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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Elliptic operators on refined Sobolev scales on vector bundles

Tetiana Zinchenko
Published Online: 2017-07-13 | DOI: https://doi.org/10.1515/math-2017-0076

Abstract

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

Keywords: Elliptic pseudodifferential operator; Vector bundle; Sobolev space; Hörmander space; Interpolation with function parameter; Fredholm property; A priori estimate of solutions; Regularity of solutions

MSC 2010: 35J48; 58J05; 46B70; 46E35

1 Introduction

It is well known [1, 2] that elliptic differential and pseudodifferential operators on a closed infinitely smooth manifold are Fredholm between appropriate Sobolev spaces. This fundamental property is used in the theory of elliptic differential equations and elliptic boundary-value problems. However, the Sobolev scale is not sufficiently finely calibrated for some mathematical problems (see monographs [310]). In this connection, Hörmander [3, 4] introduced and investigated a broad class of normed function spaces Bp,μ={wS(Rn):μw^Lp(Rn)},wBp,μ:=μw^Lp(Rn), where 1 ≤ p ≤ ∞, μ : ℝn → (0, ∞) is a weight function, and w^ is the Fourier transform of a tempered distribution w. Hörmander applied these spaces to investigation of solvability of partial differential equations given in Euclidean domains and to study of regularity of solutions to these equations.

Nevertheless, the class of all spaces 𝓑p, μ is too general for applications to differential equations on manifolds and boundary-value problems. Among these spaces, Mikhailets and Murach [1113] selected the class of inner product spaces Hs,φ : = 𝓑2,μ parametrized with the function μ(ξ) = 〈ξsφ(〈ξ〉), where s ∈ ℝ, the function φ : [1, ∞) → (0, ∞) varies slowly at infinity in the sense of Karamata [14, 15], and 〈ξ〉 = (1 + |ξ|2)1/2. This class is called the refined Sobolev scale. It contains the inner product Sobolev spaces Hs = Hs,1 and is obtained by the interpolation with a function parameter between these spaces. This interpolation property allowed Mikhailets and Murach [1113, 1621] to build the theory of solvability of general elliptic systems and elliptic boundary-value problems on the refined Sobolev scale. Their theory [7] is supplemented in [2229] for a more extensive class of Hörmander inner product spaces. The refined Sobolev scale and other classes of Hörmander spaces are applied to the spectral theory of elliptic differential operators on manifolds [7, Section 2.3], theory of interpolation of normed spaces [23, 30], to some differential-operator equations [31], parabolic initial-boundary value problems [3235], and in mathematical physics [36, 37].

However, elliptic operators on vector bundles have not been covered by this theory. These operators have important applications to elliptic boundary problems on vector bundles, elliptic complexes, spectral theory of elliptic differential operators, and others (see, e.g., [1, 2, 38]).

The goal of this paper is to introduce and investigate the refined Sobolev scale on an arbitrary vector bundle over infinitely smooth closed manifold and to give applications of this scale to general elliptic pseudodifferential operators on vector bundles.

The paper consists of eight sections. Section 1 is Introduction. In Section 2, we introduce the refined Sobolev scale on the vector bundle. Section 3 is devoted to the method of interpolation with a function parameter between Hilbert spaces. This method plays a key role in the paper. Section 4 contains main results concerning properties of the refined Sobolev scale introduced. Section 5 presents main results regarding the properties of elliptic pseudodifferential operators on this scale. Section 6 contains some auxiliary facts. The main results of the paper formulated in Sections 4 and 5 are proved in Sections 7 and 8 respectively.

2 The refined Sobolev scale on a vector bundle

The refined Sobolev scale on ℝn and smooth manifolds was introduced and investigated by Mikhalets and Murach [13, 39]. This scale consists of the inner product Hörmander spaces Hs,φ with s ∈ ℝ and φ ∈ 𝓜. Let us give the definition of the function class 𝓜 and the space Hs,φ(ℝn). The latter will be a base for our definition of the refined Sobolev scale on vector bundles.

The class 𝓜 consists of all Borel measurable functions φ : [1, ∞) → (0, ∞) that satisfy the following two conditions:

  1. both functions φ and 1/φ are bounded on each compact interval [1, b] with 1 < b < ∞;

  2. the function φ varies slowly at infinity in the sense of Karamata [14], i.e. limtφ(λt)φ(t)=1 for every λ>0.

Slowly varying functions are well investigated and play an important role in mathematical analysis and its applications (see monographs [4042]). A standard example of a function φ ∈ 𝓜 is given by a continuous function φ : [1, ∞) → (0, ∞) such that φ(t):=(logt)r1(loglogt)r2(loglogtk)rk for t1,(1) where 0 ≤ k ∈ ℤ and r1, ..., rk ∈ ℝ.

The class 𝓜 admits the following description (see, e.g., [42, Section 1.2]): φMφ(t)=exp(β(t)+1tα(τ)τdτ)for t1.

Here, α is a continuous function on [1, ∞) such that α(τ) → 0 as τ → ∞, and β is a Borel measurable function on [1, ∞) such that β(t) → l as t → ∞ for some l ∈ ℝ.

Let s ∈ ℝ and φ ∈ 𝓜. By definition, the complex linear space Hs,φ(ℝn), with 1 ≤ n ∈ ℤ, consists of all distributions w ∈ 𝓢′(ℝn) such that their Fourier transform w^ is locally Lebesgue integrable over ℝn and satisfies the condition Rnξ2sφ2(ξ)|w^(ξ)|2dξ<.

Here, 𝓢′(ℝn) is the complex linear topological space of all tempered distributions on ℝn, and 〈ξ〉 = (1 + |ξ|2)1/2. An inner product in Hs,φ(ℝn) is defined by the formula (w1,w2)s,φ;Rn:=Rnξ2sφ2(ξ)w1^(ξ)w2^(ξ)¯dξ, with w1, w2Hs,φ(ℝn). This inner product endows Hs,φ(ℝn) with the Hilbert spaces structure and induces the norm ws,φ;Rn:=(w,w)s,φ;Rn1/2.

The space Hs,φ(ℝn) is separable with respect to this norm, and the set C0(Rn) is dense in this space. Here, as usual, C0(Rn) stands for the set of all infinitely differentiable compactly supported functions w : ℝn → ℂ.

In this paper, we consider complex-valued functions and distributions; hence, all function spaces are supposed to be complex. Besides, we interpret distributions as antilinear functionals on corresponding spaces of test functions.

If φ = 1, then the space Hs,φ(ℝn) coincides with the inner product Sobolev space Hs(ℝn) of order s. Generally, we have the continuous embeddings Hs+ε(Rn)Hs,φ(Rn)Hsε(Rn) for any ε>0.(2)

They show that the function parameter φ defines a supplementary regularity with respect to the main (power) regularity s. Briefly saying, φ refines the main regularity s. Following [7, 23], we call the class of function spaces {Hs,φ(Rn):sR,φM}(3) the refined Sobolev scale on ℝn.

Let Γ be a closed (i. e. compact and without boundary) infinitely smooth real manifold of dimension n ≥ 1. We suppose that a certain C-density dx is given on Γ. Let π : V → Γ be an infinitely smooth complex vector bundle of rank p ≥ 1 on Γ. Here, V is the total space of the bundle, Γ is the base space, and π is the projector (see, e.g., [2, Chapter I, Section 2]). Let C(Γ, V) denote the complex linear space of all infinitely differentiable sections u : Γ → V. Note that u(x) ∈π− 1(x) for every x ∈Γ and that π− 1(x) is a complex vector space of dimension p (this space is called the fiber over x).

Let us introduce the Hörmander space Hs,φ(Γ, V) on this vector bundle. From the C-structure on Γ, we choose a finite atlas consisting of local charts αj : ℝn ↔ Γj with j = 1, …, ϰ. Here, the open sets Γj form a finite covering of Γ. We choose these sets so that the local trivialization βj : π− 1j) ↔ Γj\times ℂp is defined. We also choose real-valued functions χjC(Γ), j = 1, …, χ, that satisfy the condition supp χj ⊂ Γj and form a partition of unity on Γ.

Let s ∈ ℝ and φ ∈ 𝓜. We introduce the norm on C(Γ, V) by the formula us,φ;Γ,V:=(j=1ϰk=1p(Πk(βj(χju)αj)s,φ;Rn2)1/2.(4)

Here, uC(Γ, V), and the projector Πk is defined as follows: Πk : (x, a) ↦ ak for all x ∈ Γ and a = (a1, …, ap) ∈ ℂp. We put uj,k:=Πk(βj(χju)αj)(5) for arbitrary j ∈{1,… ϰ} and k ∈{1, …, p}. Note that if uC(Γ, V), then each uj,kC0(Rn); hence, the norms on the right-hand side of (4) are well defined. The norm (4) is Hilbert because it is induced by the inner product (u,v)s,φ;Γ,V:=j=1ϰk=1p(uj,k,vj,k)s,φ;Rn(6) of sections u, vC (Γ, V).

Let Hs,φ(Γ, V) be the completion of the linear space C(Γ, V) with respect to the norm (4) (and the corresponding inner product (6)). Thus, we have the Hilbert space Hs,φ(Γ, V). This space does not depend up to equivalent of norms on our choice of the atlas {αj}, partition of unity {χj}, and local trivializations {βj}. This will be proved below as Theorem 4.2. By analogy with (3) we call the class of Hilbert function spaces {Hs,φ(Γ,V):sR,φM}(7) the refined Sobolev scale on the bundle π : V → Γ.

If φ = 1, then Hs,φ(Γ, V) becomes the inner product Sobolev space Hs(Γ, V) of order s ∈ ℝ (see e.g. [2, Chapter IV, Section 1]). In the Sobolev case of φ = 1, we will omit the index φ in our designations concerning the Hörmander spaces Hs,φ(⋅). Specifically, ∥⋅∥s;Γ,V denotes the norm in the Sobolev space Hs(Γ, V).

In the case of trivial vector bundle of rank p = 1, the space Hs,φ(Γ, V) consists of distributions on Γ and is denoted by Hs,φ(Γ). The space Hs,φ(Γ) was introduced and investigated by Mikhailets and Murach [13, 39].

3 Interpolation with function parameter between Hilbert spaces

The refined Sobolev scale on the vector bundle π : V → Γ possesses an important interpolation property. Namely, every space Hs,φ(Γ, V), with s ∈ ℝ and φ ∈ 𝓜, is the result of the interpolation with an appropriate function parameter between the Sobolev spaces Hsε(Γ, V) and Hs + δ(Γ, V), where ε, δ > 0. We will systematically use this property in the paper. Therefore we recall the definition of interpolation with function parameter between Hilbert spaces and discuss some of its properties. We restrict ourselves to the case of separable complex Hilbert spaces and mainly follow monograph [7, Section 1.1]. Note that the interpolation with function parameter between normed spaces was introduced by Foiaş and Lions [43], who separately considered the case of Hilbert spaces.

Let X : = [X0, X1] be an ordered pair of separable complex Hilbert spaces X0 and X1 such that X1X0 with the continuous and dense embedding. This pair is said to be admissible. For X there exists an isometric isomorphism J : X1X0 that J is a self-adjoint positive-definite operator in X0 with the domain X1. The operator J is uniquely determined by the pair X and is called a generating operator for this pair.

Let 𝓑 denote the set of all Borel measurable functions ψ : (0, ∞) → (0, ∞) that ψ is bounded on every compact interval [a, b], with 0 < a < b < ∞, and that 1/ψ is bounded on every set [r, ∞), with r > 0. Given ψ ∈ 𝓑, consider the operator ψ(J) defined as the Borel function ψ of the self-adjoint operator J with the help of Spectral Theorem. The operator ψ(J) is (generally) unbounded and positive-definite in X0. Let [X0, X1]ψ or, simply, Xψ denote the domain of ψ(J) endowed with the inner product (u1, u2) : = (ψ(J)u1, ψ(J)u2)X0 and the corresponding norm ∥u = ∥ψ(J)uX0. The space Xψ is Hilbert and separable.

A function ψ ∈ 𝓑 is said to be an interpolation parameter if the following condition is fulfilled for each admissible pairs X = [X0, X1] and Y = [Y0, Y1] of Hilbert spaces and for an arbitrary linear mapping T given on X0: if the restriction of T to Xj is a bounded operator T : XjYj for each j ∈{0, 1}, then the restriction of T to Xψ is also a bounded operator T : XψYψ.

If ψ is an interpolation parameter, then we say that the Hilbert space Xψ is obtained by the interpolation with the function parameter ψ between X0 and X1 (or of the pair X). In this case, the continuous and dense embeddingsX1XψX0(8) hold true.

The function ψ ∈ 𝓑 is an interpolation parameter if and only if ψ is pseudoconcave in a neighborhood of +∞. The latter property means that there exists a concave function ψ1 : (b, ∞) → (0, ∞), with b ≫ 1, that both functions ψ/ψ1 and ψ1/ψ are bounded on (b, ∞). This criterion follows from Peetre’s [44, 45] description of all interpolation functions for the weighted Lebesgue spaces (see [7, Theorem 1.9]). Specifically, every function ψ ∈ 𝓑 of the form ψ(t) ≡ tθψ0(t), where 0 < θ < 1 and ψ0 varies slowly at infinity, is an interpolation parameter.

Let us formulate the above-mentioned interpolation property of the refined Sobolev scale on ℝn[7, Theorem 1.14].

Proposition 3.1

Let a function φ ∈ 𝓜 and numbers ε, δ > 0 be given. Put ψ(t):=tε/(ε+δ)φ(t1/(ε+δ))ift1,φ(1)if0<t<1.(9)

Then ψ ∈ 𝓑 is an interpolation parameter and Hs,φ(Rn)=[Hsε(Rn),Hs+δ(Rn)]ψforeverysR with equality of norms.

4 Properties of the refined Sobolev scale on vector bundle

Let us formulate the main results of the paper concerning properties of the refined Sobolev scale (7) on the vector bundle π : V → Γ.

Theorem 4.1

Let φ ∈ 𝓜 and ε, δ > 0. Define the interpolation parameter ψ by formula (9). Then [Hsε(Γ,V),Hs+δ(Γ,V)]ψ=Hs,φ(Γ,V)foreverysR(10) with equivalence of norms.

Theorem 4.2

Let s ∈ ℝ and φ ∈ 𝓜. The Hilbert space Hs,φ(Γ, V) does not depend up to equivalence of norms on the choice of the atlas {αj} and partition of unity {χj} on Γ and on the choice of the local trivializations βj of V.

Theorem 4.3

Let s ∈ ℝ and φ, φ1 ∈ 𝓜. The identity mapping uu, with uC(Γ, V), extends uniquely (by continuity) to a compact embedding Hs + ε,φ1(Γ, V)\llcorner ↪ Hs,φ(Γ, V) for every ε > 0.

Suppose now that the vector bundle π : V → Γ is Hermitian. Thus, for every x ∈ Γ, a certain inner product 〈⋅,⋅〉x is defined in the fiber π− 1(x) so that the scalar function Γ ∋ x ↦〈u(x), v(x)〉X is infinitely smooth on Γ for arbitrary sections u, vC (Γ, V). Using the C-density dx on Γ, we define the inner product of these sections by the formula u,vΓ,V:=Γu(x),v(x)xdx.(11)

Theorem 4.4

Let s ∈ ℝ and φ ∈ 𝓜. There exists a number c = c(s, φ) > 0 such that for arbitrary sections u, vC (Γ, V) we have the estimate |u,vΓ,V|cus,φ;Γ,Vvs,1/φ;Γ,V.(12)

Thus, the form (11), with u, vC (Γ, V), extends by continuity to a sesquilinear formu, vΓ,V defined for arbitrary uHs,φ(Γ, V) and vHs, 1/φ(Γ, V). Moreover, the spaces Hs,φ(Γ, V) and Hs,1/φ(Γ, V) are mutually dual (up to equivalence of norms) with respect to the latter form.

In view of this theorem note that φ ∈ 𝓜 ⇔ 1/φ ∈ 𝓜; hence, the space Hs,1/φ(Γ, V) is well defined.

Theorem 4.5

Let 0 ≤ q ∈ ℤ and φ ∈ 𝓜. Then the condition 1dttφ2(t)<(13) is equivalent to that the identity mapping uu, with uC (Γ, V), extends uniquely to a continuous embedding Hq + n/2,φ(Γ, V) ↪ Cq(Γ, V). Moreover, this embedding is compact.

Here, of course, Cq(Γ, V) denotes the Banach space of all q times continuously differentiable sections u : Γ → V. The norm in this space is defined by the formula u(q);Γ,V:=j=1ϰk=1puj,k(q);Rn,(14) where each uj,kCbq(Rn) is given by (5). Here, Cbq(Rn) denotes the Banach space of all q times continuously differentiable functions on ℝn whose partial derivatives up to the q-th order are bounded on ℝn. This space is endowed with the norm w(q);Rn:=μ1++μnqsuptRn|μ++μn1w(t)t1μ1,,tnμn| of a function w.

We will prove Theorems 4.14.5 in Section 7. In the case where π : V → Γ is a trivial vector bundle of rank p = 1, they are established by Mikhailets and Murach [39] (see also their monograph [7, Section 2.1.2]).

5 Elliptic operators on the refined Sobolev scale on a vector bundle

Consider elliptic PsDOs on a pair of vector bundles on Γ. Let π1 : V1 → Γ and π2 : V2 → Γ be two infinitely smooth complex vector bundles of the same rank p ≥ 1 on Γ. We choose the atlas {αj : ℝn → Γj} so that both the local trivializations β1,j:π11(Γj)Γj×Cp and β2,j:π21(Γj)Γj×Cp are defined. We suppose that each vector bundle πk : Vk → Γ with k ∈ {1, 2} is Hermitian. Let 〈u, vΓ,Vk denote the corresponding inner product of sections u, vC(Γ, Vk) and its extension by continuity indicated in Theorem 4.4.

Given m ∈ ℝ, we let Ψphm(Γ;V1,V2) denote the class of all polyhomogeneous (classical) PsDOs A : C(Γ, V1) → C(Γ, V2) of order m (see, e.g., [2, Chapter IV, Section 3]). Recall that if Ω is an open nonempty subset of Γ with Ω ⊂ Γj for some j ∈{1, …, ϰ}, then a PsDO AΨphm(Γ;V1,V2) is represented locally in the form Π(β2,j(φA(ψu))αj)=(φαj)AΩΠ(β1,j(ψu)αj)(15) for every section uC(Γ, V1) and arbitrary scalar functions φ, ψC(Γ) with supp φ ⊂ Ω and supp ψ ⊂ Ω. Here, AΩ is a certain p × p-matrix whose entries are polyhomogeneous PsDOs on ℝn of order m, and Π is the projector defined by the formula Π : (x, a) ↦ a for arbitrary x ∈ Γ and a ∈ ℂp. For AΨphm(Γ;V1,V2) there is a unique formally adjoint PsDO A+Ψphm(Γ;V2,V1) defined by the formula 〈Au, wΓ,V2 = 〈u, A+wΓ,V1 for all uC(Γ, V1) and wC(Γ, V2).

Hereafter we let m ∈ ℝ and suppose that A is an arbitrary elliptic PsDO from the class Ψphm(Γ;V1,V2). The ellipticity of A is equivalent to that each operator AΩ=(AΩl,r)l,r=1p from (15) is elliptic on the set αj1(Ω), i.e. det(aΩ,0l,r(x,ξ))l,r=1p0 for all xαj1(Ω) and ξRn{0}, with aΩ,0l,r(x,ξ) being the principal symbol of the scalar PsDO AΩl,r A on ℝn. Put N:={uC(Γ,V1):Au=0 on Γ},(16) N+:={wC(Γ,V2):A+w=0 on Γ}.(17)

Since the PsDOs A and A+ are elliptic, the spaces 𝔑 and 𝔑+ are finite-dimensional (see, e.g., [2, Theorem 4.8]).

Theorem 5.1

Let s ∈ ℝ and φ ∈ 𝓜. The mapping uAu, uC(Γ, V1), extends uniquely (by continuity) to a bounded linear operator A:Hs+m,φ(Γ,V1)Hs,φ(Γ,V2).(18)

This operator is Fredholm. Its kernel is 𝔑 and its domain A(Hs+m,φ(Γ,V1))={fHs,φ(Γ,V2):f,wΓ,V2=0forallwN+}.(19)

The index of operator (18) is equal to dim 𝔑-dim 𝔑+and does not dependent on s and φ.

Recall that a bounded linear operator T : E1E2 between Banach spaces E1 and E2 is called Fredholm if its kernel ker T and co-kernel coker T : = E2/T(X) are finite-dimensional. If the operator T is Fredholm, then its domain T(X) is closed in E2 and its index ind T : = dim ker T − dim coker T is finite (see, e.g. [1, Lemma 19.1.1]).

If 𝔑 = {0} and 𝔑+ = {0}, then operator (18) is an isomorphism between the spaces Hs + m,φ(Γ, V1) and Hs,φ(Γ, V2) by virtue of the Banach theorem on inverse operator. In the general situation, this operator induces an isomorphism between their certain subspaces of finite codimension. In this connection consider the following decompositions of these spaces into direct sums of their subspaces: Hs+m,φ(Γ,V1)=N{uHs+m,φ(Γ,V1):u,vΓ,V1=0 for all vN},(20) Hs,φ(Γ,V2)=N+{fHs,φ(Γ,V2):f,wΓ,V2=0 for all wN+}.(21)

These decompositions are well defined because the summands in them have the trivial intersection and the finite dimension of the first summand is equal to the codimension of the second one. This equality is due to the following fact: if we consider 𝔑 as a subspace of Hsm,1/φ(Γ, V1), then the dual of 𝔑 with respect to the form 〈⋅,⋅〉Γ,V1 coincides with the second summand in (20) according to Theorem 4.4, analogous reasoning being valid for (21).

Let P and P+ denote the oblique projectors of the spaces Hs + m,φ(Γ, V1) and Hs,φ(Γ, V2) onto the second summands parallel to the first summands in (20) and (21) respectively. These projectors are independent of s and φ.

Theorem 5.2

Let s ∈ ℝ and φ ∈ 𝓜. The restriction of operator (18) to the subspace P(Hs + m,φ(Γ, V1)) is an isomorphism A:P(Hs+m,φ(Γ,V1))P+(Hs,φ(Γ,V2)).(22)

The solutions uHs + m,φ(Γ, V1) to the elliptic equation Au = f satisfy the following a priori estimate.

Theorem 5.3

Let s ∈ ℝ, φ ∈ 𝓜, and σ < s + m. Besides, let functions χ, ηC(Γ) be chosen so that η = 1 in a neighbourhood of supp χ. Then there exists a number c > 0 that, for any sections uHs + m,φ(Γ, V1) and fHs,φ(Γ, V2) satisfying the equation Au = f on Γ, we have the estimate χus+m,φ;Γ,V1c(ηfs,φ;Γ,V2+uσ;Γ,V1).(23)

Remark 5.4

If χ = 1 on Γ, then (23) becomes the global estimate us+m,φ;Γ,V1c(fs,φ;Γ,V2+uσ;Γ,V1).(24)

If s + m − 1 < σ < s + m, then we can take η : = χ in (23); this follows in view of Theorem 4.3 from the estimate χus+m,φ;Γ,V1c(χfs,φ;Γ,V2+us+m1,φ;Γ,V1).(25)

Consider the local regularity of the solutions to the elliptic equation Au = f. Given j ∈{1, 2}, we put H(Γ,Vj):=σRHσ(Γ,Vj)=σR,ηMHσ,η(Γ,Vj) (the latter equality is due to Theorem 4.3). Assume that Γ0 is an arbitrary open nonempty subset of Γ. We let H1ocs,φ(Γ0,Vj), with s ∈ ℝ and φ ∈ 𝓜, denote the linear space of all sections vH− ∞(Γ, Vj) such that χ vHs,φ(Γ, Vj) for every function χC(Γ) with supp χ ⊂ Γ0. Here, the product χ vH− ∞(Γ, Vj) is well defined by closer.

Theorem 5.5

Let uH− ∞(Γ, V1) be a solution to the elliptic equation Au = f on Γ for a certain section fHlocs,φ(Γ0,V2).ThenuHlocs+m,φ(Γ0,V1).

As we can see, the supplementary regularity φ is inherited by the solution. If Γ0 = Γ, then the local spaces Hlocs+m,φ(Γ0,V1) and Hlocs,φ(Γ0,V2) becomes Hs + m,φ(Γ, V1) and Hs,φ(Γ, V2) respectively and then Theorem 5.5 says about the global regularity on Γ.

As an application of the refine Sobolev scale, we give the following result:

Theorem 5.6

Let 0 ≤ q ∈ ℤ. Suppose that a section uH− ∞(Γ, V1) is a solution to the elliptic equation Au = f on Γ where fHlocqm+n/2,φ(Γ0,V2) for a certain function φ ∈ 𝓜 subject to condition (13). Then uClocq(Γ0,V1).

Here, Clocq(Γ0,V1) denotes the linear space of all sections uH− ∞(Γ, V1) such that χ uCq(Γ, V1) for arbitrary χC(Γ) with supp χ ⊂ Γ0.

Remark 5.7

Let φ ∈ 𝓜. Condition (13) is sharp in Theorem 5.6. Namely, this condition is equivalent to the implication (uH(Γ,V1),AuHlocqm+n/2,φ(Γ0,V2))uClocq(Γ0,V1).(26)

We will prove Theorems 5.15.6, formula (25) in Remark 5.4, and Remark 5.7 in Section 8. In the case where both π1 : V1 → Γ and π2 : V2 → Γ are trivial vector bundles of rank p = 1, these theorems are proved by Mikhailets and Murach [39] (see also their monograph [7, Sections 2.2.2 and 2.2.3]).

6 Auxiliary results

We will use three properties of the interpolation with a function parameter. The first of them reduces the interpolation between orthogonal sums of Hilbert spaces to the interpolation between the summands (see, e.g., [7, Theorem 1.5]).

Proposition 6.1

Let [X0(j),X1(j)], with j = 1, …, r, be a finite collection of admissible couples of Hilbert spaces. Then for every function ψ ∈ 𝓑 we have [j=1rX0(j),j=1rX1(j)]ψ=j=1r[X0(j),X1(j)]ψ with equality of norms.

The second property shows that this interpolation preserves the Fredholm property of the bounded operators that have the same defect (see, e.g., [7, Theorem 1.7]).

Proposition 6.2

Let X = [X0, X1] and Y = [Y0, Y1] be admissible pairs of Hilbert spaces, and let ψ ∈ 𝓑 be an interpolation parameter. Suppose that a linear mapping T is given on X0 and satisfies the following property: the restrictions of T to the spaces Xj, where j = 0, 1, are Fredholm bounded operators T : XjYj that have a common kernel and the same index. Then the restriction of T to the space Xψ is a Fredholm bounded operator T : XψYψ with the same kernel and index and, besides, T(Xψ) = YψT(X0).

The third property reduces the interpolation between the dual or antidual spaces of given Hilbert spaces to the interpolation between these given spaces (see [7, Theorem 1.4]). We need this property in the case of antidual spaces. If H is a Hilbert space, then H′ stands for the antidual of H; namely, H′ consists of all antilinear continuous functionals l : H → ℂ. The linear space H′ is Hilbert with respect to the inner product (l1, l2)H : = (v1, v2)H of functionals l1, l2H′; here vj, with j ∈ {1, 2}, is a unique vector from H such that lj(w) = (vj, w)H for every wH. Note that we do not identify H and H′ on the base of the Riesz theorem (according to which vj exists).

Proposition 6.3

Let a function ψ ∈ 𝓑 be such that the function ψ(t)/t is bounded in a neighbourhood of infinity. Thenfor every admissible pair [X0, X1] of Hilbert spaces we have the equality [X1,X0]ψ=[X0,X1]χ with equality of norms. Here, the function χ ∈ 𝓑 is defined by the formula χ(t) : = t/ψ(t) for t > 0. If ψ is an interpolation parameter, then χ is an interpolation parameter as well.

In view of this theorem we note that if [X0, X1] is an admissible pair of Hilbert spaces, then the dual pair [X1,X0] is also admissible provided that we identify functions from X0 with their restrictions on X1.

7 Proofs of properties of the refined Sobolev scale

In this section we will prove Theorems 4.14.5.

Proof of Theorem 4.1

Let s ∈ ℝ. As is known [2, p. 110], the pair of Sobolev spaces on the left of equality (10) is admissible. We will deduce this equality from Proposition 3.1 with the help of some operators of flattening and sewing of the vector bundle π : V → Γ.

We define the flattening operator by the formula T:u(u1,1,,u1,p,,uϰ,1,,uϰ,p) for arbitraryuC(Γ,V).(27)

Here, each function uj,kC0(Rn) is defined by formula (5). According to (4), the norm of u in the space Hs,φ(Γ, V) is equal to the norm of T u in the Hilbert space (Hs,φ(ℝn)). Therefore the linear mapping uTu, with uC(Γ, V), extends continuously to the isometric operator T:Hs,φ(Γ,V)(Hs,φ(Rn))pϰ.(28)

Besides, this mapping extends continuously to the isometric operators T:Hσ(Γ,V)(Hσ(Rn))pϰ,withσR,(29) between Sobolev spaces. Since ψ is an interpolation parameter, it follows from the boundedness of the linear operators (29) with σ ∈ {sε, s + δ} that the restriction of the operator (29) with σ = sε is a bounded operator T:[Hsε(Γ,V),Hs+δ(Γ,V)]ψ[(Hsε(Rn))pϰ,(Hs+δ(Rn))pϰ]ψ.(30)

Owing to Propositions 3.1 and 6.1, the target space of (30) takes the form [(Hsε(Rn))pϰ,(Hs+δ(Rn))pϰ]ψ=([Hsε(Rn),Hs+δ(Rn)]ψ)pϰ=(Hs,φ(Rn))pϰ.(31)

Thus, (30) is a bounded operator between the spaces T:[Hsε(Γ,V),Hs+δ(Γ,V)]ψ(Hs,φ(Rn))pϰ.(32)

Consider now the mapping of sewing K:(w1,1,,w1,p,,wϰ,1,,wϰ,p)j=1ϰwj(33) defined on vectors w:=(w1,1,,w1,p,,wx,1,,wx,p)(C0(Rn))pϰ.(34)

Here, for each j ∈ {1, …, ϰ}, the section wjC(Γ, V) is defined by the formula wj(x):=βj1(x,(ηjwj,1)(αj1(x)),,(ηjwj,p)(αj1(x)))ifxΓj,0ifxΓΓj,(35) in which the function η jC0 (ℝn) is chosen so that ηj = 1 on the set αj1 (supp χj).

We have the linear mapping K:(C0(Rn))pϰC(Γ,V).(36)

It is left inverse to the flattening mapping (27). Indeed, given uC(Γ, V), we write KTu=K(u1,1,,u1,p,,uϰ,1,,uϰ,p)=j=1ϰuj, where each section ujC(Γ, V) is defined by formula (35) with u instead of w. In this formula, for arbitrary k ∈ {1, …, p} and x ∈ Γj, we have the equalities (ηjuj,k)(αj1(x))=(ηj(αj1(x)))Πk(βj((χju)(x)))=Πk(βj((χju)(x))) due to our choice of ηj. Therefore uj(x)=βj1(x,(ηjuj,1)(αj1(x)),,(ηjuj,p)(αj1(x)))=βj1(x,Π1(βj((χju)(x))),,Πp(βj((χju)(x))))=(χju)(x) for every x ∈ Γj. Note that if x ∈ Γ \ Γj, then uj(x) = 0 = (χju)(x). Thus, uj(x) = (χju)(x) for arbitrary x ∈ Γ. Hence, KTu=j=1ϰuj=j=1ϰχju=u for every uC(Γ,V).(37) Let us prove that the mapping (36) extends uniquely to a linear bounded operator between the spaces (Hs,φ(ℝn)) and Hs,φ(Γ, V). Given a vector (34), we write Kws,φ;Γ,V2=l=1ϰk=1pΠk(βl(χlKw)αl)s,φ;Rn2=l=1ϰk=1pj=1ϰΠk(βl(χlwj)αl)s,φ;Rn2.(38)

Examine the function (χlwj)∘αl : ℝnπ−1l) with l, j ∈ {1, …, x}. If t ∈ ℝn satisfies αl(t)∈ Γj, then ((χlwj)αl)(t)=(χlαl)(t)(wjαl)(t)=(χlαl)(t)βj1(αl(t),(ηjwj,1)((αj1αl)(t)),,(ηjwj,p)((αj1αl)(t)))=βj1(αl(t),(χlαl)(t)(ηjwj,1)((αj1αl)(t)),,(χlαl)(t)(ηjwj,p)((αj1αl)(t)))=βj1(αl(t),((ηj,lwj,1)αj,l)(t),,((ηj,lwj,p)αj,l)(t)). Here, ηj,l := (χlαl)ηjC0 (ℝn), whereas αj,l : ℝn↔ ℝn is an infinitely smooth diffeomorphism such that αj,l := αj1αl in a neighbourhood of supp ηj,l and that αj,l(t) = t whenever |t| ≫ 1. Then, given k ∈ {1, …, p}, we have the equalities Πk(βl(χlwj)αl)(t)=Πk(βlβj1)(αl(t),((ηj,lwj,1)αj,l)(t),,((ηj,lwj,p)αj,l)(t))=r=1pβl,jk,r(αl(t))((ηj,lwj,r)αj,l)(t).

Here, each βl,jk,r is a certain complex-valued function from C(Γ) such that the matrix-valued function (βl,jk,r(x))k,r=1p of xsupp χl supp(ηjαj1) corresponds to the transition mapping βlβj1 . Thus, Πk(βl(χlwj)αl)(t)=r=1pβl,jk,r(αl(t))((ηj,lwj,r)αj,l)(t)(39) for arbitrary t ∈ ℝ (if αl(t)∉ Γj, then this equality becomes 0 = 0).

Owing to (38) and (39) we write Kws,φ;Γ,V2=l=1ϰk=1pj=1ϰr=1p(βl,jk,rαl)((ηj,lwj,r)αj,l)s,φ;Rn2=l=1ϰk=1pj=1ϰr=1pηj,lk,r(wj,rαj,l)s,φ;Rn2; here, each ηj,lk,r:=(βl,jk,rαl)(ηj,lαj,l)C0(Rn).

Thus Kws,φ;Γ,V2l=1ϰk=1p(j=1ϰr=1pηj,lk,r(wj,rαj,l)s,φ;Rn)2.(40)

As is known [1, Theorem B.1.7, B.1.8], the operator of change of variables vvαj,l and the operator of the multiplication by a function from C0 (ℝn) are bounded on each Sobolev space Hσ(ℝn) with σ ∈ ℝ. Therefore the linear operator vηj,lk,r(vαj,l) is bounded on Hσ(ℝn). Hence, owing to Proposition 3.1, this operator is also bounded on the Hörmander space Hs,φ(ℝn). Thus, formula (40) implies that Kws,φ;Γ,V2cj=1ϰr=1pwj,rs,φ;Rn2(41) for a certain number c > 0 that does not depend on w(C0(Rn))pϰ. Therefore the mapping (36) extends uniquely (by continuity) to a linear bounded operator K:(Hs,φ(Rn))pϰHs,φ(Γ,V).(42)

Besides, this mapping extends uniquely to a bounded linear operator K:((Hσ(Rn))pϰHσ(Γ,V) for every σRn.(43) Taking here σ ∈ {sε, s + δ} and using the interpolation with the function parameter ψ, we conclude that the restriction of the operator (43) with σ = sε to the space (31) is a bounded operator K:(Hs,φ(Rn))pϰ[Hsε(Γ,V),Hs+δ(Γ,V)]ψ.(44)

It follows from equality (37) and from the boundedness of operators (44) and (28) that uXψ=KTuXψc1us,φ;Γ,V for every uC(Γ,V), where c1 is the norm of the product of these operators, and Xψ:=[Hsε(Γ,V),Hs+δ(Γ,V)]ψ. Besides, the boundedness of operators (42) and (32) implies that us,φ;Γ,V=KTus,φ;Γ,Vc2uXψ for every uC(Γ,V), with c2 being the norm of the product of the last two operators. Thus, the norms in the spaces Hs,φ(Γ, V) and Xψ are equivalent on the linear manifold C(Γ, V). Since this manifold is dense in these spaces, they coincide up to equivalence of norms (the set C(Γ, V) is dense in Xψ due to (8)). □

The proofs of Theorems 4.24.4 are quite similar to the proofs of assertions (i), (iii) and (v) of Theorem 2.3 from monograph [7, Section 2.1.2], where the case of trivial vector bundle of rank p = 1 is considered. We will give these proofs for the sake of the readers convenience and completeness of the presentation.

Proof of Theorem 4.2

Consider two triplets 𝓐1 and 𝓐2 each of which is formed by an atlas of the manifolod Γ, appropriate partition of unity on Γ, and collection of local trivializations of the total space V. Let Hs,φ (Γ, V ; 𝓐j) and Hσ(Γ, V ; 𝓐j) respectively denote the Hörmander space Hs,φ(Γ, V) and the Sobolev space Hσ(Γ, V) corresponding to the triplet 𝓐j with j ∈ {1, 2}. The conclusion of Theorem 4.2 holds true in the Sobolev case of φ ≡ 1 (see, e.g., [2, p. 110]). Hence, the identity mapping is an isomorphism I:Hσ(Γ,V;A1)Hσ(Γ,V;A2) for each σ ∈ ℝ. Considering this isomorphism for σ := {sε, s + δ} and using the interpolation with the function parameter ψ defined by formula (9), we conclude that the identity mapping is an isomorphism I:[Hsε(Γ,V;A1),Hs+δ(Γ,V;A1)]ψ[Hsε(Γ,V;A2),Hs+δ(Γ,V;A2)]ψ.

According to Theorem 4.1, [Hsε(Γ,V;Aj),Hs+δ(Γ,V;Aj)]ψ=Hs,φ(Γ,V;Aj) for each j ∈ {1, 2} with equivalence of norms in the spaces. Thus, the spaces Hs,φ (Γ, V ; 𝓐1) and Hs,φ(Γ, V ; 𝓐2) are equal up to equivalence of norms. □

Proof of Theorem 4.3

Let ε > 0. According to Theorem 4.1 there exist interpolation parameters χ, η∈ 𝓑 such that [Hs+ε/2(Γ,V),Hs+2ε(Γ,V)]χ=Hs+ε,φ1(Γ,V),[Hsε(Γ,V),Hs+ε/3(Γ,V)]η=Hs,φ(Γ,V), with equivalence of norms. Hence, owing to (8), we have the continuous embeddings Hs+ε,φ1(Γ,V)Hs+ε/2(Γ,V)Hs+ε/3(Γ,V)Hs,φ(Γ,V).

They are extensions by continuity of the identity mapping uu, uC(Γ, V). Here, the middle embedding Hs+ε/2(Γ,V)Hs+ε/3(Γ,V) is compact (see, e.g., [2, Proposition 1.2]). Therefore, the embedding Hs+ε,φ1(Γ,V)Hs,φ(Γ,V) is also compact. □

Proof of Theorem 4.4

This theorem is known in the Sobolev case of φ = 1 (see, e.g., [2, p. 110]). Hence, for every σ∈ ℝ, the lineal mapping Q : v ↦ 〈v, ⋅〉Γ, V, with vHσ(Γ, V), is an isomorphism Q : Hσ(Γ, V) ↔ (Hσ(Γ, V))…. Considering the latter for σ = s ∓ 1 and using the interpolation with the function parameter ψ defined by formula (9) with ε = δ = 1, we obtain an isomorphism Q:[Hs1(Γ,V),Hs+1(Γ,V)]ψ[(Hs+1(Γ,V)),(Hs1(Γ,V))]ψ.(45)

Here, [Hs1(Γ,V),Hs+1(Γ,V)]ψ=Hs,φ(Γ,V) by Theorem 4.1. Besides, according to Proposition 6.3 we have [(Hs+1(Γ,V)),(Hs1(Γ,V))]ψ=[Hs1(Γ,V),Hs+1(Γ,V)]χ=(Hs,1/φ(Γ,V)). The latter equality is true due to Theorem 4.1 because χ(t) := t/ψ(t) = t1/2/φ(t1/2) for t ≥ 1. Thus, isomorphism (45) acts between the spaces Q:Hs,φ(Γ,V)(Hs,1/φ(Γ,V)).

This means that the spaces Hs,φ(Γ, V) and Hs,1/φ(Γ, V) are mutually dual (up to equivalence of norms) with respect to the sesquilinear form 〈u, vΓ,V of uHs,φ(Γ, V) and vHs,1/φ(Γ, V). This form is an extension by continuity of the form 〈u, vΓ,V of uHs,φ(Γ, V)↪ Hs−1 (Γ, V) and vHs+1(Γ, V). Since the first form is continuous in their arguments separately, estimate (12) holds true (see, e.g. [46, Chapter II, Section 4, Exercise 4]). □

Our proof of the next Theorem 4.5 is based on the following result.

Proposition 7.1

Let φ ∈ 𝓜 and 0 ≤ q ∈ ℤ. Then condition (13) implies the continuous embedding Hq+n/2,φ(ℝn) Cbq(Rn). Conversely, if {wHq+n/2,φ(Rn):supp wG}Cq(Rn)(46) for some open nonempty set G ⊂ ℝn, then condition (13) is satisfied.

This proposition follows from Hörmander’s embedding theorem [3, Theorem 2.2.7] in the same way as [7, Theorem 1.15(iii)].

Proof of Theorem 4.5

Let us deduce Theorem 4.5 from Proposition 7.1. First, suppose that condition (13) is fulfilled. Then for an arbitrary section uC(Γ, V) we have the inequality u(q);Γ,V=j=1ϰk=1puj,k(q);Rncj=1ϰk=1puj,kq+n/2,φ;Rn2(pϰ1)/2c(j=1ϰk=1puj,kq+n/2,φ;Rn2)1/2=2(pϰ1)/2cuq+n/2,φ;Γ,V.

Here, the first equality is due to (14), and c is the norm of the continuous embedding operator Hq+n/2,φ (ℝn)↪ Cbq(Rn), which holds due to (13) and Proposition 7.1. Hence, the identity mapping I : uu, with uC(Γ, V), extends uniquely (by continuity) to a linear bounded operator I:Hq+n/2,φ(Γ,V)Cq(Γ,V).(47)

If this operator is injective, then it sets the continuous embedding of Hq+n/2,φ(Γ, V) in Cq(Γ, V). Let us prove the injectivity of (47). Consider the isometric flattening operator (28) with s = q + n/2. It is an extension by continuity of mapping (27). This mapping is well defined on functions uCq(Γ, V) and sets an isometric operator T:Cq(Γ,V)(Cbq(Rn))pϰ.

Therefore the equality T I u = T u extends by closer from functions uC(Γ, V) to all functions uHq+n/2,φ(Γ, V). Now, if a section uHq+n/2,φ(Γ, V) satisfies I u = 0, then T u = T I u = 0 and therefore u = 0. Thus, operator (47) is injective, and hence it sets the continuous embedding Hq+n/2,φ(Γ,V)Cq(Γ,V).(48)

Let us now prove that this embedding is compact. Without loss of generality we may consider φ ∈ 𝓜 as a continuous function on [1, ∞). Indeed, as is known [42, Section 1.4], there exists a continuous function φ1 ∈ 𝓜 that both functions φ/φ1 and φ1/φ are bounded on [1, ∞). Therefore the spaces Hq+n/2,φ(Γ, V) and Hq+n/2,φ1(Γ, V) are equal up to equivalence of norms. Then we may use the second space instead of the first in our reasoning.

We put φ0(t):=φ(t)(tdttφ(t))1/2 for arbitrary t1.(49)

Owing to [7, Lemma 1.4], the function φ0 belongs to 𝓜 and has the following two properties: limtφ0(t)φ(t)0and1dttφ0(t)<.

It follows from the first property that we have the compact embedding Hq+n/2,φ(Γ,V)Hq+n/2,φ0(Γ,V).

This is demonstrated in the same way as in the proof of [7, Theorem 2.3(iv)]. According to the second property, the continuous embedding Hq+n/2,φ0(Γ,V)Cq(Γ,V), holds true, as we have just proved. Hence, embedding (48) is compact as a composition of compact and continuous embeddings.

It remains to prove that condition (13) follows from embedding (48). Assume that this embedding holds true. Without loss of generality we may suppose that Γ1⊄(Γ2∪⋯∪Γϰ). Therefore, there exists a nonempty open set U ⊂Γ1 such that χ1(x) = 1 for every xU. We arbitrarily choose a function wHq+n/2,φ(ℝn) such that supp wα11 (U). Turn to the operator K defined by formulas (33)(35). According to (42) with s = q + n/2 and owing to our assumption, we have the inclusion K(w,0,,0pϰ1)Hq+n/2,φ(Γ,V)Cq(Γ,V).(50) Let us deduce from this inclusion that w∈ Cq(ℝn).

To this end we introduce the linear mapping T1:uΠ1(β1(χ1u)α1),withuCq(Γ,V).

It acts continuously from Cq(Γ, V) to Cbq(Rn). Besides, the operator K acts continuously from (Hq+n/2,φ(ℝn))^{px} to Cq(Γ, V) according to (42) with s = q+n/2 and our assumption. Hence, T1K(w, 0, \ldots, 0)=w; this equality is evident if additionally w∈ C0 (ℝn) and then extends by closure over each function w chosen above. Now w=T1K(w,0,,0)Cq(Rn)

Thus, we obtain embedding (46) with G := α11 (U). It implies condition (13) due to Proposition 7.1. □

8 Proofs of properties of elliptic operators on the refined Sobolev scale

Beforehand we will prove the following result:

Lemma 8.1

Let r ∈ ℝ and LΨphr (Γ;V1, V2). Then the mapping uLu, with uC(Γ, V1), extends uniquely (by continuity) to a bounded linear operator L:Hσ,φ(Γ,V1)Hσr,φ(Γ,V2)(51) for all σ ∈ ℝ and φ ∈ 𝓜.

Proof

Let σ ∈ ℝ and φ ∈ 𝓜. This lemma is known in the Sobolev case of φ = 1 (see, e.g., [1, p. 92]). Thus, the mapping u ↦ Lu, with uC(Γ, V1), extends uniquely to bounded linear operators L:Hσ1(Γ,V1)Hσ1r(Γ,V2).

Using the interpolation with the function parameter ψ defined by formula (9) with ε=δ = 1, we conclude by Theorem 4.1 that the restriction of the first operator to the space Hs,φ(Γ, V1) is a bounded operator between the spaces L:Hσ,φ(Γ,V1)=[Hσ1(Γ,V1),Hσ+1(Γ,V1)]ψ[Hσr1(Γ,V2),Hσr+1(Γ,V2)]ψ=Hσr,φ(Γ,V2). □

Proof of Theorem 5.1

According to Lemma 8.1 the mapping uAu, with uC(Γ, V1), extends by continuity to the bounded linear operator (18). Let us prove that this operator is Fredholm. Theorem 5.1 is known for Sobolev spaces, where φ = 1 (see, e.g., [1, Theorem 19.2.1]). Therefore the bounded linear operators A:Hs1+m(Γ,V1)Hs1(Γ,V2)(52) are Fredholm with the kernel 𝔑, index dim 𝔑− dim 𝔑+, and range A(Hs1+m(Γ,V1))={fHs1(Γ,V2):f,wΓ,V2=0 for all wN+}.(53)

Using the interpolation with the function parameter ψ defined by formula (9) with ε=δ = 1, we conclude by Proposition 6.2 that the bounded operator A:[Hs1+m(Γ,V1),Hs+1+m(Γ,V1)]ψ[Hs1(Γ,V2),Hs+1(Γ,V2)]ψ is also Fredholm. According to Proposition 3.1 this operator coincides with (18). Moreover, owing to Proposition 6.2, the kernel of the Fredholm operator (18) equals 𝔑, the index equals dim 𝔑− dim 𝔑+, and the range is Hs,φ(Γ,V2)A(Hs1+m(Γ,V1))={fHs,φ(Γ,V2):f,wΓ,V2=0 for all wN+} in view of (53). □

Proof of Theorem 5.2

Owing to Theorem 4.1, 𝔑 is the kernel and P+(Hs,φ(Γ, V2)) is the range of the operator (18). Hence, the restriction of (18) to the subspace P(Hs+m,φ(Γ, V1)) is the bijective linear bounded operator (22). This operator is an isomorphism by the Banach theorem on inverse operator. □

Proof of Theorem 5.3

The global estimate (24) is a direct consequence of Theorem 5.1 and Peetre’s lemma [47, p. 728, Lemma 3]. (Of course, in (24) we may take not only σ<s + m but also arbitrary σ∈ ℝ.) We will deduce (23) from (24). Beforehand, let us prove the following result: for each integer r ≥ 1 and for arbitrary functions χ, η from Theorem 5.3 there exists a number c > 0 such that χus+m,φ;Γ,V1c(ηAus,φ;Γ,V2+ηus+mr,φ;Γ,V1+uσ;Γ,V1)(54) for every uHs+m,φ(Γ, V1).

According to (24) there exists a number c0>0 such that χus+m,φ;Γ,V1c0(A(χu)s,φ;Γ,V2+χuσ;Γ,V1)(55) for arbitrary uHs+m,φ(Γ, V1). Rearranging the PsDO A and the operator of the multiplication by χ, we arrived at the formula A(χu)=A(χηu)=χA(ηu)+A(ηu)=χAu+χA((η1)u)+A(ηu).(56)

Here, A… is a certain PsDO from Ψphm1 (Γ;V1, V2) (see, e.g., [38, p. 13]), and the PsDO uχ A((η−1)u) belongs to each class Ψphλ (Γ;V1, V2) with λ ∈ ℝ because supp χ ∩ supp(η−1) = ∅. Therefore, owing to Lemma 8.1, we obtain the inequalities A(χu)s,φ;Γ,V2χAus,φ;Γ,V2+χA((η1)u)s,φ;Γ,V2+A(ηu)s,φ;Γ,V2χAus,φ;Γ,V2+c1uσ1,φ;Γ,V1+c2ηus+m1,φ;Γ,V1χAus,φ;Γ,V2+c1c3uσ;Γ,V1+c2ηus+m1,φ;Γ,V1.(57)

Here, c1 is the norm of the operator uχ A((η−1)u) that acts continuously from Hσ−1,φ(Γ, V1) to Hs,φ(Γ, V2), and c2 is the norm of the bounded operator A… from Hs+m−1,φ(Γ, V1) to Hs,φ(Γ, V2). Besides, c3 is the norm of the operator of the continuous embedding Hσ(Γ, V1) ↪ Hσ−1,φ(Γ, V1).

Formulas (55) and (57) yield the inequalities χus+m,φ;Γ,V1c0(χAus,φ;Γ,V2+c1c3uσ;Γ,V1+c2ηus+m1,φ;Γ,V1+χuσ;Γ,V1)c0(χAus,φ;Γ,V2+c1c3uσ;Γ,V1+c2ηus+m1,φ;Γ,V1+c4uσ;Γ,V1).(58)

Here, c4 is the norm of the bounded operator uχ u on the space Hσ(Γ, V1). Note that χAus,φ;Γ,V2=χηAus,φ;Γ,V2c~ηAus,φ;Γ,V2, with c~ being the norm of the bounded operator vχ v on the space Hs,φ(Γ, V2). Thus, we have proved (54) for r = 1.

Choose an integer k ≥ 1 arbitrarily and assume that (54) is true for r = k. Let us prove that (54) is also true for r = k + 1. We choose a function η1C(Γ) such that η1 = 1 in a neighbourhood of supp χ and that η = 1 in a neighbourhood of supp η1. According to our assumption, there exists a number c5 > 0 such that χus+m,φ;Γ,V1c5(η1Aus,φ;Γ,V2+η1us+mk,φ;Γ,V1+uσ;Γ,V1)(59) for arbitrary uHs+m,φ(Γ, V1). Owing to (24) we write η1us+mk,φ;Γ,V1c6(A(η1u)sk,φ;Γ,V2+η1uσ;Γ,V1);(60) here, c6 is a certain positive number that does not depend on u. Rearranging the PsDO A and the operator of the multiplication by η1, we obtain A(η1u)=A(η1ηu)=η1A(ηu)+A1(ηu)=η1Au+η1A((η1)u)+A1(ηu).(61)

Here, A1 is a certain PsDO from Ψphm1 (Γ;V1, V2), and the PsDO uη1A((η − 1)u) belongs to each class Ψphλ (Γ;V1, V2) with λ∈ ℝ because supp η1 ∩ supp(η−1) = ∅. Therefore, owing to Lemma 8.1, we obtain the inequalities A(η1u)sk,φ;Γ,V2η1Ausk,φ;Γ,V2+η1A((η1)u)sk,φ;Γ,V2+A1(ηu)sk,φ;Γ,V2η1Ausk,φ;Γ,V2+c7uσ1,φ;Γ,V1+c8ηusk+m1,φ;Γ,V1η1Ausk,φ;Γ,V2+c7c3uσ;Γ,V1+c8ηus+m(k+1),φ;Γ,V1.(62)

Here, c7 is the norm of the operator uη1A((η−1)u) that acts continuously from Hσ−1,φ(Γ, V1) to Hsk,φ(Γ, V2), and c8 is the norm of the bounded operator A1 from Hsk+m−1, φ(Γ, V1) to Hsk,φ(Γ, V2).

Now formulas (59), (60), and (62) yield the inequalities χus+m,φ;Γ,V1c5(η1Aus,φ;Γ,V2+c6(A(η1u)sk,φ;Γ,V2+η1uσ;Γ,V1)+uσ;Γ,V1)c5(η1Aus,φ;Γ,V2+c6(η1Ausk,φ;Γ,V2+c7c3uσ;Γ,V1+c8ηus+m(k+1),φ;Γ,V1)+uσ;Γ,V1).(63)

Since η1 = η1η, we have η1Aus,φ;Γ,V2+c6η1Ausk,φ;Γ,V2(1+c6)η1ηAus,φ;Γ,V2(1+c6)c9ηAus,φ;Γ,V2.(64)

Here, c9 is the norm of the bounded operator vη1v on the space Hs,φ(Γ, V2). Now formulas (63) and (64) give the inequality (54) with r = k + 1. Owing to the principle of mathematical induction, this inequality is true for each integer r ≥ 1.

The required estimate (23) follows from the inequality (54), where r ∈ ℤ such that s + mr <σ, in view of ηus+mr,φ;Γ,V1c10ηuσ;Γ,V1c10c11uσ;Γ,V1.

Here, c10 is the norm of the embedding operator Hσ(Γ, V1)↪Hs+mr,φ(Γ, V1), and c11 is the norm of the operator uη u on the space Hσ(Γ, V1) □

As to Remark 5.4 note that inequality (25) follows from (58) with σ <s+m−1 in view of Theorem 4.3.

Proof of Theorem 5.5

Since uH−∞(Γ, V1), there exists an integer r ≥ 0 such that uHs+mr,φ(Γ, V1). Let us first prove this theorem in the global case where Γ0 = Γ. In this case, Au=fHs,φ(Γ,V2)A(Hs+mr,φ(Γ,V1))=A(Hs+m,φ(Γ,V1)) by the condition and Theorem 5.1. Hence, there exists a section vHs+m,φ(Γ, V1) such that Av = f on Γ. Since A(uv) = 0 on Γ and uvHs+mr,φ(Γ, V1), we conclude by Theorem 5.1 that w:=uvNC(Γ,V1).

Thus, u=v+wHs+m,φ(Γ,V1). Theorem 5.5 is proved in the case of Γ0 = Γ.

We now deduce this theorem in the general situation from the case just considered. Beforehand, let us prove that for every integer k ≥ 1 the following implication holds for u: uHlocs+mk,φ(Γ0,V1)uHlocs+mk+1,φ(Γ0,V1).(65) Assume that uHlocs+mk,φ0, V1). We arbitrarily choose a function χC(Γ) such that supp χ⊂Γ0. Let a function ηC(Γ) satisfy the conditions supp η ⊂ Γ0 and η = 1 in a neighbourhood of supp χ. According to (56) we have the equality A(χu)=χf+χA((η1)u)+A(ηu).(66) Here, χ fHs,φ(Γ, V2) by the condition; χ A((η−1)u)∈ Hs,φ(Γ, V2) because uHs+mr,φ(Γ, V1) and the PsDO uχ A((η−1)u) belongs to Ψphmr (Γ;V1, V2), and A…(η u)∈ Hsk+1,φ(Γ, V2) because η uHs+mk,φ(Γ, V1) by our assumption and because the inclusion A…∈ Ψphm1 (Γ;V1, V2). Hence, the right-hand side of equality (66) belongs to Hsk+1,φ(Γ, V2). Therefore χ uHs+mk+1,φ(Γ, V1) by what we have proved in the previous paragraph. Thus, uHlocs+mk+1,φ0, V1) in view of our choice of χ. Implication (65) is proved.

Applying this implication successively for k = r, r−1, …, 1, we conclude that uHs+mr,φ(Γ,V1)Hlocs+mr,φ(Γ0,V1)uHlocs+mr+1,φ(Γ0,V1)uHlocs+m,φ(Γ0,V1).

Thus, we have proved the required inclusion uHlocs+m,φ(Γ0,V1). □

Proof of Theorem 5.6

Owing to Theorem 5.5 where s := qm+n/2 we have the inclusion uHlocq+n/2,φ(Γ0,V1). We arbitrarily choose a function χC(Γ) such that supp χ ⊂ Γ0. Then χuHq+n/2,φ(Γ,V1)Cq(Γ,V1) due to condition (13) and Theorem 4.5. Therefore uCq0, V1). □

Proof of Remark 5.7

If condition (13) is satisfied, then we have implication (26) according to Theorem 5.6. Assume now that this implication is valid and prove that φ satisfies condition (13). Without loss of generality we may suppose that Γ0∩Γ1≠∅. We choose a nonempty open set U⊂Γ0∩Γ1 and a function χC(Γ) such that supp χ⊂Γ0 and χ = 1 on U. Turn to the operator K defined by formulas (33)(35), where βj := β1,j and the function η1 additionally satisfies the equality η1 = 1 on the set α11 (U). We arbitrarily choose a function wHq+n/2,φ(ℝn) such that supp wα11 (U). Then u:=K(w,0,,0pϰ1)Hq+n/2,φ(Γ,V1) according to (42) with s := q + n/2 and V := V1. The premise of implication (26) holds true for the section u. Hence, uClocq0, V1) according to this implication. Therefore u = χ uCq(Γ, V1) due to our choice of χ. Let us use the operator T1, with β1 := β1,1, introduced in the proof of Theorem 4.5. Owing to the properties of T1 mentioned therein, we write w=T1K(w,0,,0)Cq(Rn).

Thus, we obtain embedding (46) with G := α11 (U). It implies condition (13) by Proposition 7.1. □

Acknowledgement

The author is grateful to A. A. Murach for his big help in preparing of the paper. The author thanks Referees for their remarks and suggestions about improving the language of the paper.

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About the article

Received: 2016-11-29

Accepted: 2017-05-29

Published Online: 2017-07-13


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 907–925, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0076.

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© 2017 Zinchenko. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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