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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 575819, 22 pages
http://dx.doi.org/10.1155/2012/575819
Research Article

Variable Exponent Spaces of Differential Forms on Riemannian Manifold

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 30 May 2012; Accepted 22 July 2012

Academic Editor: AlbertoΒ Fiorenza

Copyright Β© 2012 Yongqiang Fu and Lifeng Guo. 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

We introduce the Lebesgue space and the exterior Sobolev space for differential forms on Riemannian manifold 𝑀 which are the Lebesgue space and the Sobolev space of functions on 𝑀, respectively, when the degree of differential forms to be zero. After discussing the properties of these spaces, we obtain the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous 𝑝(π‘š)-harmonic equations with variable growth in π‘Š01,𝑝(π‘š)(Ξ›π‘˜π‘€).

1. Introduction

Gol'dshteΔ­n et al. introduced spaces of differential forms on Riemannian manifold in [1–3]. The study of spaces for differential forms has been developed rapidly. For example, 𝐿𝑝-Cohomology and 𝐿𝑝,π‘ž-Cohomology and applications to some nonlinear PDE were studied in [4–6]; 𝐿𝑝 Hodge decomposition theory on the compact and complete Riemannian manifold were discussed in [7, 8]; properties of Riesz transforms of differential forms on complete Riemannian manifold were discussed in [9, 10]; the existence of minima of certain mean-coercive functionals is established in [11]. Many interesting results concerning 𝐴-harmonic equations have been established recently (see [12, 13] and the references therein).

After KovÑčik and RΓ‘kosnΓ­k first discussed the 𝐿𝑝(π‘₯)(Ξ©) and π‘Š1,𝑝(π‘₯)(Ξ©) spaces in [14], a lot of research has been done concerning these kinds of variable exponent spaces (see [15–19] and the references therein). The existence and uniqueness of solutions for 𝑝(π‘₯)-Laplacian Dirichlet problems with different types on bounded domains in ℝ𝑛 have been greatly discussed under various conditions (see [20] for the existence and [21] for the uniqueness). In recent years, the theory on problems with variable exponential growth conditions has important applications in nonlinear elastic mechanics (see [22]), electrorheological fluids (see [23, 24]).

The paper is organized as follows. In Section 2, we give the necessary definitions and some elementary properties of differential forms on Riemannian manifold. Moreover, we introduce the functional πœŒπ‘(π‘š),Ξ›π‘˜π‘€ on Ξ›π‘˜π‘€ and the spaces of differential forms 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) and π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€), then discuss some important properties. In Section 3, we show the existence and uniqueness of weak solution for Dirichlet problems of nonhomogeneous 𝑝(π‘š)-harmonic equations with variable growth in π‘Š01,𝑝(π‘š)(Ξ›π‘˜π‘€).

2. Preliminaries

  Let 𝑀 be an arbitrary smooth 𝑛-dimensional manifold (Hausdorff and with countable basis). Let π‘‡βˆ—π‘€=βˆͺπ‘šβˆˆπ‘€π‘‡βˆ—π‘šπ‘€ be the cotangent bundle on M and Ξ›π‘˜π‘‡βˆ—π‘€ (or Ξ›π‘˜π‘€) be the bundles of the exterior π‘˜-forms. We will call each fiber 𝑒 of the bundle Ξ›π‘˜π‘‡βˆ—π‘€ a exterior form of degree π‘˜ on the manifold M. Here, Ξ›0𝑀=ℝ and Ξ›π‘˜π‘€={0} in the case π‘˜>𝑛 or π‘˜<0. Given a exterior π‘˜-form 𝑒(π‘š) and a local chart π‘“π›ΌβˆΆπ‘ˆπ›Ό(βŠ‚π‘€)→ℝ𝑛, around π‘šβˆˆπ‘ˆπ›Ό, we define the representation of 𝑒(π‘š) in this local coordinates system as the exterior π‘˜-forms 𝑒𝛼 on 𝑓𝛼(π‘ˆπ›Ό)βŠ‚β„π‘› given by 𝑒𝛼𝑓𝛼𝑋(π‘š)ξ€Έξ€·1,𝑋2,…,π‘‹π‘˜ξ€Έ=ξ‚€ξ€·π‘“π›Όβˆ’1ξ€Έβˆ—π‘’ξ‚ξ€·π‘“π›Όπ‘‹(π‘š)ξ€Έξ€·1,𝑋2,…,π‘‹π‘˜ξ€Έξ€·=𝑒(π‘š)π‘‘π‘“π›Όβˆ’1𝑋1ξ€Έ,π‘‘π‘“π›Όβˆ’1𝑋2ξ€Έ,…,π‘‘π‘“π›Όβˆ’1ξ€·π‘‹π‘˜,ξ€Έξ€Έ(2.1) for any 𝑋1,𝑋2,…,π‘‹π‘˜βˆˆβ„π‘›, where π‘‘π‘“π›Όβˆ’1 is the induced map by π‘“π›Όβˆ’1 that takes vectors on T𝑓𝛼(π‘š)ℝ𝑛 into vectors on π‘‡π‘šπ‘€ and (π‘“π›Όβˆ’1)βˆ— is the induced map by π‘“π›Όβˆ’1 that takes exterior forms on π‘‡π‘šπ‘€ into exterior forms on 𝑇𝑓𝛼(π‘š)ℝ𝑛 (see [25]).

In this paper we will always assume (𝑀,𝑔) is an 𝑛-dimensional smooth orientable complete Riemannian manifold and βˆšπ‘‘πœ‡=det(𝑔𝑖𝑗)𝑑π‘₯ is the Riemannian volume element on (𝑀,𝑔), where the 𝑔𝑖𝑗 are the components of the Riemannian metric 𝑔 in the chart and 𝑑π‘₯ is the Lebesgue volume element of ℝ𝑛. A Riemannian metric 𝑔 on 𝑀 induces a scalar product on each fiber of the bundle Ξ›π‘˜π‘€. Hence for any exterior forms 𝑒 and 𝑣 of the same degree π‘˜, the scalar product βŸ¨π‘’,π‘£βŸ©=βŸ¨π‘’(π‘š),𝑣(π‘š)⟩ is defined at each point π‘šβˆˆπ‘€ and the norm of 𝑒 is given by the formula √|𝑒|=βŸ¨π‘’,π‘’βŸ©. Let π›ΎβˆΆ[a,𝑏]→𝑀 be a curve of class 𝐢1, the length of 𝛾 is ξ€œπΏ(𝛾)=π‘π‘ŽξƒŽπ‘”(𝛾(𝑑))𝑑𝛾𝑑𝑑(𝑑),𝑑𝛾𝑑𝑑(𝑑)π‘‘πœ‡.(2.2) For π‘š1,π‘š2βˆˆπ‘€, let 𝐢1π‘š1,π‘š2 be the space of piecewise 𝐢1 curves π›ΎβˆΆ[π‘Ž,𝑏]→𝑀 such that 𝛾(π‘Ž)=π‘š1 and 𝛾(𝑏)=π‘š2. One can define a distance 𝑑𝑔(π‘š1,π‘š2)=inf𝐢1π‘š12,π‘šπΏ(𝛾) on 𝑀.

The Grassman algebra Ξ›βˆ—π‘€=βŠ•Ξ›π‘˜π‘€ is a graded algebra with respect to the exterior products. We denote by 𝐿1loc(Ξ›π‘˜π‘€) the space of locally integrable exterior forms of degree π‘˜ (i.e., differential π‘˜-forms) on 𝑀. The local integrability of an exterior π‘˜-form means the local integrability of the components of its coordinate representation in each chart of the Riemannian manifold 𝑀. We denote by πΆβˆžπ‘(Ξ›π‘˜π‘€) the vector space of smooth differential forms of degree π‘˜ with compact support on 𝑀.

Let (𝑀,𝑔) be is an 𝑛-dimensional smooth orientable Riemannian manifold. We define the integral of 𝑒, a exterior 𝑛-form 𝑒 with compact support on 𝑀 (see [26]). Let (π‘ˆπ›Ό,𝑓𝛼) be a local chart of (𝑀,𝑔), we have a partition of unity {πœ‹π›Ό} subordinate to this cover. Recall that supp(πœ‹π›Ό)βŠ†π‘ˆπ›Ό and βˆ‘π›Όπœ‹π›Ό=1. Thus, every πœ‹π›Όπ‘’ is an exterior 𝑛-form whose support is a subset of π‘ˆπ›Ό and we may write βˆ‘π‘’=π›Όπœ‹π›Όπ‘’. By definition ξ€œπ‘€ξ“π‘’=π›Όξ€œπ‘ˆπ›Όπœ‹π›Όξ“π‘’=π›Όξ€œπ‘“π›Ό(π‘ˆπ›Ό)ξ€·π‘“π›Όβˆ’1ξ€Έβˆ—ξ€·πœ‹π›Όπ‘’ξ€Έ=ξ“π›Όξ€œπ‘“π›Ό(π‘ˆπ›Ό)𝑔detπ‘–π‘—ξ€Έπœ‹π›Όπ‘’ξ‚Άβˆ˜π‘“π›Όβˆ’1𝑑π‘₯.(2.3)

We will identify each exterior form of degree π‘˜ on the 𝑛-dimensional Riemannian manifold 𝑀 with an exterior (π‘›βˆ’π‘˜)-form on 𝑀 (see [27]). Using this identification, we can assume that each exterior form 𝑒 has a weak exterior differential 𝑑𝑒.

Definition 2.1 (see [6]). We say that an exterior form π‘£βˆˆπΏ1loc(Ξ›π‘˜π‘€) is the weak exterior differential of a form π‘’βˆˆπΏ1loc(Ξ›π‘˜βˆ’1𝑀) and we write 𝑑𝑒=𝑣 if for each πœ‘βˆˆπΆβˆžπ‘(Ξ›π‘˜π‘€), one has ξ€œπ‘€π‘£βˆ§πœ‘=(βˆ’1)π‘˜ξ€œπ‘€π‘’βˆ§π‘‘πœ‘.(2.4)

The operator β‹†βˆΆΞ›π‘˜π‘€β†’Ξ›π‘›βˆ’π‘˜π‘€, also called Hodge star operator (see [27]), has the following properties: for 𝑒,π‘£βˆˆΞ›π‘˜π‘€ and πœ‘,πœ“βˆˆπΆβˆž(𝑀)(π‘Ž1)⋆(πœ‘π‘’+πœ“π‘£)=πœ‘β‹†π‘’+πœ“β‹†π‘£,(π‘Ž2)⋆⋆𝑒=(βˆ’1)π‘˜(π‘›βˆ’π‘˜)𝑒,(π‘Ž3)β‹†πœ‘=πœ‘π‘‘πœ‡,(π‘Ž4)βŸ¨π‘’,π‘£βŸ©=⋆(π‘’βˆ§β‹†π‘£)=βŸ¨β‹†π‘’,β‹†π‘£βŸ©,(π‘Ž5)π‘’βˆ§β‹†π‘£=βŸ¨π‘’,π‘£βŸ©π‘‘πœ‡.

By the operator ⋆ and the exterior differentiation 𝑑 we define the codifferential operator 𝛿 by the formula 𝛿𝑒=(βˆ’1)𝑛(π‘˜+1)+1β‹†π‘‘β‹†π‘’βˆˆπΏ1locξ€·Ξ›π‘˜βˆ’1𝑀,(2.5) for any differential form π‘’βˆˆπΏ1loc(Ξ›π‘˜π‘€).

The Riemannian measure and the characteristic function of a set π΄βŠ†π‘€ will be denoted by πœ‡(𝐴) and πœ’π΄, respectively.

Let 𝒫(𝑀) be the set of all measurable functions π‘βˆΆπ‘€β†’[1,∞]. For π‘βˆˆπ’«(𝑀) we put 𝑀1=𝑀𝑝1={π‘šβˆˆπ‘€βˆΆπ‘(π‘š)=1},π‘€βˆž=π‘€π‘βˆž={π‘šβˆˆπ‘€βˆΆπ‘(π‘š)=∞}, 𝑀0=𝑀⧡(𝑀1βˆͺπ‘€βˆž), π‘βˆ—=essinf𝑀0𝑝(π‘š) and π‘βˆ—=esssup𝑀0𝑝(π‘š) if πœ‡(𝑀0)>0, π‘βˆ—=π‘βˆ—=1 if πœ‡(𝑀0)=0,𝑐𝑝=β€–πœ’π‘€0β€–πΏβˆž(𝑀)+β€–πœ’π‘€1β€–πΏβˆž(𝑀)+β€–πœ’π‘€βˆžβ€–πΏβˆž(𝑀) and π‘Ÿπ‘=𝑐𝑝+1/π‘βˆ—+1/π‘βˆ—. We always assume that π‘βˆˆπ’«(𝑀), 𝒫1(𝑀)=𝒫(𝑀)∩𝐿∞(𝑀) and 𝒫2(𝑀)={π‘βˆˆπ’«1(𝑀)∢1<essinf𝑀𝑝(π‘š)}. We use the convention 1/∞=0.

For a differential π‘˜-form 𝑒 on 𝑀 we define the functional πœŒπ‘(π‘š),Ξ›π‘˜π‘€ by πœŒπ‘(π‘š),Ξ›π‘˜π‘€(ξ€œπ‘’)=π‘€β§΅π‘€βˆž|𝑒|𝑝(π‘š)π‘‘πœ‡+esssupπ‘€βˆž|𝑒|.(2.6)

The Lebesgue space 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) is the space of differential forms 𝑒 in 𝐿1loc(Ξ›π‘˜π‘€) such that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(πœ†π‘’)<∞forsomeπœ†=πœ†(𝑒)>0,(2.7) with the following norm ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)=infπœ†>0βˆΆπœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ‚€π‘’πœ†ξ‚ξ‚‡.≀1(2.8)

The exterior Sobolev space π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) consists of such forms π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€) for which π‘‘π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜+1𝑀). The norm is defined by β€–π‘’β€–π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€)=‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)+‖𝑑𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜+1𝑀).(2.9)

The space π‘Š01,𝑝(π‘š)(Ξ›π‘˜π‘€) is defined as the closure of πΆβˆžπ‘(Ξ›π‘˜π‘€) in π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€).

Note that 𝐿𝑝(π‘š)(Ξ›0𝑀), π‘Š1,𝑝(π‘š)(Ξ›0𝑀) and π‘Š01,𝑝(π‘š)(Ξ›0𝑀) are spaces of functions on 𝑀. In this paper we denote them by 𝐿𝑝(π‘š)(𝑀), π‘Š1,𝑝(π‘š)(𝑀) and π‘Š01,𝑝(π‘š)(𝑀).

Given π‘βˆˆπ’«(𝑀) we define the conjugate function π‘ξ…ž(π‘š)βˆˆπ’«(𝑀) by π‘ξ…žβŽ§βŽͺ⎨βŽͺ⎩(π‘š)=∞ifπ‘šβˆˆπ‘€1,1ifπ‘šβˆˆπ‘€βˆž,𝑝(π‘š)𝑝(π‘š)βˆ’1ifπ‘šβˆˆπ‘€0.(2.10)

Similar to the proof of properties of πœŒπ‘(π‘š),Ξ© and 𝐿𝑝(π‘š)(Ξ©) for Ξ©βŠ‚β„π‘› (see [15, 16, 18]), it is easy to see that πœŒπ‘(π‘š),Ξ›π‘˜π‘€ and 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) has the following properties:(𝑏1)πœŒπ‘(π‘š),Ξ›π‘˜π‘€ is convex.(𝑏2)πœŒπ‘(π‘š),Ξ›π‘˜π‘€(π‘’πœ’π΄)β‰€πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒) for every subset π΄βŠ‚π‘€ and differential forms 𝑒.(𝑏3) If |𝑒(π‘š)|β‰₯|𝑣(π‘š)| for a.e. π‘šβˆˆπ‘€ and if πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)<∞, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)β‰₯πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑣), the last inequality is strict if |𝑒|β‰ |𝑣|.(𝑏4) If 0<πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)<∞, then the function πœ†β†’πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/πœ†) is continuous and decreasing on the interval [1,∞).(𝑏5) If 0<‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€))≀1.(𝑏6) If π‘βˆ—<∞, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€))=1 for every differential forms 𝑒 with 0<‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞.(𝑏7) If ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀1, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)≀‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).(𝑏8) If π‘βˆˆπ’«1(𝑀) and ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)>1, then β€–π‘’β€–π‘βˆ—πΏπ‘(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έβ‰€πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)β‰€β€–π‘’β€–π‘βˆ—πΏπ‘(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έ.(2.11)(𝑏9) If π‘βˆˆπ’«1(𝑀) and ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<1, then β€–π‘’β€–π‘βˆ—πΏπ‘(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έβ‰₯πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)β‰₯β€–π‘’β€–π‘βˆ—πΏπ‘(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έ.(2.12)

Lemma 2.2. If 𝑝(π‘š)βˆˆπ’«(𝑀), then the inequality ξ€œπ‘€||||βŸ¨π‘’,π‘£βŸ©π‘‘πœ‡β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)(2.13) holds for every π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€), π‘£βˆˆπΏπ‘β€²(π‘š)(Ξ›π‘˜π‘€).

Proof. Obviously, we can suppose that ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰ 0, ‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)β‰ 0 and πœ‡(𝑀0)>0. We have ||||||||1<𝑝(π‘š)<∞,𝑒(π‘š)<∞,𝑣(π‘š)<∞a.e.π‘šβˆˆπ‘€0.(2.14) By Young inequality, we have ||||βŸ¨π‘’,π‘£βŸ©β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)≀1𝑝(π‘š)|𝑒|‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)𝑝(π‘š)+1π‘ξ…žξƒ©(π‘š)|𝑣|‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)ξƒͺ𝑝′(π‘š).(2.15) Integrating over 𝑀0 we obtain ξ€œπ‘€0||||βŸ¨π‘’,π‘£βŸ©β€–π‘£β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)≀1π‘‘πœ‡π‘βˆ—ξ€œπ‘€0ξ‚΅|𝑒|‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)𝑝(π‘₯)ξ‚΅1π‘‘πœ‡+1βˆ’π‘βˆ—ξ‚Άξ€œπ‘€0|𝑣|‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)ξƒͺ𝑝′(π‘š)1π‘‘πœ‡β‰€1+π‘βˆ—βˆ’1π‘βˆ—.(2.16) Then by (𝑏2), we have ξ€œπ‘€||||ξ‚΅1βŸ¨π‘’,π‘£βŸ©π‘‘πœ‡β‰€1+π‘βˆ—βˆ’1π‘βˆ—ξ‚Άβ€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€)β€–β€–πœ’π‘€0β€–β€–πΏβˆž(𝑀)+β€–β€–π‘’πœ’π‘€1‖‖𝐿1(Ξ›π‘˜π‘€)β€–β€–π‘£πœ’π‘€1β€–β€–πΏβˆž(Ξ›π‘˜π‘€)+β€–β€–π‘’πœ’π‘€βˆžβ€–β€–πΏβˆž(Ξ›π‘˜π‘€)β€–β€–π‘£πœ’π‘€βˆžβ€–β€–πΏ1(Ξ›π‘˜π‘€)β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘˜π‘€),(2.17)

For differential π‘˜-forms 𝑒 on 𝑀, we define β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)=supπœŒπ‘β€²π‘€(π‘š),Ξ›π‘›βˆ’π‘˜(𝑣)≀1ξ€œπ‘€π‘’βˆ§π‘£.(2.18)

We denote by Ξ›(π‘˜,𝑛) the set of ordered multi-indices (𝑖1,𝑖2,…,π‘–π‘˜) of integers 1≀𝑖1<𝑖2<β‹―<π‘–π‘˜β‰€π‘›. Let 𝐼=(𝑖1,𝑖2,…,π‘–π‘˜) be a multi-index from Ξ›(π‘˜,𝑛). The complement πΌβˆ— of the multi-index 𝐼 is the multi-index πΌβˆ—=(π‘–π‘˜+1,π‘–π‘˜+2,…,𝑖𝑛) in Ξ›(π‘›βˆ’π‘˜,𝑛) where the components 𝑖𝑙 are in {1,…,𝑛}⧡{𝑖1,𝑖2,…,π‘–π‘˜} for all 𝑙=π‘˜+1,…,𝑛.

Let π‘₯1,…,π‘₯𝑛 be the orientable coordinates on 𝑀. Each differential π‘˜-form 𝑒 can be written as the linear combination 𝑒=1≀𝑖1<β‹―<π‘–π‘˜β‰€π‘›π‘’π‘–1,…,π‘–π‘˜π‘‘π‘₯𝑖1βˆ§β‹―βˆ§π‘‘π‘₯π‘–π‘˜=ξ“πΌβˆˆΞ›(π‘˜,𝑛)𝑒𝐼𝑑π‘₯𝐼.(2.19) Here 𝑒𝐼 are the components of 𝑒 with respect to natural basis 𝑑π‘₯𝐼=𝑑π‘₯𝑖1βˆ§β‹―βˆ§π‘‘π‘₯π‘–π‘˜ξ€·π‘–,𝐼=1,𝑖2,…,π‘–π‘˜ξ€ΈβˆˆΞ›(π‘˜,𝑛).(2.20) For a differential (π‘›βˆ’π‘˜)-form βˆ‘π‘£=πΏβˆˆΞ›(π‘˜,𝑛)π‘£πΏβˆ—π‘‘π‘₯πΏβˆ—, we have π‘’βˆ§π‘£=(βˆ’1)π‘˜(π‘›βˆ’π‘˜)π‘’βˆ§β‹†β‹†π‘£=(βˆ’1)π‘˜(π‘›βˆ’π‘˜)βŸ¨π‘’,β‹†π‘£βŸ©π‘‘πœ‡=βŸ¨β‹†π‘’,π‘£βŸ©π‘‘πœ‡.(2.21) Note that ⋆𝑑π‘₯𝐼=√det(𝑔𝑖𝑗)βˆ‘π½βˆˆΞ›(π‘˜,𝑛)βˆπ‘˜π›Ύ=1π‘”π‘–π›Ύπ‘—π›ΎπœŽ(𝐽)𝑑π‘₯π½βˆ—, and hence ξ”βŸ¨β‹†π‘’,π‘£βŸ©=𝑔detπ‘–π‘—ξ€Έξ“π‘˜πΌ,𝐽,πΏβˆˆΞ›(π‘˜,𝑛)𝛾=1𝑔𝑖𝛾𝑗𝛾𝑛𝛽=π‘˜+1π‘”π‘—π›½π‘™π›½πœŽ(𝐽)π‘’πΌπ‘£πΏβˆ—on𝑀,(2.22) where 𝑔𝑖𝑗 are the components of the inverse matrix of (𝑔𝑖𝑗) and 𝜎(𝐽) is the signature of the permutation (𝑗1⋯𝑗𝑛) in the set {1⋯𝑛}.

We consider an arbitrary local chart π‘“βˆΆπ‘‰(βŠ‚π‘€)→ℝ𝑛 on 𝑀. Let π‘ˆ be any open set in 𝑀, whose closure is compact and is contained in 𝑉. Note that the components 𝑔𝑖𝑗 of 𝑔 in (π‘ˆ,𝑓) satisfy 1/2𝛿𝑖𝑗≀𝑔𝑖𝑗≀2𝛿𝑖𝑗 as bilinear forms. Then ξ„Άξ„΅ξ„΅βŽ·βŸ¨β‹†π‘’,π‘£βŸ©=𝑛𝑙=1π‘”π‘™π‘™ξ“πΌβˆˆΞ›(π‘˜,𝑛)𝜎(𝐼)π‘’πΌπ‘£πΌβˆ—on𝑀.(2.23) Thus, if sgnπ‘£πΌβˆ—=𝜎(𝐼)sgn𝑒𝐼, βˆ‘πœ”=πΌβˆˆΞ›(π‘˜,𝑛)πœ”πΌβˆ—π‘‘π‘₯πΌβˆ— with πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(πœ”)≀1 and πœ”πΌβˆ—=Β±π‘£πΌβˆ—, we have βŸ¨β‹†π‘’,πœ”βŸ©β‰€βŸ¨β‹†π‘’,π‘£βŸ©,2βˆ’π‘›/2𝐼||𝑒𝐼||||π‘£πΌβˆ—||β‰€βŸ¨β‹†π‘’,π‘£βŸ©β‰€2𝑛/2𝐼||𝑒𝐼||||π‘£πΌβˆ—||on𝑀.(2.24) Integrating on 𝐾 and 𝑀, by (2.18) we have β€–β€–||0β‰€π‘’πœ’πΎ||‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀‖|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€),(2.25) for any compact subset 𝐾 on 𝑀. Furthermore, It is easy to see that it is a norm on the class of differential π‘˜-forms 𝑒 with β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞.

Lemma 2.3. Let β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞ and πœŒπ‘ξ…ž(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)<∞. Then ||||ξ€œπ‘€||||β‰€ξ‚»π‘’βˆ§π‘£β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)π‘–π‘“πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€πœŒ(𝑣)≀1,𝑝′(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)π‘–π‘“πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)>1.(2.26)

Proof. The first case follows from (2.18). Assume that πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)>1, we have πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€ξ‚΅π‘£πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€ξ‚Άβ‰€πœŒ(𝑣)𝑝′(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)=1,(2.27) and so ||||ξ€œπ‘€||||π‘’βˆ§π‘£=πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€||||ξ€œ(𝑣)π‘€π‘£π‘’βˆ§πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€||||(𝑣)β‰€πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€β€–β€–(𝑣)|𝑒|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).(2.28)

Lemma 2.4. If πœ‡(𝑀)=πœ‡(𝑀0),πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)<∞ and β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀1, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)≀1.

Proof. If this is not true, we may assume that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)>1, by (𝑏4) there exist πœ†>1 such that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/πœ†)=1. Set 𝑣=|𝑒|𝑝(π‘š)βˆ’2πœ†π‘(π‘š)βˆ’1(⋆𝑒),π‘šβˆˆπ‘€,(2.29) we have πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)=πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/πœ†)=1 and so β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰₯ξ€œπ‘€π‘’βˆ§π‘£=πœ†πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ‚€π‘’πœ†ξ‚=πœ†>1,(2.30) which is a contradiction.

Lemma 2.5. If β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀1, then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)≀𝑐𝑝‖|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).

Proof. First, suppose that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)<∞. We have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)=𝑗=0,1,βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’π‘—ξ€Έ,(2.31) where 𝑒𝑗=π‘’πœ’π‘€π‘—, 𝑗=0,1,∞. Set 𝑣1=ξ‚»|𝑒|βˆ’1⋆𝑒1𝑣if|𝑒|β‰ 0,0if|𝑒|=0,0=ξ‚»|𝑒|𝑝(π‘š)βˆ’2⋆𝑒0ξ€Έif|𝑒|β‰ 0,0if|𝑒|=0.(2.32) Then πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣1)=esssup𝑀1|𝑣1|=1 and due to Lemma 2.4, πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€ξ€·π‘£0ξ€Έ=ξ€œπ‘€0||𝑒0||𝑝(π‘š)π‘‘πœ‡β‰€1.(2.33) Hence, Lemma 2.3 yields πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’π‘—ξ€Έ=ξ€œπ‘€β§΅π‘€βˆžπ‘’βˆ§π‘£π‘—β‰€β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€),𝑗=0,1.(2.34) If πœ‡(π‘€βˆž)>0, then for every πœ€βˆˆ(0,1) there exists a set π·βŠ‚π‘€βˆž such that 0<πœ‡(𝐷)<∞ and |𝑒(π‘š)|β‰₯esssupπ‘€βˆž|𝑒|πœ€, π‘šβˆˆπ·. Take π‘£βˆž=ξ‚»πœ‡(𝐷)βˆ’1πœ’π·|𝑒|βˆ’1(⋆𝑒)if|𝑒|β‰ 0,0if|𝑒|=0,(2.35) we have πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(π‘£βˆžβˆ«)=π·πœ‡(𝐷)βˆ’1|𝑒|βˆ’1|⋆𝑒|π‘‘πœ‡β‰€1 and so β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰₯ξ€œπ‘€π‘’βˆ§π‘£βˆž=πœ‡(𝐷)βˆ’1ξ€œπ·|𝑒|π‘‘πœ‡β‰₯πœ€esssupπ‘€βˆž|𝑒|=πœ€πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’βˆžξ€Έ.(2.36) Letting πœ€β†’1 we obtain πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’βˆžξ€Έβ€–β‰€β€–|𝑒|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).(2.37) Hence, (2.31)–(2.37) yield the desired results.
To avoid the assumption πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)<∞ we define differential π‘˜-forms 𝑒𝑑=⎧βŽͺ⎨βŽͺβŽ©π‘’πœ’πΊπ‘‘if|𝑒|≀𝑑,π‘‘π‘’πœ’πΊπ‘‘|𝑒|if|𝑒|>𝑑,(2.38) where {𝐺𝑑} is a sequence of compact sets such that πΊπ‘‘βŠ‚πΊπ‘‘+1βŠ‚π‘€, πœ‡(𝐺𝑑)<∞ for π‘‘βˆˆβ„• and 𝑀=βˆͺβˆžπ‘‘=1𝐺𝑑. Then for every 𝑒𝑑 we have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)<∞,β€–|𝑒𝑑‖|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀‖|𝑒‖|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀1. By the first part of the proof, πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)≀𝑐𝑝‖|𝑒‖|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€). It follows let π‘‘β†’βˆž.

Lemma 2.6. For every π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€), the following inequalities hold π‘π‘βˆ’1‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀‖|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€).(2.39) Furthermore, we have 𝐿𝑝(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έ=ξ€½π‘’βˆˆπΏ1locξ€·Ξ›π‘˜π‘€ξ€ΈβˆΆβ€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)ξ€Ύ.<∞(2.40)

Proof. Let π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€). If πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)≀1, then ‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)≀1 and HΓΆlder inequality yields ξ€œπ‘€π‘’βˆ§π‘£β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€).(2.41) This gives the second inequality in (2.39) and, consequently, β€–|𝑒‖|𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞.
Conversely, we can suppose that 0<β€–|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)<∞. By Lemma 2.5 and following inequalitiy β€–β€–β€–β€–||||𝑒0<𝑐𝑝‖|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)ξ€Έ||||‖‖‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)=π‘π‘βˆ’1≀1,(2.42) we get πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/(𝑐𝑝‖|𝑒|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)))β‰€π‘π‘π‘π‘βˆ’1=1. The first inequality in (2.39) follows and then π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€).

We shall say that differential π‘˜-forms π‘’π‘‘βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€)converge modularly to a differential π‘˜-form π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€) if limπ‘‘β†’βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€(π‘’π‘‘βˆ’π‘’)=0.

Next, we consider the relationship between convergence in norm, convergence in modular, and convergence in measure. For the corresponding results for domains in ℝ𝑛, readers can be referred to [15, 16].

Lemma 2.7. If π‘βˆˆπ’«1(𝑀), then πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)β†’0 if and only if ‖𝑒𝑑‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β†’0.

Proof. According to Lemmas 2.5 and 2.6, the norm convergence is stronger than the modular convergence. Suppose that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)β†’0, and take πœ€βˆˆ(0,1]. For sufficiently large 𝑑 we have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)<πœ€β‰€1 and so πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξƒ©π‘’π‘‘ξ€·πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’π‘‘ξ€Έξ€Έ1/π‘βˆ—ξƒͺβ‰€πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’π‘‘ξ€ΈπœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ€·π‘’π‘‘ξ€Έ=1,(2.43) that is, ‖𝑒𝑑‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀(πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑))1/π‘βˆ—β‰€πœ€1/π‘βˆ—. Hence, ‖𝑒𝑑‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β†’0.

Lemma 2.8. If π‘βˆˆπ’«1(𝑀) and πœ‡(𝑀)<∞, then β€–π‘’π‘‘βˆ’π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β†’0 if and only if 𝑒𝑑 converges to 𝑒 on 𝑀 in measure and limπ‘‘β†’βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)=πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒).

Proof. If β€–π‘’π‘‘βˆ’π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β†’0, by Lemma 2.7limπ‘‘β†’βˆžξ€œπ‘€||𝑒𝑑||βˆ’π‘’π‘(π‘š)π‘‘πœ‡=0,(2.44) then it is easy to see that 𝑒𝑑 converges to 𝑒 on 𝑀 in measure. Hence by πœ‡(𝑀)<∞, |𝑒𝑑|𝑝(π‘š) converges to |𝑒|𝑝(π‘š) on 𝑀 in measure and the integrals of the functions |π‘’π‘‘βˆ’π‘’|𝑝(π‘š) possess absolutely equicontinuity on 𝑀. Since ||𝑒𝑑||𝑝(π‘š)≀2π‘βˆ—βˆ’1ξ‚€||𝑒𝑑||βˆ’π‘’π‘(π‘š)+|𝑒|𝑝(π‘š),(2.45) the integrals of the |𝑒𝑑|𝑝(π‘š) are also absolutely equicontinuous on 𝑀. By Vitali convergence theorem (see [28]), we deduce that limπ‘‘β†’βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)=πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒).
Conversely, if 𝑒𝑑 converges to 𝑒 on 𝑀 in measure, we can deduce that |π‘’π‘‘βˆ’π‘’|𝑝(π‘š) converges to 0 on 𝑀 in measure. Similar to the above proof, by the inequality ||𝑒𝑑||βˆ’π‘’π‘(π‘š)≀2π‘βˆ—βˆ’1ξ‚€||𝑒𝑑||𝑝(π‘š)+|𝑒|𝑝(π‘š),(2.46) and limπ‘‘β†’βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒𝑑)=πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒), we get limπ‘‘β†’βˆžπœŒπ‘(π‘š),Ξ›π‘˜π‘€(π‘’π‘‘βˆ’π‘’)=0.

Lemma 2.9. If π‘βˆˆπ’«1(𝑀), then 𝐿∞(Ξ›π‘˜π‘€)βˆ©πΏπ‘(π‘š)(Ξ›π‘˜π‘€) is dense in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).

Proof. Let π‘š0 be some point of 𝑀, 𝑑𝑔 be the distance associated to 𝑔 and 𝐺𝑑={π‘šβˆˆπ‘€βˆΆπ‘‘π‘”(π‘š0,π‘š)<𝑑,π‘‘βˆˆβ„•}. Given π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€), we define sequence of differential π‘˜-forms by 𝑒𝑑=⎧βŽͺ⎨βŽͺβŽ©π‘’πœ’πΊπ‘‘if|𝑒|≀𝑑,π‘‘π‘’πœ’πΊπ‘‘|𝑒|if|𝑒|>𝑑.(2.47) Then π‘’π‘‘βˆˆπΏβˆž(Ξ›π‘˜π‘€) and by Lebesgue dominated convergence theorem, we have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(π‘’βˆ’π‘’π‘‘)β†’0. Hence, by Lemma 2.7β€–π‘’βˆ’π‘’π‘‘β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β†’0.

Lemma 2.10. If π‘βˆˆπ’«1(𝑀), then πΆβˆžπ‘(Ξ›π‘˜π‘€) is dense in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).

Proof. Since π‘βˆˆπ’«1(𝑀), we have πΆβˆžπ‘(Ξ›π‘˜π‘€)βŠ‚πΏπ‘(π‘š)(Ξ›π‘˜π‘€). By Lemma 2.9, there is a differential π‘˜-form 𝑒𝑑0∈𝐿∞(Ξ›π‘˜π‘€)βˆ©πΏπ‘(π‘š)(Ξ›π‘˜π‘€) such that β€–β€–π‘’βˆ’π‘’π‘‘0‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.48)
By Luzin theorem there exists a continuous π‘˜-form πœ‘βˆˆπΆ(Ξ›π‘˜π‘€) and an open set π·βŠ‚π‘€ such that ⎧βŽͺ⎨βŽͺβŽ©ξƒ©πœ€πœ‡(𝐷)<min1,2‖‖𝑒𝑑0β€–β€–πΏβˆž(Ξ›π‘˜π‘€)ξƒͺπ‘βˆ—βŽ«βŽͺ⎬βŽͺ⎭,(2.49)πœ‘=𝑒𝑑0 on 𝑀⧡𝐷 and sup𝑀|πœ‘|=sup𝑀⧡𝐷|𝑒𝑑0|≀‖𝑒𝑑0β€–πΏβˆž(Ξ›π‘˜π‘€). Thus, πœŒπ‘(π‘š),Ξ›π‘˜π‘€ξ‚΅π‘’π‘‘0βˆ’πœ‘πœ€ξ‚ΆβŽ§βŽͺ⎨βŽͺβŽ©ξƒ©2‖‖𝑒≀max1,𝑑0β€–β€–πΏβˆž(Ξ›π‘˜π‘€)πœ€ξƒͺπ‘βˆ—βŽ«βŽͺ⎬βŽͺβŽ­πœ‡(𝐷)≀1,(2.50) that is, ‖‖𝑒𝑑0β€–β€–βˆ’πœ‘πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.51)
Since πœ‘βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€), we have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(πœ‘)<∞ and there exists a bounded open set πΊβŠ‚π‘€ such that πœŒπ‘(π‘š),Ξ›π‘˜π‘€(πœ‘πœ’π‘€β§΅πΊ/πœ€)≀1, that is, β€–β€–πœ‘βˆ’πœ‘πœ’πΊβ€–β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.52)
Let β„Ž be a polynomial differential π‘˜-form with sup𝐺|πœ‘βˆ’β„Ž|<πœ€min{1,πœ‡(𝐺)βˆ’1}. The polynomial differential π‘˜-form means the components of its coordinate representation in each chart of the manifold 𝑀 are polynomial functions. Then πœŒπ‘(π‘š),Ξ›π‘˜π‘€((πœ‘πœ’πΊβˆ’β„Žπœ’πΊ)/πœ€)≀min{1,πœ‡(𝐺)βˆ’1}πœ‡(𝐺)≀1, that is, β€–β€–πœ‘πœ’πΊβˆ’β„Žπœ’πΊβ€–β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.53) Finally, there exists a compact set πΎβŠ‚πΊ such that β€–β„Žπœ’πΊβˆ’β„Žπœ’πΎβ€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€. Let πœ‹βˆˆπΆβˆžπ‘(𝐺) with 0β‰€πœ‹β‰€1 in 𝐺 and πœ‹=1 on 𝐾 we obtain the estimate β€–β€–β„Žπœ’πΊβ€–β€–βˆ’πœ‹β„ŽπΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€β€–β€–β„Žπœ’πΊβˆ’β„Žπœ’πΎβ€–β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.54) From (2.48)–(2.54), we get β€–π‘’βˆ’πœ‹β„Žβ€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)≀5πœ€.(2.55) Obviously, πœ‹β„ŽβˆˆπΆβˆžπ‘(Ξ›π‘˜π‘€).

Theorem 2.11. If π‘βˆˆπ’«1(𝑀), then the space 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) is separable.

Proof. Let π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€),πœ€>0. By the proof of Lemma 2.10, we can fine a continuous π‘˜-form πœ‘βˆˆπΆ(Ξ›π‘˜π‘€) and a set 𝐺𝑑0={π‘šβˆˆπ‘€βˆΆπ‘‘π‘”(π‘š0,π‘š)<𝑑0} such that β€–π‘’βˆ’πœ‘β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)β€–β€–β‰€πœ€,πœ‘πœ’π‘€β§΅πΊπ‘‘0‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€,(2.56) Let β„Ž be a polynomial differential π‘˜-form with sup𝐺𝑑0|πœ‘βˆ’β„Ž|<πœ€min{1,πœ‡(𝐺𝑑0)βˆ’1}, 𝑣 be a polynomial differential π‘˜-form with rational coefficients and sup𝐺𝑑0|β„Žβˆ’π‘£|<πœ€min{1,πœ‡(𝐺𝑑0)βˆ’1}. Then we have β€–β€–πœ‘πœ’πΊπ‘‘0βˆ’β„Žπœ’πΊπ‘‘0‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β€–β€–β‰€πœ€,π‘£πœ’πΊπ‘‘0βˆ’β„Žπœ’πΊπ‘‘0‖‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)β‰€πœ€.(2.57) Thus, β€–β€–π‘£πœ’πΊπ‘‘0β€–β€–βˆ’π‘’πΏπ‘(π‘š)(Ξ›π‘˜π‘€)≀4πœ€.(2.58) Therefore, we conclude that the set of all differential π‘˜-forms π‘£πœ’πΊπ‘‘ is dense in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€).

Theorem 2.12. If π‘βˆˆπ’«(𝑀), then the space 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) is complete.

Proof. Let {π‘’π‘‘βˆΆπ‘’π‘‘=βˆ‘πΌ(𝑒𝑑)𝐼𝑑π‘₯𝐼} be a Cauchy sequence of differential π‘˜-forms in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) and πœ€>0. Let {𝐺𝑙} be a sequence of compact sets such that πΊπ‘™βŠ‚πΊπ‘™+1βŠ‚π‘€ for π‘™βˆˆβ„• and 𝑀=βˆͺβˆžπ‘™=1𝐺𝑙. There exists 𝑑0βˆˆβ„• such that supπœŒπ‘β€²π‘€(π‘š),Ξ›π‘›βˆ’π‘˜(𝑣)≀1ξ€œπΊπ‘™ξ€·π‘’π‘‘βˆ’π‘’πœξ€Έβˆ§π‘£β‰€πœ€,(2.59) for every 𝑑,𝜏β‰₯𝑑0 and π‘™βˆˆβ„•. By (2.24) we have ξ€œπΊπ‘™ξ“πΌ||ξ€·π‘’π‘‘βˆ’π‘’πœξ€ΈπΌ||||π‘£πΌβˆ—||π‘‘πœ‡β‰€2𝑛/2πœ€,(2.60) for every βˆ‘π‘£=πΌπ‘£πΌβˆ—π‘‘π‘₯πΌβˆ—, πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€(𝑣)≀1 and sgnπ‘£πΌβˆ—=𝜎(𝐼)sgn(π‘’π‘‘βˆ’π‘’πœ)𝐼. We define 𝑣𝑙=πœ‘π‘™πœ’πΊπ‘™ where |πœ‘π‘™|=(1+πœ‡(𝐺𝑙))βˆ’1 for π‘™βˆˆβ„•. Then πœŒπ‘β€²(π‘š),Ξ›π‘›βˆ’π‘˜π‘€ξ€·π‘£π‘™ξ€Έβ‰€ξ€œπΊπ‘™ξ€·ξ€·πΊ1+πœ‡π‘™ξ€Έξ€Έβˆ’π‘β€²(π‘š)ξ€·ξ€·πΊπ‘‘πœ‡+1+πœ‡π‘™ξ€Έξ€Έβˆ’1≀1,(2.61) thus, by (2.60) we get ξ€œπΊπ‘™||π‘’π‘‘βˆ’π‘’πœ||π‘‘πœ‡β‰€2π‘˜/2ξ€œπΊπ‘™ξ“πΌ||ξ€·π‘’π‘‘βˆ’π‘’πœξ€ΈπΌ||π‘‘πœ‡β‰€πœ€2𝑛𝐺1+πœ‡π‘™ξ€Έξ€Έ,for𝑑,𝜏β‰₯𝑑0,π‘™βˆˆβ„•.(2.62) This means that the sequence {𝑒𝑑} is Cauchy in each 𝐿1(Ξ›π‘˜πΊπ‘™). By induction we may find subsequences {𝑒𝑑(𝑙)}𝑑 and differential π‘˜-forms 𝑒(𝑙)∈𝐿1(Ξ›π‘˜πΊπ‘™) such that 𝑒𝑑(𝑙)→𝑒(𝑙) a.e.  on𝐺𝑙 for π‘™βˆˆβ„•, and 𝑒(𝑙+1)πœ’πΊπ‘™=𝑒(𝑙). Thus, limπœβ†’βˆžπ‘’πœ(𝜏)=limπœβ†’βˆžπ‘’(𝜏)πœ’πΊπœ=𝑒 a.e.  on 𝑀. Replacing π‘’πœ by π‘’πœ(𝜏) in (2.60) and using the Fatou lemma we obtain ξ€œπΊπ‘™ξ“πΌ||ξ€·π‘’π‘‘ξ€Έβˆ’π‘’πΌ||||π‘£πΌβˆ—||π‘‘πœ‡β‰€sup𝜏>𝑑0ξ€œπΊπ‘™ξ“πΌ|||ξ‚€π‘’π‘‘βˆ’π‘’πœ(𝜏)𝐼|||||π‘£πΌβˆ—||π‘‘πœ‡β‰€2𝑛/2πœ€.(2.63) Let π‘™β†’βˆž, together with (2.24) we have ξ€œπ‘€ξ€·π‘’π‘‘ξ€Έβˆ’π‘’βˆ§π‘£β‰€2π‘›πœ€.(2.64) Therefore, by (2.18) and (2.24), we obtain β€–|π‘’π‘‘βˆ’π‘’|‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀2π‘›πœ€.

Theorem 2.13. If π‘βˆˆπ’«2(𝑀), then the space 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) is reflexive.

Proof. Let [𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)]ξ…ž denote the dual space to 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€). We will show that [𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)]ξ…ž=𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€) in steps.
(i) For fixed π‘£βˆˆπΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€), we define a linear functional 𝐹𝑣 on 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)𝐹𝑣(ξ€œπ‘’)=π‘€ξ€œπ‘’βˆ§π‘£=π‘€βŸ¨β‹†π‘’,π‘£βŸ©π‘‘πœ‡.(2.65) By Lemma 2.2, we have |𝐹𝑣(𝑒)|β‰€π‘Ÿπ‘β€–π‘’β€–πΏπ‘(π‘š)(Ξ›π‘˜π‘€)‖𝑣‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€), that is, β€–β€–πΉπ‘£β€–β€–β‰€π‘Ÿπ‘β€–π‘£β€–πΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€).(2.66) Thus, 𝐹𝑣 is a bounded linear functional on 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) and so 𝐹𝑣 belongs to [𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)]ξ…ž.
(ii) We consider an arbitrary local chart π‘“βˆΆπ‘‰(βŠ‚π‘€)→ℝ𝑛 on 𝑀. Let π‘ˆ be any open set in 𝑀, whose closure is compact and contained in 𝑉. We define β„ŽπΌξ€·πœ‘π‘‘π‘₯𝐼=πœ‘forπΌβˆˆΞ›(π‘˜,𝑛),πœ‘βˆˆπΏπ‘(π‘“βˆ’1(π‘₯))(𝑓(π‘ˆ)).(2.67) Since each continuous linear functional ξ‚π‘“βˆˆ[𝐿𝑝(π‘“βˆ’1(π‘₯))(𝑓(π‘ˆ))]ξ…ž can be represented uniquely in the form ξ‚βˆ«π‘“(πœ‘)=𝑓(π‘ˆ)πœ‘πœ“ξ‚π‘“π‘‘π‘₯ for some πœ“ξ‚π‘“βˆˆπΏπ‘ξ…ž(π‘“βˆ’1(π‘₯))(𝑓(π‘ˆ)), then for each continuous linear functional π‘“βˆˆ[𝐿𝑝(π‘“βˆ’1(π‘₯))(Ξ›π‘˜π‘“(π‘ˆ))]ξ…ž, we have 𝑓(πœ”)=πΌβˆˆΞ›(π‘˜,𝑛)π‘“ξ€·πœ”πΌπ‘‘π‘₯𝐼=ξ“πΌβˆˆΞ›(π‘˜,𝑛)π‘“βˆ˜β„ŽπΌβˆ’1ξ€·πœ”πΌξ€Έ=ξ“πΌβˆˆΞ›(π‘˜,𝑛)ξ€œπ‘“(π‘ˆ)πœ”πΌπœ“π‘“βˆ˜β„ŽπΌβˆ’1=ξ€œπ‘‘π‘₯𝑓(π‘ˆ)ξƒ©ξ“πœ”βˆ§πΌβˆˆΞ›(π‘˜,𝑛)𝜎(𝐼)πœ“π‘“βˆ˜β„ŽπΌβˆ’1𝑑π‘₯πΌβˆ—ξƒͺ,(2.68) that is, 𝑓 can be represented in the form ξ€œπ‘“(πœ”)=𝑓(π‘ˆ)πœ”βˆ§πœ›π‘“,(2.69) where πœ›π‘“=βˆ‘πΌβˆˆΞ›(π‘˜,𝑛)𝜎(𝐼)πœ“π‘“βˆ˜β„ŽπΌβˆ’1𝑑π‘₯πΌβˆ—βˆˆπΏπ‘β€²(π‘“βˆ’1(π‘₯))(𝑓(π‘ˆ)). If πœ›1=βˆ‘πΌπœ›1𝐼𝑑π‘₯πΌβˆ—,πœ›2=βˆ‘πΌπœ›2𝐼𝑑π‘₯πΌβˆ— such that ξ€œπ‘“(πœ”)=𝑓(π‘ˆ)πœ”βˆ§πœ›1=ξ€œπ‘“(π‘ˆ)πœ”βˆ§πœ›2,(2.70) for every πœ”βˆˆπΏπ‘(π‘“βˆ’1(π‘₯))(Ξ›π‘˜π‘“(π‘ˆ)). Taking πœ”=πœ‘π‘‘π‘₯𝐼 for πΌβˆˆΞ›(π‘˜,𝑛), we have π‘“βˆ˜β„ŽπΌβˆ’1(πœ‘)=βˆ«π‘“(πœ”)=𝑓(π‘ˆ)πœ‘πœ›1πΌβˆ«π‘‘π‘₯=𝑓(π‘ˆ)πœ‘πœ›2𝐼𝑑π‘₯, then πœ›1𝐼=πœ›2𝐼, that is, πœ›1=πœ›2. Hence πœ›π‘“ is uniquely determined.
For fixed 𝐹∈[𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)]ξ…ž and any π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€) with compact support we have πΉξ€·πœ’π‘ˆπ‘’ξ€Έ=πΉβˆ˜π‘“βˆ—ξ‚€ξ€·π‘“βˆ’1ξ€Έβˆ—ξ€·πœ’π‘ˆπ‘’ξ€Έξ‚=ξ€œπ‘“(π‘ˆ)ξ€·π‘“βˆ’1ξ€Έβˆ—ξ€·πœ’π‘ˆπ‘’ξ€Έβˆ§π‘£πΉβˆ˜π‘“βˆ—=ξ€œπ‘ˆπœ’π‘ˆπ‘’βˆ§π‘“βˆ—ξ€·π‘£πΉβˆ˜π‘“βˆ—ξ€Έ,(2.71) where π‘£π‘ˆ=π‘“βˆ—(π‘£πΉβˆ˜π‘“βˆ—)βˆˆπΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜π‘ˆ) is uniquely determined. For any two sets π‘ˆ1 and π‘ˆ2, the differential forms π‘£π‘ˆ1 and π‘£π‘ˆ2 coincide on π‘ˆ1βˆ©π‘ˆ2 because of the uniqueness of the differential form π‘£π‘ˆ1βˆ©π‘ˆ2. Thus, all the differential forms π‘£π‘ˆ, defined for different π‘ˆ, are compatible with one another, and hence defines a differential form 𝑣𝐹 on 𝑀. The differential form 𝑣𝐹 locally belongs to the space 𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘ˆ) and satisfies ξ€œπΉ(𝑒)=π‘€π‘’βˆ§π‘£πΉ,(2.72) for every π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€) with compact support, and is uniquely determined.
Let {𝐺𝑑} be a sequence of compact sets such that πΊπ‘‘βŠ‚πΊπ‘‘+1βŠ‚π‘€ for π‘‘βˆˆβ„• and 𝑀=βˆͺβˆžπ‘‘=1𝐺𝑑. Then 𝐹(𝑒)=𝐹limπ‘‘β†’βˆžπœ’πΊπ‘‘π‘’ξ‚Ά=limπ‘‘β†’βˆžπΉξ€·πœ’πΊπ‘‘π‘’ξ€Έ=limπ‘‘β†’βˆžξ€œπ‘€πœ’πΊπ‘‘π‘’βˆ§π‘£πΉ=ξ€œπ‘€π‘’βˆ§π‘£πΉ.(2.73) If 𝑣1,𝑣2 such that ξ€œπΉ(𝑒)=π‘€π‘’βˆ§π‘£1=ξ€œπ‘€π‘’βˆ§π‘£2,(2.74) for every π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€). Then for any π‘ˆ, we have 𝐹(πœ’π‘ˆβˆ«π‘’)=π‘€πœ’π‘ˆπ‘’βˆ§π‘£1=βˆ«π‘€πœ’π‘ˆπ‘’βˆ§π‘£2. Thus πœ’π‘ˆπ‘£1=πœ’π‘ˆπ‘£2 for any π‘ˆ, that is, 𝑣1=𝑣2.
Therefore, we conclude that each continuous linear functional 𝐹∈[𝐿𝑝(π‘₯)(Ξ›π‘˜π‘€)]ξ…ž can be uniquely represented in the form (2.72).
(iii) We shall show ‖𝑣𝐹‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)≀𝐢‖𝐹‖ with the constant 𝐢 dependent only on 𝑝(π‘š). We define a differential form 𝑒 on 𝑀‖‖𝑣𝑒(π‘š)=𝐹‖‖(π‘š)𝐿1/(1βˆ’π‘(π‘š))𝑝′(π‘š)ξ€·Ξ›π‘›βˆ’π‘˜π‘€ξ€Έ||𝑣𝐹||(π‘š)𝑝′(π‘š)βˆ’2⋆𝑣𝐹||𝑣(π‘š)if𝐹||||𝑣(π‘š)β‰ 0,0if𝐹||(π‘š)=0,(2.75) then by (𝑏4) and (𝑏6), we have ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)⎧βŽͺ⎨βŽͺβŽ©ξ€œ=infπœ†>0βˆΆπ‘€ξƒ©||𝑣𝐹||πœ†π‘(π‘š)βˆ’1‖‖𝑣𝐹‖‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)ξƒͺ𝑝′(π‘š)⎫βŽͺ⎬βŽͺβŽ­π‘‘πœ‡β‰€1=1.(2.76) Moreover ||||=||||ξ€œπΉ(𝑒)π‘€π‘’βˆ§π‘£πΉ||||=ξ€œπ‘€ξƒ©||𝑣𝐹||‖‖𝑣𝐹‖‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)ξƒͺ𝑝′(π‘š)‖‖𝑣𝐹‖‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)β‰₯β€–β€–π‘£π‘‘πœ‡πΉβ€–β€–πΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)2π‘βˆ—/(π‘βˆ—βˆ’1)ξ€œπ‘€ξƒ©||𝑣𝐹||‖‖𝑣(1/2)𝐹‖‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)ξƒͺ𝑝′(π‘š)β‰₯β€–β€–π‘£π‘‘πœ‡πΉβ€–β€–πΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)2π‘βˆ—/(π‘βˆ—βˆ’1).(2.77) Hence, we assert that ‖𝑣𝐹‖𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€)≀2π‘βˆ—/(π‘βˆ—βˆ’1)‖𝐹‖.
Now we reach the conclusion [𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)]ξ…ž=𝐿𝑝′(π‘š)(Ξ›π‘›βˆ’π‘˜π‘€), and hence 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€) is reflexive.

Theorem 2.14. If π‘βˆˆπ’«2(𝑀), then the exterior Sobolev space π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) is a separable, reflexive Banach space.

Proof. We treat π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) in a natural way as a subspace of the Cartesian product space 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)×𝐿𝑝(π‘š)(Ξ›π‘˜+1𝑀). Then we need only to show that π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) is a closed subspace of 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)×𝐿𝑝(π‘š)(Ξ›π‘˜+1𝑀). Let {𝑒𝑑}βŠ‚π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) be a convergent sequence. Then {𝑒𝑑} is a convergent sequence in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€). In view of Theorem 2.12, there exists π‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜π‘€) such that 𝑒𝑑→𝑒 in 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€). Similarly there exists Μƒπ‘’βˆˆπΏπ‘(π‘š)(Ξ›π‘˜+1𝑀) such that 𝑑𝑒𝑑→̃𝑒 in 𝐿𝑝(π‘š)(Ξ›π‘˜+1𝑀). Then it is easy to see that 𝑒𝑑 converges to 𝑒 and 𝑑𝑒𝑑 converges to ̃𝑒 on 𝑀 in measure. For any πœ‘βˆˆπΆβˆžπ‘(Ξ›π‘›βˆ’π‘˜βˆ’1𝑀)βŠ‚πΏπ‘β€²(π‘š)(Ξ›π‘›βˆ’π‘˜βˆ’1𝑀), we have ξ€œπ‘€π‘’π‘‘βˆ§π‘‘πœ‘=(βˆ’1)π‘˜+1ξ€œπ‘€π‘‘π‘’π‘‘βˆ§πœ‘.(2.78) Obviously, |π‘’π‘‘βˆ§π‘‘πœ‘|≀|(π‘’π‘‘βˆ’π‘’)βˆ§π‘‘πœ‘|+|π‘’βˆ§π‘‘πœ‘| and |π‘‘π‘’π‘‘βˆ§πœ‘|≀|(π‘‘π‘’π‘‘βˆ’Μƒπ‘’)βˆ§πœ‘|+|Μƒπ‘’βˆ§πœ‘|, then integrals of the functions |π‘’π‘‘βˆ§π‘‘πœ‘| and |π‘‘π‘’π‘‘βˆ§πœ‘| possess absolutely equicontinuity on 𝑀. Hence, by Vitali convergence theorem (see [28]), we get ξ€œπ‘€π‘’βˆ§π‘‘πœ‘=(βˆ’1)π‘˜+1ξ€œπ‘€Μƒπ‘’βˆ§πœ‘.(2.79) Thus, we obtain that 𝑑𝑒=̃𝑒. Then it is immediate that π‘Š1,𝑝(π‘š)(Ξ›π‘˜π‘€) is a closed subspace of 𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)×𝐿𝑝(π‘š)(Ξ›π‘˜+1𝑀).

Given two Banach spaces 𝑋 and π‘Œ, the symbol Xβ†·π‘Œ means that 𝑋 is continuously embedded in π‘Œ.

Theorem 2.15. Let 0<πœ‡(𝑀)<∞. If 𝑝(π‘š),π‘ž(π‘š)βˆˆπ’«(𝑀) and 𝑝(π‘š)β‰€π‘ž(π‘š) a.e. π‘šβˆˆπ‘€, then πΏπ‘ž(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έβ†·πΏπ‘(π‘š)ξ€·Ξ›π‘˜π‘€ξ€Έ.(2.80) The norm of the embedding operator (2.80) does not exceed πœ‡(𝑀)+1.

Proof. Since 𝑝(π‘š)β‰€π‘ž(π‘š) a.e. π‘šβˆˆπ‘€, then π‘€π‘βˆžβŠ‚π‘€π‘žβˆž. We may assume that π‘’βˆˆπΏπ‘ž(π‘š)(Ξ›π‘˜π‘š) with β€–π‘’β€–πΏπ‘ž(π‘š)(Ξ›π‘˜π‘€)≀1. Otherwise we can consider 𝑒/β€–π‘’β€–πΏπ‘ž(π‘š)(Ξ›π‘˜π‘€). By (𝑏7) we have πœŒπ‘ž(π‘š),Ξ›π‘˜π‘€(𝑒)≀1, in particular, |𝑒(π‘š)|≀1 a.e. π‘šβˆˆπ‘€π‘žβˆž. Then we can write πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)β‰€πœ‡ξ€·ξ€½π‘šβˆˆπ‘€β§΅π‘€π‘žβˆž+ξ€œβˆΆ|𝑒|≀1ξ€Ύξ€Έπ‘€β§΅π‘€π‘žβˆž|𝑒|π‘ž(π‘š)ξ€·π‘€π‘‘πœ‡+πœ‡π‘žβˆžβ§΅π‘€π‘βˆžξ€Έ+esssupπ‘€π‘βˆž|𝑒|β‰€πœ‡(𝑀)+1.(2.81) Thus, we have πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒/(πœ‡(𝑀)+1))≀(πœ‡(𝑀)+1)βˆ’1πœŒπ‘(π‘š),Ξ›π‘˜π‘€(𝑒)≀1. Therefore ‖𝑒‖𝐿𝑝(π‘š)(Ξ›π‘˜π‘€)≀(πœ‡(𝑀)+1)β€–π‘’β€–πΏπ‘ž(π‘š)(Ξ›π‘˜π‘€).(2.82)

Theorem 2.16. Let 𝑀 be a compact Riemannian manifold with a smooth boundary or without boundary and 𝑝(π‘š),π‘ž(π‘š)∈𝐢(𝑀)βˆ©π’«1(𝑀). Assume that 𝑝(π‘š)<𝑛,π‘ž(π‘š)<𝑛𝑝(π‘š)π‘›βˆ’π‘(π‘š),forπ‘šβˆˆπ‘€.(2.83) Then π‘Š1,𝑝(π‘š