋), by the Stone-Weierstrass theorem. We know that if 𝐴 is a Banach function algebra on a compact Hausdorff space 𝑋 such that 𝐴 is self-adjoint and 1/π‘“βˆˆπ΄ whenever π‘“βˆˆπ΄ and 𝑓(π‘₯)β‰ 0 for each π‘₯βˆˆπ‘‹, then 𝐴 is natural, that is, the maximal ideal space of 𝐴 coincides with 𝑋. Hence, if 𝑋 is infinite, then the Lipschitz algebras Lip(𝑋,𝛼) for 𝛼⩽1 and lip(𝑋,𝛼) for 𝛼<1, are natural.

Extensive study of Lipschitz algebras started with Sherbert [1, 2]. Honary and Moradi introduced new classes of analytic Lipschitz algebras on compact plane sets and determined their maximal ideal spaces [3].

Bade et al. have obtained a sufficient condition for density of a linear subspace 𝑃 of lip(𝑋,𝛼) in this algebra as follows.

Theorem 1.1 (see [4, Theorem  3.6]). Let (𝑋,𝑑) be a compact metric space, and let 𝑃 be a linear subspace of lip(𝑋,𝛼). Suppose that there is a constant 𝐢 such that for each finite subset 𝐸 of 𝑋 and each π‘“βˆˆlip(𝑋,𝛼), there exists π‘”βˆˆπ‘ƒ with 𝑔|𝐸=𝑓|𝐸 and with ‖𝑔‖𝛼⩽𝐢‖𝑓‖𝛼. Then 𝑃 is dense in lip(𝑋,𝛼).

They also showed that Lip(𝑋,1) is dense in lip(𝑋,𝛼) [4, Corollary  3.7]. We extend the above results for the more general classes of the Lipschitz algebras by generalizing and using some results that have been given by them.

Throughout this work we always assume that (𝑋,𝑑) is a compact metric space, 𝐾 is nonempty compact subset of 𝑋, and 𝛼 is a positive number.

Definition 1.2. The algebra of all continuous complex-valued functions 𝑓 on 𝑋 for which 𝑝𝛼,𝐾||||=sup𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)∢π‘₯,π‘¦βˆˆπΎ,π‘₯≠𝑦<∞(1.4) is denoted by Lip(𝑋,𝐾,𝛼), and the subalgebra of those π‘“βˆˆLip(𝑋,𝐾,𝛼) for which |𝑓(π‘₯)βˆ’π‘“(𝑦)|/𝑑𝛼(π‘₯,𝑦)β†’0 as 𝑑(π‘₯,𝑦)β†’0, when π‘₯,π‘¦βˆˆπΎ, is denoted by lip(𝑋,𝐾,𝛼). The algebras Lip(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) are called extended Lipschitz algebra and extended little Lipschitz algebra of order 𝛼 on (𝑋,𝑑) with respect to 𝐾, respectively.

It is easy to see that these extended Lipschitz algebras are both Banach algebras under the norm ‖𝑓‖𝛼,𝐾=‖𝑓‖𝑋+𝑃𝛼,𝐾(𝑓). In fact, lip(𝑋,𝐾,𝛼) is a Banach function algebra on 𝑋 for π›Όβˆˆ(0,1], and lip(𝑋,𝐾,𝛼) is a Banach function algebra on 𝑋 for π›Όβˆˆ(0,1). Note that if 0<𝛼<𝛽⩽1, then Lip(𝑋,𝐾,𝛽)βŠ†lip(𝑋,𝐾,𝛼). We always assume that 0<𝛼⩽1 for Lip and 0<𝛼<1 for lip. Note that lip(𝑋,𝐾,𝛼) is a proper subalgebra of Lip(𝑋,𝐾,𝛼) when 𝐾 is infinite. Because if π‘¦βˆˆπΎ, then function π‘“βˆΆπ‘‹β†’β„‚ defined by 𝑓(π‘₯)=𝑑𝛼(π‘₯,𝑦) is an element of Lip(𝑋,𝐾,𝛼) but does not belong to lip(𝑋,𝐾,𝛼).

It is clear that whenever 𝐾=𝑋, the new classes of Lipschitz algebras coincide with the standard Lipschitz algebras. Also Lip(𝑋,𝐾,𝛼)=lip(𝑋,𝐾,𝛼)=𝐢(𝑋), whenever 𝐾 is finite. Hence, we may assume that 𝐾 is infinite.

By the Stone-Weierstrass theorem, Lip(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) are both uniformly dense in 𝐢(𝑋).

Let 𝐴 be Lip(𝑋,𝐾,𝛼) or lip(𝑋,𝐾,𝛼). For each π‘“βˆˆπ΄ and for all π‘›βˆˆβ„•, we have‖𝑓𝑛‖𝛼,𝐾⩽‖𝑓𝑛‖𝑋‖+π‘›π‘“β€–π‘‹ξ€Έπ‘›βˆ’1𝑝𝛼,𝐾(𝑓).(1.5) By the spectral radius theorem,‖‖𝑓‖‖ℳ(𝐴)=limπ‘›β†’βˆžξ€·β€–π‘“π‘›β€–π›Ό,𝐾1/𝑛⩽limπ‘›β†’βˆžβ€–π‘“β€–π‘‹ξ‚΅π‘1+𝛼,𝐾(𝑓)‖𝑓‖𝑋𝑛1/𝑛=‖𝑓‖𝑋,(1.6) where β„³(𝐴) is the maximal ideal space of 𝐴 and 𝑓 is the Gelfand transform of 𝑓 on β„³(𝐴). Hence, by applying the main theorem in [5], we can show that 𝐴 is natural, that is, β„³(𝐴) coincides with 𝑋. We can prove this fact with another way. Since every self-adjoint inverse-closed Banach function algebra 𝐴 on a compact Hausdorff space 𝑋 is natural, and Banach function algebras Lip(𝑋,𝐾,𝛼) and lip(𝑋,𝐾,𝛼) have the mentioned properties, they are natural.

In this paper we obtain a sufficient condition for density of a linear subspace 𝑃 of lip(𝑋,𝐾,𝛼) that is dense in this algebra by generalizing some results in [4]. In particular, we show that Lip(𝑋,𝐾,1) is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).

2. Representing Measure

We denoteΞ”(𝐾)={(π‘₯,𝑦)βˆˆπΎΓ—πΎβˆΆπ‘₯=𝑦},𝑉(𝐾)=(𝐾×𝐾)⧡0π‘₯00082Ξ”(𝐾),π‘Š(𝑋,𝐾)=𝑋βˆͺ𝑉(𝐾),π‘Š(𝐾)=π‘Š(𝐾,𝐾).(2.1) Obviously, π‘Š(𝑋,𝐾) is a locally compact Hausdorff space. We define the norm |β€–β‹…β€–| on 𝐢𝑏(π‘Š(𝑋,𝐾)) by||||=β€–β€–β€–β„Žβ€–β„Ž|𝑋‖‖𝑋+β€–β€–β„Ž|𝑉(𝐾)‖‖𝑉(𝐾).(2.2) Then 𝐢𝑏(π‘Š(𝑋,𝐾) is a Banach space under the norm |β€–β‹…β€–|, sinceβ€–β„Žβ€–π‘Š(𝑋,𝐾)β©½β€–β€–||β„Ž||β€–β€–β©½2β€–β„Žβ€–π‘Š(𝑋,𝐾),(2.3) for all β„ŽβˆˆπΆπ‘(π‘Š(𝑋,𝐾)). Moreover, 𝐢0(π‘Š(𝑋,𝐾)) is a closed linear subspace of (𝐢𝑏(π‘Š(𝑋,𝐾)),|β€–β‹…β€–|).

We define the norm |β€–β‹…β€–| on 𝑀(π‘Š(𝑋,𝐾)) byβ€–β€–||πœ‡||β€–β€–ξ€½||πœ‡||||πœ‡||ξ€Ύ=max(𝑋),(𝑉(𝐾)).(2.4) Then 𝑀(π‘Š(𝑋,𝐾)) is a Banach space under the norm β€–|β‹…|β€–, sinceβ€–β€–β€–||πœ‡||β€–β€–πœ‡β€–β©½β©½2β€–πœ‡β€–,(2.5) for all πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾)). By applying the Riesz representation theorem, we obtain the following result which is a generalization of Theorem 𝐴 in [6], and one can prove it by the same method.

Theorem 2.1. For each Ψ∈(𝐢0(π‘Š(𝑋,𝐾)),β€–|β‹…|β€–)βˆ—, there exists a unique measure πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾)) with β€–|πœ‡β€–|=β€–|Ξ¨β€–| such that ξ€œΞ¨(β„Ž)=π‘Š(𝑋,𝐾)ξ€·β„Žπ‘‘πœ‡β„ŽβˆˆπΆ0(ξ€Έ,π‘Š(𝑋,𝐾))(2.6) where β€–|Ξ¨|β€–=sup{|Ξ¨(β„Ž)|βˆΆβ„ŽβˆˆπΆ0(π‘Š(𝑋,𝐾)),|β€–β„Žβ€–|β©½1}.

Definition 2.2. For π‘“βˆˆπΆ(𝑋), the function 𝑇𝑋,𝐾(𝑓)βˆΆπ‘Š(𝑋,𝐾)β†’β„‚ defined by 𝑇𝑋,𝐾𝑇(𝑓)(π‘₯)=𝑓(π‘₯)(π‘₯βˆˆπ‘‹),𝑋,𝐾(𝑓)(π‘₯,𝑦)=𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)((π‘₯,𝑦)βˆˆπ‘‰(𝐾))(2.7) is called Leeuw’s extension of 𝑓 on π‘Š(𝑋,𝐾).

It is obvious that 𝑇𝑋,𝐾(𝑓)βˆˆπΆπ‘(π‘Š(𝑋,𝐾)) for each π‘“βˆˆLip(𝑋,𝐾,𝛼).

Theorem 2.3. Take 0<𝛼<1.(i)𝑇𝑋,𝐾 is a linear isometry from the extended Lipschitz algebra (Lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾) into (𝐢𝑏(π‘Š(𝑋,𝐾)),β€–|β‹…|β€–).(ii)𝑇𝑋,𝐾(Lip(𝑋,𝐾,𝛼)) is a closed linear subspace of (𝐢0(π‘Š),β€–|β‹…|β€–).(iii)For each Φ∈(lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾)βˆ—, there exists πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾)) such thatξ€œΞ¦(𝑓)=π‘Š(𝑋,𝐾)𝑇𝑋,𝐾(β€–β€–||πœ‡||‖‖𝑓)π‘‘πœ‡(π‘“βˆˆlip(𝑋,𝐾,𝛼)),=β€–Ξ¦β€–.(2.8)

Proof. (i) It is immediate.
(ii) Since lip(𝑋,𝐾,𝛼) is a linear subspace Lip(𝑋,𝐾,𝛼), 𝑇𝑋,𝐾(lip(𝑋,𝐾,𝛼)) is a linear subspace of 𝐢𝑏(π‘Š(𝑋,𝐾)) by (𝑖). Let π‘“βˆˆlip(𝑋,𝐾,𝛼), and let πœ€ be an arbitrary positive number. There exists 𝛿>0 such that||||𝑓(π‘₯)βˆ’π‘“(𝑦)𝑑𝛼(π‘₯,𝑦)<πœ€,(2.9) for all π‘₯,π‘¦βˆˆπΎ with 0<𝑑(π‘₯,𝑦)<𝛿. Set πΈπœ€=𝑋βˆͺ{(π‘₯,𝑦)βˆˆπΎβˆΆπ‘‘(π‘₯,𝑦)⩾𝛿}. Clearly, πΈπœ€ is a compact subset of π‘Š(𝑋,𝐾) and ||𝑇𝑋,𝐾||(𝑓)(π‘₯,𝑦)<πœ€,(2.10) for all (π‘₯,𝑦)βˆˆπ‘Š(𝑋,𝐾)β§΅πΈπœ€. It follows that 𝑇𝑋,𝐾(𝑓)∈𝐢0(π‘Š(𝑋,𝐾)). Therefore, 𝑇𝑋,𝐾(lip(𝑋,𝐾,𝛼)) is a subset 𝐢0(π‘Š(𝑋,𝐾)). Since 𝑇𝑋,𝐾 is a linear isometry and 𝐢0(π‘Š(𝑋,𝐾)) is a closed linear subspace of (𝐢𝑏(π‘Š(𝑋,𝐾)),β€–|β‹…|β€–), we conclude that 𝑇𝑋,𝐾(lip(𝑋,𝐾,𝛼)) is a closed linear subspace of (𝐢0(π‘Š(𝑋,𝐾)),β€–|β‹…|β€–).
(iii) Let Φ∈(Lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾)βˆ— and define πœ‚π‘‹,𝐾∢lip(𝑋,𝐾,𝛼)→𝑇𝑋,𝐾(lip(𝑋,𝐾,𝛼)) by πœ‚π‘‹,𝐾(𝑓)=𝑇𝑋,𝐾(𝑓)(π‘“βˆˆlip(𝑋,𝐾,𝛼)).(2.11) Then Ξ¦π‘œ(πœ‚π‘‹,𝐾)βˆ’1∈(𝑇𝑋,𝐾(lip(𝑋,𝐾,𝛼)),β€–β‹…β€–)βˆ—. By the Hahn-Banach extension theorem, there exists Ψ∈𝐢0(π‘Š(𝑋,𝐾),β€–|β‹…|β€–)βˆ— with β€–|Ξ¨β€–|=β€–|Ξ¦π‘œ(πœ‚π‘‹,𝐾)βˆ’1β€–| such that ξ€·πœ‚Ξ¨(β„Ž)=Ξ¦π‘œπ‘‹,πΎξ€Έβˆ’1ξ€·(β„Ž)β„Žβˆˆπ‘‡π‘‹,𝐾.(lip(𝑋,𝐾,𝛼))(2.12) By Theorem 2.1, there exists πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾)) with β€–|πœ‡β€–|=β€–|Ξ¨β€–| such thatξ€œΞ¨(β„Ž)=π‘Š(𝑋,𝐾)ξ€·β„Žπ‘‘πœ‡β„ŽβˆˆπΆ0(ξ€Έ.π‘Š(𝑋,𝐾))(2.13) Therefore, ξ‚€ξ€·πœ‚Ξ¦(𝑓)=Ξ¦π‘œπ‘‹,πΎξ€Έβˆ’1𝑇𝑋,𝐾=ξ€œ(𝑓)π‘Š(𝑋,𝐾)𝑇𝑋,𝐾‖‖||πœ‡||β€–β€–=β€–β€–||(𝑓)π‘‘πœ‡(π‘“βˆˆlip(𝑋,𝐾,𝛼)),Ξ¦π‘œ(πœ‚π‘‹,𝐾)βˆ’1||β€–β€–.(2.14) On the other hand, β€–β€–β€–|||ξ€·πœ‚Ξ¦π‘œπ‘‹,πΎξ€Έβˆ’1|||β€–β€–β€–ξ€½||||β€–β€–||𝑇=sup(Ξ¦(𝑓))βˆΆπ‘“βˆˆlip(𝑋,𝐾,𝛼),𝑋,𝐾||β€–β€–ξ€Ύ(𝑓)β©½1=β€–Ξ¦β€–.(2.15) It follows that β€–|πœ‡β€–|=β€–Ξ¦β€–. This completes the proof.

Note that the map 𝑇𝑋,𝐾 is not an algebra homomorphism and that its image is not a subalgebra of 𝐢𝑏(π‘Š(𝑋,𝐾)).

Definition 2.4. For Φ∈(lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾)βˆ—, a measure πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾))withβ€–|πœ‡β€–|=β€–Ξ¦β€– and with ξ€œΞ¦(𝑓)=π‘Š(𝑋,𝐾)𝑇𝑋,𝐾(𝑓)π‘‘πœ‡(π‘“βˆˆlip(𝑋,𝐾,𝛼))(2.16) is called a representing measure for Ξ¦ on π‘Š(𝑋,𝐾).

Note that a representing measure for Ξ¦ on π‘Š(𝑋,𝐾) is not unique.

3. Main Results

In this section, by generalizing Theorem 1.1, we obtain a sufficient condition for which a linear subspace of lip(𝑋,𝐾,𝛼) is dense in this algebra. In particular, we show that Lip(𝑋,𝐾,1) is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).

Theorem 3.1. let 𝑃 be a linear subspace of lip(𝑋,𝐾,𝛼) which satisfies the following conditions:(a)if β„ŽβˆˆπΆ(𝑋) with β„Ž|𝐾=0, then β„Žβˆˆπ‘ƒ, where 𝑃 is the closure of 𝑃 in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).(b)there is a constant 𝐢 such that for each finite subset 𝐸 of 𝐾 and each π‘“βˆˆlip(𝑋,𝐾,𝛼), there exists π‘”βˆˆπ‘ƒ with 𝑔|𝐸=𝑓|𝐸 and with ‖𝑔‖𝛼,𝐾⩽𝐢‖𝑓‖𝛼,𝐾. Then 𝑃 is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).

Proof. We first show that if 𝑃𝐾={π‘”βˆˆπΆ(𝐾)βˆΆπ‘”=𝑓|𝐾forsomeπ‘“βˆˆπ‘ƒ}, then 𝑃𝐾 is dense in the little Lipschitz algebra lip(𝐾,𝛼). Clearly, 𝑃𝐾 is a linear subspace of lip(𝐾,𝛼). Let 𝐸 be a Finite subset of 𝐾, and let π‘“βˆˆlip(𝐾,𝛼). By Tietze’s extension theorem [7, Theorem  20.4], there exists 𝐹∈𝐢(𝑋) such that 𝐹|𝐾=𝑓and‖𝐹‖𝑋=‖𝑓‖𝐾. Clearly, 𝐹∈lip(𝑋,𝐾,𝛼). by (b), there exists πΊβˆˆπ‘ƒ such that 𝐺|𝐸=𝐹|𝐸and‖𝐺‖𝛼,𝐾⩽𝐢‖𝐹‖𝛼,𝐾. We define 𝑔=𝐺|𝐾. Then π‘”βˆˆπ‘ƒπΎ, 𝑔|𝐸=𝑓|𝐸 and ‖𝑔‖𝛼=‖𝑔‖𝐾+𝑝𝛼(𝑔)⩽‖𝐺‖𝑋+𝑝𝛼(𝑔)=‖𝐺‖𝑋+𝑝𝛼,𝐾(𝐺)=‖𝐺‖𝛼,𝐾⩽𝐢‖𝐹‖𝛼,𝐾=𝐢‖𝐹‖𝑋+𝑝𝛼,𝐾(𝐹)=𝐢‖𝑓‖𝐾+𝑝𝛼(𝑓)=𝐢‖𝑓‖𝛼.(3.1) Thus 𝑃𝐾 is dense in (lip(𝐾,𝛼),‖⋅‖𝛼) by Theorem 1.1.
To prove the density of 𝑃 in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾), it is enough to show that if Φ∈(lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾)βˆ— with Ξ¦(𝑓)=0 for all π‘“βˆˆπ‘ƒ, then Ξ¦(𝑓)=0 for all π‘“βˆˆlip(𝑋,𝐾,𝛼).
Let Φ∈(lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾)βˆ— such that Ξ¦(𝑓)=0 for all π‘“βˆˆπ‘ƒ. Continuity of Ξ¦ implies that Ξ¦(𝑓)=0 for all π‘“βˆˆπ‘ƒ. By Theorem 2.3, there exists πœ‡βˆˆπ‘€(π‘Š(𝑋,𝐾)) such that ξ€œΞ¦(𝐹)=π‘Š(𝑋,𝐾)𝑇𝑋,𝐾(𝐹)π‘‘πœ‡(𝐹∈lip(𝑋,𝐾,𝛼)),(3.2) where 𝑇𝑋,𝐾(𝐹) is Leeuw’s extension of 𝐹∈lip(𝑋,𝐾,𝛼) on π‘Š(𝑋,𝐾). We claim that ξ€œΞ¦(𝐹)=π‘Š(𝐾)𝑇𝑋,𝐾(𝐹)π‘‘πœ‡(𝐹∈lip(𝑋,𝐾,𝛼)).(3.3) Let 𝐹∈lip(𝑋,𝐾,𝛼). We define the sequence {π‘Œπ‘›}βˆžπ‘›=1 of the subsets of 𝑋 by π‘Œπ‘›=1π‘₯βˆˆπ‘‹βˆΆπ‘‘(π‘₯,𝐾)⩾𝑛.(3.4) Then π‘Œπ‘› is a compact subset of 𝑋 for each π‘›βˆˆβ„•, π‘Œ1βŠ†π‘Œ2βŠ†β‹―βŠ†π‘‹β§΅πΎ, and β‹ƒβˆžπ‘›=1π‘Œπ‘›=𝑋⧡K. Let π‘›βˆˆβ„•. By Urysohn’s lemma, there exists πΉπ‘›βˆˆπΆ(𝑋) such that ‖𝐹𝑛‖𝑋=1, 𝐹𝑛|𝐾=0, and 𝐹𝑛|π‘Œπ‘›=1. Define 𝐺𝑛=𝐹𝑛𝐹. Then πΊπ‘›βˆˆπΆ(𝑋) and 𝐺𝑛|𝐾=0. Hence, πΊπ‘›βˆˆπ‘ƒ by (a) and so Ξ¦(𝐺𝑛)=0. Thus ξ€œπ‘Š(𝑋,𝐾)𝑇𝑋,πΎξ€·πΊπ‘›ξ€Έπ‘‘πœ‡=0.(3.5) Let ξ‚πœ’π‘‹β§΅πΎ be the characteristic function of 𝑋⧡𝐾 on π‘Š(𝑋,𝐾). It is easy to see that limπ‘›β†’βˆžπ‘‡π‘‹,𝐾𝐺𝑛𝑇(𝑀)=𝑋,𝐾(𝐹)β‹…ξ‚πœ’π‘‹β§΅πΎξ€Έ(𝑀),(3.6) for all π‘€βˆˆπ‘Š(𝑋,𝐾). Since ‖𝑇𝑋,𝐾(𝐺𝑛)β€–π‘Š(𝑋,𝐾)=1 for all π‘›βˆˆβ„• and 1∈𝐿1(π‘Š(𝑋,𝐾),|πœ‡|), we conclude that limπ‘›β†’βˆžξ€œπ‘Š(𝑋,𝐾)𝑇𝑋,πΎξ€·πΊπ‘›ξ€Έξ€œπ‘‘πœ‡=π‘Š(𝑋,𝐾)𝑇𝑋,𝐾(𝐹)β‹…ξ‚πœ’π‘‹β§΅πΎπ‘‘πœ‡,(3.7) by Lebesgue’s dominated convergence theorem. Thus ξ€œπ‘Š(𝑋,𝐾)𝑇𝑋,𝐾(𝐹)β‹…ξ‚πœ’π‘‹β§΅πΎπ‘‘πœ‡=0,(3.8) by (3.5) and (3.7). It follows that ξ€œπ‘‹β§΅πΎπ‘‡π‘‹,𝐾(𝐹)π‘‘πœ‡=0.(3.9) Thus (3.3) is justified, by (3.2) and (3.9).
We now define the function Ψ∢lip(𝐾,𝛼)β†’β„‚, byξ€œΞ¨(𝑔)=π‘Š(𝐾)𝑇𝐾,𝐾(𝑔)π‘‘πœ‡|π‘Š(𝐾).(3.10) Clearly, Ξ¨ is a linear functional on lip(𝐾,𝛼). Since ||||⩽‖‖𝑇Ψ(𝑔)𝐾,𝐾‖‖(𝑔)π‘Š(𝐾)||πœ‡||β€–(π‘Š(𝐾))⩽𝑔‖𝛼||πœ‡||(π‘Š(𝐾)),(3.11) for all π‘”βˆˆlip(𝐾,𝛼), we deduce that Ψ∈(lip(𝐾,𝛼),‖⋅‖𝛼)βˆ—. We claim that Ξ¨(𝑔)=0 for all π‘”βˆˆπ‘ƒπΎ. If π‘”βˆˆπ‘ƒπΎ, there exists π‘“βˆˆπ‘ƒ such that 𝑓|𝐾=𝑔, and so ξ€œΞ¨(𝑔)=π‘Š(𝐾)𝑇𝐾,𝐾(𝑔)π‘‘πœ‡|π‘Š(𝐾)=ξ€œπ‘Š(𝐾)𝑇𝐾,𝐾(𝑓)π‘‘πœ‡=Ξ¦(𝑓)=0.(3.12) Therefore, our claim is justified. It follows that Ξ¨(𝑔)=0 for all π‘”βˆˆlip(𝐾,𝛼), by the density of 𝑃𝐾 in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾) and continuity of Ξ¨ on lip(𝐾,𝛼). Let π‘“βˆˆlip(𝑋,𝐾,𝛼). If 𝑔=𝑓|𝐾, then π‘”βˆˆlip(𝐾,𝛼), and so Ξ¨(𝑔)=0. Therefore, ξ€œΞ¦(𝑓)=π‘Š(𝐾)𝑇𝑋,𝐾(ξ€œπ‘“)π‘‘πœ‡=π‘Š(𝐾)𝑇𝐾,𝐾(𝑔)π‘‘πœ‡|π‘Š(𝐾)=Ξ¨(𝑔)=0,(3.13) by (3.3). This completes the proof.

By applying the above result, we show that Lip(𝑋,𝐾,1) is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).

Bade et al. obtained the following result.

Lemma 3.2 (see [4, Lemma  3.3]). For each finite subset 𝐸 of 𝑋 and each β„ŽβˆˆLip(𝑋,𝛼), there exists π‘“βˆˆLip(𝑋,1) with 𝑓|𝐸=β„Ž|𝐸 and with ‖𝑓‖𝛼⩽2β€–β„Žβ€–π›Ό.

We now generalize the above lemma by applying it and Tietze’s extension theorem as follows.

Lemma 3.3. For each finite subset 𝐸 of 𝑋 and each β„ŽβˆˆLip(𝑋,𝐾,𝛼), there exists π‘“βˆˆLip(𝑋,𝐾,1) with 𝑓|𝐸=β„Ž|𝐸 and with ‖𝑓‖𝛼,𝐾⩽3β€–β„Žβ€–π›Ό,𝐾.

Proof. Let 𝐸 be a finite subset of 𝑋, and let β„ŽβˆˆLip(𝑋,𝐾,𝛼). Define 𝑔=β„Ž|𝐾. Then π‘”βˆˆLip(𝐾,𝛼). By Lemma 3.2, there exists 𝑔0∈Lip(𝐾,1) with 𝑔0|𝐸∩𝐾=𝑔|𝐸∩𝐾 and with ‖𝑔0‖𝛼⩽2‖𝑔‖𝛼. We now define the function 𝑔1∢𝐸βˆͺ𝐾→ℂ by 𝑔1𝑔(π‘₯)=0(π‘₯),π‘₯∈𝐾,β„Ž(π‘₯),π‘₯∈𝐸⧡𝐾.(3.14) Clearly, 𝐸βˆͺ𝐾 is a compact subset of 𝑋 and 𝑔1∈𝐢(𝐸βˆͺ𝐾). By Tietze’s extension theorem, there exists π‘“βˆˆπΆ(𝑋) such that 𝑓|𝐸βˆͺ𝐾=𝑔1 and ‖𝑓‖𝑋=‖𝑔1‖𝐸βˆͺ𝐾. It follows that π‘“βˆˆLip(𝑋,𝐾,1) and 𝑓|𝐸=β„Ž|𝐸. Furthermore, ‖𝑓‖𝛼,𝐾=‖‖𝑔1‖‖𝐸βˆͺ𝐾+𝑝𝛼𝑔0⩽‖‖𝑔1‖‖𝐾+‖‖𝑔1‖‖𝐸⧡𝐾+𝑃𝛼𝑔0ξ€Έ=‖‖𝑔0‖‖𝐾+β€–β„Žβ€–πΈβ§΅πΎ+𝑝𝛼𝑔0ξ€Έ=‖‖𝑔0‖‖𝛼+β€–β„Žβ€–πΈβ§΅πΎβ©½2‖𝑔‖𝛼+β€–β„Žβ€–πΈβ§΅πΎ=‖𝑔‖𝐾+𝑝𝛼(𝑔)+β€–β„Žβ€–πΈβ§΅πΎβ©½2β€–β„Žβ€–π‘‹+β€–β„Žβ€–πΈβ§΅πΎ+2𝑃𝛼,𝐾(β„Ž)β©½3β€–β„Žβ€–π›Ό,𝐾.(3.15) This completes the proof.

Theorem 3.4. Lip(𝑋,𝐾,1) is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾).

Proof. Take 𝑃=Lip(𝑋,𝐾,1). Then 𝑃 is a linear subspace of lip(𝑋,𝐾,𝛼) and β„Žβˆˆπ‘ƒ for all β„ŽβˆˆπΆ(𝑋) with β„Ž|𝐾=0. Let 𝐸 be a finite subset of 𝐾 and π‘“βˆˆlip(𝑋,𝐾,𝛼). By Lemma 3.3, there exists π‘”βˆˆπ‘ƒ with 𝑔|𝐸=𝑓|𝐸 and with ‖𝑔‖𝛼,𝐾⩽3‖𝑓‖𝛼,𝐾. Therefore, 𝑃 is dense in (lip(𝑋,𝐾,𝛼),‖⋅‖𝛼,𝐾), by Theorem 3.1.

Corollary 3.5. Lip(𝑋,1) is dense in (lip(𝑋,𝛼),‖⋅‖𝛼).

Proof. It is enough to take 𝐾=𝑋 in Theorem 3.4.

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