#### 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
$$\begin{array}{}T:u\mapsto ({u}_{1,1},\dots ,{u}_{1,p},\dots ,{u}_{\varkappa ,1},\dots ,{u}_{\varkappa ,p})\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}for arbitrary}\phantom{\rule{1em}{0ex}}u\in {C}^{\mathrm{\infty}}(\mathrm{\Gamma},V).\end{array}$$(27)

Here, each function *u*_{j,k} ∈
$\begin{array}{}{C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n})\end{array}$ is defined by formula (5). According to (4), the norm of *u* in the space *H*^{s,φ}(Γ, *V*) is equal to the norm of *T u* in the Hilbert space (*H*^{s,φ}(ℝ^{n}))^{pϰ}. Therefore the linear mapping *u* ↦ *Tu*, with *u* ∈ *C*^{∞}(Γ, *V*), extends continuously to the isometric operator
$$\begin{array}{}T:{H}^{s,\phi}(\mathrm{\Gamma},V)\to ({H}^{s,\phi}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$$(28)

Besides, this mapping extends continuously to the isometric operators
$$\begin{array}{}T:{H}^{\sigma}(\mathrm{\Gamma},V)\to ({H}^{\sigma}({\mathbb{R}}^{n}){)}^{p\varkappa},\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}\sigma \in \mathbb{R},\end{array}$$(29)
between Sobolev spaces. Since *ψ* is an interpolation parameter, it follows from the boundedness of the linear operators (29) with *σ* ∈ {*s* − *ε*, *s* + *δ*} that the restriction of the operator (29) with *σ* = *s* − *ε* is a bounded operator
$$\begin{array}{}T:[{H}^{s-\epsilon}(\mathrm{\Gamma},V),{H}^{s+\delta}(\mathrm{\Gamma},V){]}_{\psi}\to [({H}^{s-\epsilon}({\mathbb{R}}^{n}){)}^{p\varkappa},({H}^{s+\delta}({\mathbb{R}}^{n}){)}^{p\varkappa}{]}_{\psi}.\end{array}$$(30)

Owing to Propositions 3.1 and 6.1, the target space of (30) takes the form
$$\begin{array}{}[({H}^{s-\epsilon}({\mathbb{R}}^{n}){)}^{p\varkappa},({H}^{s+\delta}({\mathbb{R}}^{n}){)}^{p\varkappa}{]}_{\psi}=([{H}^{s-\epsilon}({\mathbb{R}}^{n}),{H}^{s+\delta}({\mathbb{R}}^{n}){]}_{\psi}{)}^{p\varkappa}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=({H}^{s,\phi}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$$(31)

Thus, (30) is a bounded operator between the spaces
$$\begin{array}{}T:[{H}^{s-\epsilon}(\mathrm{\Gamma},V),{H}^{s+\delta}(\mathrm{\Gamma},V){]}_{\psi}\to ({H}^{s,\phi}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$$(32)

Consider now the mapping of sewing
$$\begin{array}{}K:({w}_{1,1},{\displaystyle \dots ,{w}_{1,p},\dots ,{w}_{\varkappa ,1},\dots ,{w}_{\varkappa ,p})\mapsto \sum _{j=1}^{\varkappa}{w}_{j}}\end{array}$$(33)
defined on vectors
$$\begin{array}{}\mathbf{w}:=({w}_{1,1},\dots ,{w}_{1,p},\dots ,{w}_{x,1},\dots ,{w}_{x,p})\in ({C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$$(34)

Here, for each *j* ∈ {1, …, *ϰ*}, the section *w*_{j} ∈ *C*^{∞}(Γ, *V*) is defined by the formula
$$\begin{array}{}{w}_{j}(x):=\left\{\begin{array}{lll}{\beta}_{j}^{-1}(x,({\eta}_{j}{w}_{j,1})({\alpha}_{j}^{-1}(x)),\dots ,({\eta}_{j}{w}_{j,p})({\alpha}_{j}^{-1}(x)))& \text{if}& x\in {\mathrm{\Gamma}}_{j},\\ 0& \text{if}& x\in \mathrm{\Gamma}\mathrm{\setminus}{\mathrm{\Gamma}}_{j},\end{array}\right.\end{array}$$(35)
in which the function *η j* ∈
$\begin{array}{}{C}_{0}^{\mathrm{\infty}}\end{array}$
(ℝ^{n}) is chosen so that *η*_{j} = 1 on the set
$\begin{array}{}{\alpha}_{j}^{-1}\end{array}$
(supp *χj*).

We have the linear mapping
$$\begin{array}{}K:({C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n}){)}^{p\varkappa}\to {C}^{\mathrm{\infty}}(\mathrm{\Gamma},V).\end{array}$$(36)

It is left inverse to the flattening mapping (27). Indeed, given *u* ∈ *C*^{∞}(Γ, *V*), we write
$$\begin{array}{}KTu=K({u}_{1,1},\dots ,{u}_{1,p},\dots ,{u}_{\varkappa ,1},\dots ,{u}_{\varkappa ,p})={\displaystyle \sum _{j=1}^{\varkappa}{u}_{j},}\end{array}$$
where each section *u*_{j} ∈ *C*^{∞}(Γ, *V*) is defined by formula (35) with *u* instead of *w*. In this formula, for arbitrary *k* ∈ {1, …, *p*} and *x* ∈ Γ_{j}, we have the equalities
$$\begin{array}{}({\eta}_{j}{u}_{j,k})({\alpha}_{j}^{-1}(x))=({\eta}_{j}({\alpha}_{j}^{-1}(x))){\mathrm{\Pi}}_{k}({\beta}_{j}(({\chi}_{j}u)(x)))={\mathrm{\Pi}}_{k}({\beta}_{j}(({\chi}_{j}u)(x)))\end{array}$$
due to our choice of *η*_{j}. Therefore
$$\begin{array}{}{u}_{j}(x)={\beta}_{j}^{-1}(x,({\eta}_{j}{u}_{j,1})({\alpha}_{j}^{-1}(x)),\dots ,({\eta}_{j}{u}_{j,p})({\alpha}_{j}^{-1}(x)))\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\beta}_{j}^{-1}(x,{\mathrm{\Pi}}_{1}({\beta}_{j}(({\chi}_{j}u)(x))),\dots ,{\mathrm{\Pi}}_{p}({\beta}_{j}(({\chi}_{j}u)(x))))\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=({\chi}_{j}u)(x)\end{array}$$
for every *x* ∈ Γ_{j}. Note that if *x* ∈ Γ \ Γ_{j}, then *u*_{j}(*x*) = 0 = (*χ*_{j}u)(*x*). Thus, *u*_{j}(*x*) = (*χ*_{j}u)(*x*) for arbitrary *x* ∈ Γ. Hence,
$$\begin{array}{}KTu={\displaystyle \sum _{j=1}^{\varkappa}{u}_{j}=\sum _{j=1}^{\varkappa}{\chi}_{j}u=u\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}for every\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u\in {C}^{\mathrm{\infty}}(\mathrm{\Gamma},V).}\end{array}$$(37)
Let us prove that the mapping (36) extends uniquely to a linear bounded operator between the spaces (*H*^{s,φ}(ℝ^{n}))^{pϰ} and *H*^{s,φ}(Γ, *V*). Given a vector (34), we write
$$\begin{array}{}\parallel K\mathbf{w}{\parallel}_{s,\phi ;\mathrm{\Gamma},V}^{2}{\displaystyle =\sum _{l=1}^{\varkappa}\sum _{k=1}^{p}\parallel {\mathrm{\Pi}}_{k}({\beta}_{l}\circ ({\chi}_{l}K\mathbf{w})\circ {\alpha}_{l}){\parallel}_{s,\phi ;{\mathbb{R}}^{n}}^{2}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle =\sum _{l=1}^{\varkappa}\sum _{k=1}^{p}\parallel \sum _{j=1}^{\varkappa}{\mathrm{\Pi}}_{k}({\beta}_{l}\circ ({\chi}_{l}{w}_{j})\circ {\alpha}_{l}){\parallel}_{s,\phi ;{\mathbb{R}}^{n}}^{2}.}\end{array}$$(38)

Examine the function (*χ*_{l}w_{j})∘*α*_{l} : ℝ^{n}↦*π*^{−1}(Γ_{l}) with *l*, *j* ∈ {1, …, *x*}. If *t* ∈ ℝ^{n} satisfies *α*_{l}(*t*)∈ Γ_{j}, then
$$\begin{array}{}(({\chi}_{l}{w}_{j})\circ {\alpha}_{l})(t)=({\chi}_{l}\circ {\alpha}_{l})(t)\cdot ({w}_{j}\circ {\alpha}_{l})(t)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=({\chi}_{l}\circ {\alpha}_{l})(t)\cdot {\beta}_{j}^{-1}({\alpha}_{l}(t),({\eta}_{j}{w}_{j,1})(({\alpha}_{j}^{-1}\circ {\alpha}_{l})(t)),\dots ,({\eta}_{j}{w}_{j,p})(({\alpha}_{j}^{-1}\circ {\alpha}_{l})(t)))\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\beta}_{j}^{-1}({\alpha}_{l}(t),({\chi}_{l}\circ {\alpha}_{l})(t)\cdot ({\eta}_{j}{w}_{j,1})(({\alpha}_{j}^{-1}\circ {\alpha}_{l})(t)),\dots ,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}({\chi}_{l}\circ {\alpha}_{l})(t)\cdot ({\eta}_{j}{w}_{j,p})(({\alpha}_{j}^{-1}\circ {\alpha}_{l})(t)))\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\beta}_{j}^{-1}({\alpha}_{l}(t),(({\eta}_{j,l}{w}_{j,1})\circ {\alpha}_{j,l})(t),\dots ,(({\eta}_{j,l}{w}_{j,p})\circ {\alpha}_{j,l})(t)).\end{array}$$
Here, *η*_{j,l} := (*χ*_{l} ∘ *α*_{l})*η*_{j} ∈
$\begin{array}{}{C}_{0}^{\mathrm{\infty}}\end{array}$
(ℝ^{n}), whereas *α*_{j,l} : ℝ^{n}↔ ℝ^{n} is an infinitely smooth diffeomorphism such that *α*_{j,l} :=
$\begin{array}{}{\alpha}_{j}^{-1}\end{array}$
∘ *α*_{l} in a neighbourhood of supp *η*_{j,l} and that *α*_{j,l}(*t*) = *t* whenever |*t*| ≫ 1. Then, given *k* ∈ {1, …, *p*}, we have the equalities
$$\begin{array}{}\phantom{\rule{1em}{0ex}}{\mathrm{\Pi}}_{k}({\beta}_{l}\circ ({\chi}_{l}{w}_{j})\circ {\alpha}_{l})(t)\\ ={\mathrm{\Pi}}_{k}({\beta}_{l}\circ {\beta}_{j}^{-1})({\alpha}_{l}(t),(({\eta}_{j,l}{w}_{j,1})\circ {\alpha}_{j,l})(t),\dots ,(({\eta}_{j,l}{w}_{j,p})\circ {\alpha}_{j,l})(t))\\ ={\displaystyle \sum _{r=1}^{p}{\beta}_{l,j}^{k,r}({\alpha}_{l}(t))\cdot (({\eta}_{j,l}{w}_{j,r})\circ {\alpha}_{j,l})(t).}\end{array}$$

Here, each
$\begin{array}{}{\beta}_{l,\phantom{\rule{thinmathspace}{0ex}}j}^{k,\phantom{\rule{thinmathspace}{0ex}}r}\end{array}$
is a certain complex-valued function from *C*^{∞}(Γ) such that the matrix-valued function
$\begin{array}{}({\beta}_{l,j}^{k,r}(x){)}_{k,r=1}^{p}\text{\hspace{0.17em}of\hspace{0.17em}}x\in \text{supp\hspace{0.17em}}{\chi}_{l}\cap \text{\hspace{0.17em}supp}({\eta}_{j}\circ {\alpha}_{j}^{-1})\end{array}$
corresponds to the transition mapping *β*_{l} ∘
$\begin{array}{}{\beta}_{j}^{-1}\end{array}$
. Thus,
$$\begin{array}{}{\displaystyle {\mathrm{\Pi}}_{k}({\beta}_{l}\circ ({\chi}_{l}{w}_{j})\circ {\alpha}_{l})(t)=\sum _{r=1}^{p}{\beta}_{l,j}^{k,r}({\alpha}_{l}(t))\cdot (({\eta}_{j,l}{w}_{j,r})\circ {\alpha}_{j,l})(t)}\end{array}$$(39)
for arbitrary *t* ∈ ℝ (if *α*_{l}(*t*)∉ Γ_{j}, then this equality becomes 0 = 0).

Owing to (38) and (39) we write
$$\begin{array}{}{\displaystyle \parallel K\mathbf{w}{\parallel}_{s,\phi ;\mathrm{\Gamma},V}^{2}=\sum _{l=1}^{\varkappa}\sum _{k=1}^{p}\parallel \sum _{j=1}^{\varkappa}\sum _{r=1}^{p}({\beta}_{l,j}^{k,r}\circ {\alpha}_{l})\cdot (({\eta}_{j,l}{w}_{j,r})\circ {\alpha}_{j,l}){\parallel}_{s,\phi ;{\mathbb{R}}^{n}}^{2}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle =\sum _{l=1}^{\varkappa}\sum _{k=1}^{p}\parallel \sum _{j=1}^{\varkappa}\sum _{r=1}^{p}{\eta}_{j,l}^{k,r}\cdot ({w}_{j,r}\circ {\alpha}_{j,l}){\parallel}_{s,\phi ;{\mathbb{R}}^{n}}^{2};}\end{array}$$
here, each
$$\begin{array}{}{\eta}_{j,l}^{k,r}:=({\beta}_{l,j}^{k,r}\circ {\alpha}_{l})\cdot ({\eta}_{j,l}\circ {\alpha}_{j,l})\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n}).\end{array}$$

Thus
$$\begin{array}{}{\displaystyle \parallel K\mathbf{w}{\parallel}_{s,\phi ;\mathrm{\Gamma},V}^{2}\le \sum _{l=1}^{\varkappa}\sum _{k=1}^{p}(\sum _{j=1}^{\varkappa}\sum _{r=1}^{p}\parallel {\eta}_{j,l}^{k,r}\cdot ({w}_{j,r}\circ {\alpha}_{j,l}){\parallel}_{s,\phi ;{\mathbb{R}}^{n}}{)}^{2}.}\end{array}$$(40)

As is known [1, Theorem B.1.7, B.1.8], the operator of change of variables *v* ↦ *v* ∘ *α*_{j,l} and the operator of the multiplication by a function from
$\begin{array}{}{C}_{0}^{\mathrm{\infty}}\end{array}$
(ℝ^{n}) are bounded on each Sobolev space *H*^{σ}(ℝ^{n}) with *σ* ∈ ℝ. Therefore the linear operator
$\begin{array}{}v\mapsto {\eta}_{j,l}^{k,r}\cdot (v\circ {\alpha}_{j,l})\end{array}$
is bounded on *H*^{σ}(ℝ^{n}). Hence, owing to Proposition 3.1, this operator is also bounded on the Hörmander space *H*^{s,φ}(ℝ^{n}). Thus, formula (40) implies that
$$\begin{array}{}{\displaystyle \parallel K\mathbf{w}{\parallel}_{s,\phi ;\mathrm{\Gamma},V}^{2}\le c\sum _{j=1}^{\varkappa}\sum _{r=1}^{p}\parallel {w}_{j,r}{\parallel}_{s,\phi ;{\mathbb{R}}^{n}}^{2}}\end{array}$$(41)
for a certain number *c* > 0 that does not depend on
$\begin{array}{}\mathbf{w}\in ({C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$
Therefore the mapping (36) extends uniquely (by continuity) to a linear bounded operator
$$\begin{array}{}K:({H}^{s,\phi}({\mathbb{R}}^{n}){)}^{p\varkappa}\to {H}^{s,\phi}(\mathrm{\Gamma},V).\end{array}$$(42)

Besides, this mapping extends uniquely to a bounded linear operator
$$\begin{array}{}K:(({H}^{\sigma}({\mathbb{R}}^{n}){)}^{p\varkappa}\to {H}^{\sigma}(\mathrm{\Gamma},V)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}for every\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sigma \in {\mathbb{R}}^{n}.\end{array}$$(43)
Taking here *σ* ∈ {*s* − *ε*, *s* + *δ*} and using the interpolation with the function parameter *ψ*, we conclude that the restriction of the operator (43) with *σ* = *s* − *ε* to the space (31) is a bounded operator
$$\begin{array}{}K:({H}^{s,\phi}({\mathbb{R}}^{n}){)}^{p\varkappa}\to [{H}^{s-\epsilon}(\mathrm{\Gamma},V),{H}^{s+\delta}(\mathrm{\Gamma},V){]}_{\psi}.\end{array}$$(44)

It follows from equality (37) and from the boundedness of operators (44) and (28) that
$$\begin{array}{}\parallel u{\parallel}_{{X}_{\psi}}=\parallel K\mathit{T}\mathit{u}{\parallel}_{{X}_{\psi}}\le {c}_{1}\parallel u{\parallel}_{s,\phi ;\mathrm{\Gamma},V}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}for every\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u\in {C}^{\mathrm{\infty}}(\mathrm{\Gamma},V),\end{array}$$
where *c*_{1} is the norm of the product of these operators, and
$$\begin{array}{}{X}_{\psi}:=[{H}^{s-\epsilon}(\mathrm{\Gamma},V),{H}^{s+\delta}(\mathrm{\Gamma},V){]}_{\psi}.\end{array}$$
Besides, the boundedness of operators (42) and (32) implies that
$$\begin{array}{}\parallel u{\parallel}_{s,\phi ;\mathrm{\Gamma},V}=\parallel KTu{\parallel}_{s,\phi ;\mathrm{\Gamma},V}\le {c}_{2}\parallel u{\parallel}_{{X}_{\psi}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}for every\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}u\in {C}^{\mathrm{\infty}}(\mathrm{\Gamma},V),\end{array}$$
with *c*_{2} being the norm of the product of the last two operators. Thus, the norms in the spaces *H*^{s,φ}(Γ, *V*) and *X*_{ψ} 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 *X*_{ψ} due to (8)). □

The proofs of Theorems 4.2–4.4 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 4.4

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 *v* ∈ *H*^{σ}(Γ, *V*), is an isomorphism *Q* : *H*^{σ}(Γ, *V*) ↔ (*H*^{−σ}(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
$$\begin{array}{c}Q:[{H}^{s-1}(\mathrm{\Gamma},V),{H}^{s+1}(\mathrm{\Gamma},V){]}_{\psi}\leftrightarrow [({H}^{-s+1}(\mathrm{\Gamma},V){)}^{\prime},({H}^{-s-1}(\mathrm{\Gamma},V){)}^{\prime}{]}_{\psi}.\end{array}$$(45)

Here,
$$\begin{array}{c}[{H}^{s-1}(\mathrm{\Gamma},V),{H}^{s+1}(\mathrm{\Gamma},V){]}_{\psi}={H}^{s,\phi}(\mathrm{\Gamma},V)\end{array}$$
by Theorem 4.1. Besides, according to Proposition 6.3 we have
$$\begin{array}{}[({H}^{-s+1}(\mathrm{\Gamma},V){)}^{\prime},({H}^{-s-1}(\mathrm{\Gamma},V){)}^{\prime}{]}_{\psi}=[{H}^{-s-1}(\mathrm{\Gamma},V),{H}^{-s+1}(\mathrm{\Gamma},V){]}_{\chi}^{\prime}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=({H}^{-s,1/\phi}(\mathrm{\Gamma},V){)}^{\prime}.\end{array}$$
The latter equality is true due to Theorem 4.1 because *χ*(*t*) := *t*/*ψ*(*t*) = *t*^{1/2}/*φ*(*t*^{1/2}) for *t* ≥ 1. Thus, isomorphism (45) acts between the spaces
$$\begin{array}{}Q:{H}^{s,\phi}(\mathrm{\Gamma},V)\leftrightarrow ({H}^{-s,1/\phi}(\mathrm{\Gamma},V){)}^{\prime}.\end{array}$$

This means that the spaces *H*^{s,φ}(Γ, *V*) and *H*^{−s,1/φ}(Γ, *V*) are mutually dual (up to equivalence of norms) with respect to the sesquilinear form 〈*u*, *v*〉_{Γ,V} of *u* ∈ *H*^{s,φ}(Γ, *V*) and *v* ∈ *H*^{−s,1/φ}(Γ, *V*). This form is an extension by continuity of the form 〈*u*, *v*〉_{Γ,V} of *u* ∈ *H*^{s,φ}(Γ, *V*)↪ *H*^{s−1} (Γ, *V*) and *v* ∈ *H*^{−s+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]). □

#### 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 *u* ∈ *C*^{∞}(Γ, *V*) we have the inequality
$$\begin{array}{}\parallel u{\parallel}_{(q);\mathrm{\Gamma},V}={\displaystyle \sum _{j=1}^{\varkappa}\sum _{k=1}^{p}\parallel {u}_{j,k}{\parallel}_{(q);{\mathbb{R}}^{n}}\le c\sum _{j=1}^{\varkappa}\sum _{k=1}^{p}\parallel {u}_{j,k}{\parallel}_{q+n/2,\phi ;{\mathbb{R}}^{n}}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le {2}^{(p\varkappa -1)/2}c{\displaystyle (\sum _{j=1}^{\varkappa}\sum _{k=1}^{p}\parallel {u}_{j,k}{\parallel}_{q+n/2,\phi ;{\mathbb{R}}^{n}}^{2}{)}^{1/2}={2}^{(p\varkappa -1)/2}c\parallel u{\parallel}_{q+n/2,\phi ;\mathrm{\Gamma},V}.}\end{array}$$

Here, the first equality is due to (14), and *c* is the norm of the continuous embedding operator *H*^{q+n/2,φ} (ℝ^{n})↪
$\begin{array}{}{C}_{\text{b}}^{q}({\mathbb{R}}^{n}),\end{array}$
which holds due to (13) and Proposition 7.1. Hence, the identity mapping *I* : *u* ↦ *u*, with *u* ∈ *C*^{∞}(Γ, *V*), extends uniquely (by continuity) to a linear bounded operator
$$\begin{array}{}I:{H}^{q+n/2,\phi}(\mathrm{\Gamma},V)\to {C}^{q}(\mathrm{\Gamma},V).\end{array}$$(47)

If this operator is injective, then it sets the continuous embedding of *H*^{q+n/2,φ}(Γ, *V*) in *C*^{q}(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 *u* ∈ *C*^{q}(Γ, *V*) and sets an isometric operator
$$\begin{array}{}T:{C}^{q}(\mathrm{\Gamma},V)\to ({C}_{\text{b}}^{q}({\mathbb{R}}^{n}){)}^{p\varkappa}.\end{array}$$

Therefore the equality *T I u* = *T u* extends by closer from functions *u* ∈ *C*^{∞}(Γ, *V*) to all functions *u* ∈ *H*^{q+n/2,φ}(Γ, *V*). Now, if a section *u* ∈ *H*^{q+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
$$\begin{array}{}{H}^{q+n/2,\phi}(\mathrm{\Gamma},V)\hookrightarrow {C}^{q}(\mathrm{\Gamma},V).\end{array}$$(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 *φ*_{1} ∈ 𝓜 that both functions *φ*/*φ*_{1} and *φ*_{1}/*φ* are bounded on [1, ∞). Therefore the spaces *H*^{q+n/2,φ}(Γ, *V*) and *H*^{q+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
$$\begin{array}{}{\phi}_{0}(t):={\displaystyle \phi (t)(\underset{t}{\overset{\mathrm{\infty}}{\int}}\frac{dt}{t\phi (t)}{)}^{1/2}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}for arbitrary\hspace{0.17em}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge 1.}\end{array}$$(49)

Owing to [7, Lemma 1.4], the function *φ*_{0} belongs to 𝓜 and has the following two properties:
$$\begin{array}{}{\displaystyle \underset{t\to \mathrm{\infty}}{lim}\frac{{\phi}_{0}(t)}{\phi (t)}\to 0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{1}{\overset{\mathrm{\infty}}{\int}}\frac{dt}{t{\phi}_{0}(t)}<\mathrm{\infty}.}\end{array}$$

It follows from the first property that we have the compact embedding
$$\begin{array}{}{H}^{q+n/2,\phi}(\mathrm{\Gamma},V)\hookrightarrow {H}^{q+n/2,{\phi}_{0}}(\mathrm{\Gamma},V).\end{array}$$

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
$$\begin{array}{}{H}^{q+n/2,{\phi}_{0}}(\mathrm{\Gamma},V)\hookrightarrow {C}^{q}(\mathrm{\Gamma},V),\end{array}$$
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 Γ_{1}⊄(Γ_{2}∪⋯∪Γ_{ϰ}). Therefore, there exists a nonempty open set *U* ⊂Γ_{1} such that *χ*_{1}(*x*) = 1 for every *x* ∈ *U*. We arbitrarily choose a function *w* ∈ *H*^{q+n/2,φ}(ℝ^{n}) such that supp *w* ⊂
$\begin{array}{}{\alpha}_{1}^{-1}\end{array}$
(*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
$$\begin{array}{}K(w,{\displaystyle \underset{p\varkappa -1}{\underset{\u23df}{0,\dots ,0}})\in {H}^{q+n/2,\phi}(\mathrm{\Gamma},V)\subset {C}^{q}(\mathrm{\Gamma},V).}\end{array}$$(50)
Let us deduce from this inclusion that w∈ *C*^{q}(ℝ^{n}).

To this end we introduce the linear mapping
$$\begin{array}{}{T}_{1}:u\mapsto {\mathrm{\Pi}}_{1}({\beta}_{1}\circ ({\chi}_{1}u)\circ {\alpha}_{1}),\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}u\in {C}^{q}(\mathrm{\Gamma},V).\end{array}$$

It acts continuously from *C*^{q}(Γ, *V*) to
$\begin{array}{}{C}_{\text{b}}^{q}({\mathbb{R}}^{n}).\end{array}$
Besides, the operator *K* acts continuously from (*H*^{q+n/2,φ}(ℝ^{n}))^{px} to *C*^{q}(Γ, *V*) according to (42) with s = q+n/2 and our assumption. Hence, T_{1}K(w, 0, \ldots, 0)=w; this equality is evident if additionally w∈
$\begin{array}{}{C}_{0}^{\mathrm{\infty}}\end{array}$
(ℝ^{n}) and then extends by closure over each function *w* chosen above. Now
$$\begin{array}{}w={T}_{1}K(w,0,\dots ,0)\in {C}^{q}({\mathbb{R}}^{n})\end{array}$$

Thus, we obtain embedding (46) with *G* :=
$\begin{array}{}{\alpha}_{1}^{-1}\end{array}$
(*U*). It implies condition (13) due to Proposition 7.1. □

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