To deduce Theorems 4.1 and 4.2 from their known counterparts in the Sobolev case, we need to prove a version of Proposition 5.5 (with *λ =* 0) for the target spaces of isomorphisms (24) and (31). This proof will be based on the following lemma about properties of the operator that assigns the Cauchy data to an arbitrary function *g* ∈ *H*^{s,s/2;φ}(*S*).

#### Proof

We first prove an analog of this lemma for Hörmander spaces defined on ℝ^{n} and ℝ^{n-}^{1} instead of *S* and Γ. Then we deduce the lemma with the help of the special local charts on *S*. Consider the linear mapping
$${R}_{0}:w\mapsto (w{|}_{t=0},{\mathrm{\partial}}_{t}w{|}_{t=0,\cdots ,}{\mathrm{\partial}}_{t}^{r-1}w{|}_{t=0}),\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}w\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n}).$$(38)

Here, we interpret *w* as a function *w*(*x,t*)of *x*∈ ℝ^{n}^{-1}and*t*∈ ℝ so that${R}_{0}w\in ({C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n-1}){)}^{r}.$Choose s >*2r —* 1 and *φ*∈ 𝓜 arbitrarily, and prove that the mapping (38) extends uniquely (by continuity) to a bounded linear operator
$${R}_{0}:{H}^{s,s/2;\phi}({\mathbb{R}}^{n})\to \underset{k=0}{\overset{r-1}{\u2a01}}{H}^{s-2k-1;\phi}({\mathbb{R}}^{n-1})=:{\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1})$$(39)

This fact is known in the Sobolev case of *φ* ≡ 1 due to [38, Chapter II, Theorem 7]. Using the interpolation with a function parameter between Sobolev spaces, we can deduce this fact in the general situation of arbitrary *φ* ∈ 𝓜. Namely, choose*s*_{0}*, s*_{1}∈ ℝ such that *2r —* 1*<s*_{0}*<s < s*_{1}and consider the bounded linear operators
$${R}_{0}:{H}^{{s}_{j},{s}_{j}/2}({\mathbb{R}}^{n})\to {\mathbb{H}}^{{s}_{j}}({\mathbb{R}}^{n-1}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\in \{0,1\}.$$(40)

Let *ψ* be the interpolation parameter (32). Then the restriction of the mapping (40) with *j* = 0 to the space
$$[{H}^{{s}_{0},{s}_{0}/2}({\mathbb{R}}^{n}),{H}^{{s}_{1},{s}_{1}/2}({\mathbb{R}}^{n}){]}_{\psi}={H}^{s,s/2;\phi}({\mathbb{R}}^{n})$$(41)

is a bounded operator
$${R}_{0}:{H}^{s,s/2;\phi}({\mathbb{R}}^{n})\to [{\mathbb{H}}^{{s}_{0}}({\mathbb{R}}^{n-1}),{\mathbb{H}}^{{s}_{1}}({\mathbb{R}}^{n-1}){]}_{\psi}.$$(42)

The latter equality is due to Proposition 5.5. This operator is an extension by continuity of the mapping (38) because the set ${C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n})$is dense in *H*^{s,s/2;φ}(ℝ^{n}). Owing to Propositions 5.2 and 5.4, we get
$$\begin{array}{ll}{[{\mathbb{H}}^{{s}_{0}}({\mathbb{R}}^{n-1}),{\mathbb{H}}^{{s}_{1}}({\mathbb{R}}^{n-1})]}_{\psi}& =\underset{k=0}{\overset{r-1}{\u2a01}}{[{H}^{{s}_{0}-2k-1}({\mathbb{R}}^{n-1}),{H}^{{s}_{1}-2k-1}({\mathbb{R}}^{n-1})]}_{\psi}\\ & =\underset{k=0}{\overset{r-1}{\u2a01}}{H}^{s-2k-1;\phi}({\mathbb{R}}^{n-1})={\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1})\phantom{\rule{thickmathspace}{0ex}}.\end{array}$$(43)

Hence, the linear bounded operator (42) is the required operator (39). Let us now build a linear mapping
$${T}_{0}:({L}_{2}({\mathbb{R}}^{n-1}){)}^{r}\to {L}_{2}({\mathbb{R}}^{n})$$(44)

that its restriction to each space ℍ^{s;φ}(ℝ^{n}^{-1} ) with *s > 2r* — 1 and *φ* ∈ 𝓜 is a bounded operator between the spaces ℍ^{s;φ}(ℝ^{n}^{-1} )and *H*^{s;s/}^{2;φ}ℝ^{n})and that this operator is right inverse to (39).

Similarly to Hörmander [10, Proof of Theorem 2.5.7] we define the linear mapping
$${T}_{0}:v{\displaystyle \mapsto {F}_{\xi \mapsto x}^{-1}[\beta (\u3008{\xi}^{2}t\u3009)\sum _{k=0}^{r-1}\frac{1}{k!}\hat{{v}_{k}}(\xi )\times {t}^{k}](x,t)}$$(45)

on the linear topological space of vectors
$$v:=({v}_{0,\cdots ,}{v}_{r-1})\in ({\mathcal{S}}^{\prime}({\mathbb{R}}^{n-1}){)}^{r}.$$

We consider *T*_{0}υ as a distribution on the Euclidean space ℝ^{n} of points (*x, t*), with *x=* {*x*_{1}*,..., x*_{n}_{-1})∈ ℝ^{n}^{-1} and *t* ∈ ℝ. In (45), the function $\beta \in {C}_{0}^{\mathrm{\infty}}(\mathbb{R})$ is chosen so that *β =* 1 in a certain neighbourhood of zero. As usual, ${F}_{\xi \mapsto x}^{-1}$denotes the inverse Fourier transform with respect to ξ* =* (ξ_{1},... ,ξ_{n}_{-1}) ∈ ℝ^{n}^{-1},and (ξ) := (1 + |ξ|^{2})^{1//2}. The variable *ξ* is dual to *x*relative to the direct Fourier transform $\hat{w}(\xi )=(Fw)(\xi )$ of a function *w(x*).

Obviously, the mapping (45) is well defined and acts continuously between (*S*′(ℝ^{n}^{-1})^{r} and *S*′(ℝ^{n}). It is also evident that the restriction of this mapping to the space (*L*_{2}(ℝ^{n}^{-1}))^{r} is a bounded operator between (*L*_{2}(ℝ^{n}^{-1}))^{r} and (*L*_{2}(ℝ^{n}^{-1}))^{r}.

We assert that
$${R}_{0}{T}_{0}v=v\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for every}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v\in (S({\mathbb{R}}^{n-1}){)}^{r}.$$(46)

Here, as usual, *S*(ℝ^{n}^{-1})denotes the linear topological space of all rapidly decreasing infinitely smooth functions on ℝ^{n}^{-1}. Since *v*∈ *S*(ℝ^{n}^{-1}) implies *T*_{0}v ∈ *S*(ℝ^{n}^{-1}), the left-hand side of the equality (46) is well defined. Let us prove this equality.

Choosing *j* ∈ {0,..., *r —* 1} and *v*= (*v*_{0}*, ..., v*_{r}_{-1}) ∈ *S*(ℝ^{n}^{-1})^{r}arbitrarily, we get
$$\begin{array}{ll}F[{\mathrm{\partial}}_{t}^{j}{T}_{0}v{|}_{t=0}](\xi )& ={\mathrm{\partial}}_{t}^{j}{F}_{x\mapsto \xi}[{T}_{0}v](\xi ,t){|}_{t=0}={\mathrm{\partial}}_{t}^{j}(\beta ({\u3008\xi \u3009}^{2}t)\sum _{k=0}^{r-1}\frac{1}{k!}\hat{{v}_{k}}(\xi ){t}^{k}){|}_{t=0}\\ & =\beta (0)({\mathrm{\partial}}_{t}^{j}\sum _{k=0}^{r-1}\frac{1}{k!}\hat{{v}_{k}}(\xi ){t}^{k}){|}_{t=0}=\beta (0)j!\frac{\text{1}}{j!}\hat{{\text{v}}_{\text{j}}}(\xi )=\hat{{v}_{j}}(\xi )\end{array}$$

for every *ξ* ∈ ℝ^{n}^{-1}. In the fourth equality, we have used the fact that *β =* 1 in a neighbourhood of zero. Thus, the Fourier transforms of all components of the vectors *R*_{0}* T*_{0}*υ* and υ coincide, which is equivalent to (46). Let us now prove that the restriction of the mapping (45) to each space
$${\mathbb{H}}^{2m}({\mathbb{R}}^{n-1})=\underset{k=0}{\overset{r-1}{\u2a01}}{H}^{2m-2k-1}({\mathbb{R}}^{n-1})$$(47)

with 0 ≤ *m* ∈ ℤis a bounded operator between ℍ^{2m}(ℝ^{n}^{-1}) and *H*^{2m, m}(ℝ^{n}). Note that the integers *2m-2k-1* may be negative in (47).

Let an integer *m* ≥ 0. We make use of the fact that the norm in the space *H*^{2m,m}(ℝ^{n})is equivalent to the norm
$$\parallel w{\parallel}_{2m,m}:={\left(\parallel w{\parallel}^{2}+\sum _{j=1}^{n-1}\parallel {\mathrm{\partial}}_{{x}_{j}}^{2m}w{\parallel}^{2}+\parallel {\mathrm{\partial}}_{t}^{m}w{\parallel}^{2}\right)}^{1/2}$$

(see, e.g., [44, Section 9.1]). Here and below in this proof, || · || stands for the norm in the Hilbert space *L*_{2}(ℝ^{n}). Of course, ${\mathrm{\partial}}_{{x}_{j}}u$and ∂_{t}denote the operators of generalized partial derivatives with respect to *x*_{j}and *t* respectively. Choosing *v*= (*v*_{0}*, ..., v*_{r}_{-1})∈(*S*(ℝ^{n}^{-1}))^{r}arbitrarily and using the Parseval equality, we obtain the following:
$$\begin{array}{ll}{T}_{0}{v}_{2m,m}^{2}& ={T}_{0}v{}^{2}+\sum _{j=1}^{n-1}{\mathrm{\partial}}_{{x}_{\dot{j}}}^{2m}{T}_{0}v{}^{2}+{\mathrm{\partial}}_{t}^{m}{T}_{0}v{}^{2}\\ & =\hat{{T}_{0}v}{}^{2}+\sum _{j=1}^{n-1}{\xi}_{j}^{2m}\hat{{T}_{0}v}{}^{2}+{\mathrm{\partial}}_{t}^{m}\hat{{T}_{0}v}{}^{2}\\ & \u2a7d\sum _{k=0}^{r-1}\frac{1}{k!}\underset{{\mathbb{R}}^{n}}{\int}|\beta ({\u3008\xi \u3009}^{2}t)\hat{{v}_{k}}(\xi ){t}^{k}{|}^{2}d\xi dt\\ & +\sum _{j=1}^{n-1}\sum _{k=0}^{r-1}\frac{1}{k!}\underset{{\mathbb{R}}^{n}}{\int}|{\xi}_{j}^{2m}\beta ({\u3008\xi \u3009}^{2}t)\hat{{v}_{k}}(\xi ){t}^{k}{|}^{2}d\xi dt\\ & +\sum _{k=0}^{r-1}\frac{1}{k!}{\int}_{{\text{R}}^{n}}|{\mathrm{\partial}}_{t}^{m}(\beta ({\u3008\xi \u3009}^{2}t){t}^{k})\hat{{v}_{k}}(\xi ){|}^{2}d\xi dt.\end{array}$$

Let us estimate each of these three integrals separately. We begin with the third integral. Changing the variable τ = 〈ξ〉^{2}*t*in the interior integral with respect to *t*, we get the equalities
$$\begin{array}{ll}\underset{{\mathbb{R}}^{n}}{\int}|{\mathrm{\partial}}_{t}^{m}(\beta ({\{\xi \}}^{2}t){t}^{k})\hat{{v}_{k}}(\xi ){|}^{2}d\xi dt& =\underset{{\mathbb{R}}^{n-1}}{\int}|\hat{{v}_{k}}(\xi ){|}^{2}d\xi \underset{\mathbb{R}}{\int}|{\mathrm{\partial}}_{t}^{m}(\beta ({\u3008\xi \u3009}^{2}t){t}^{k}){|}^{2}dt\\ & =\underset{{\mathbb{R}}^{n-1}}{\int}{\u3008\xi \u3009}^{4m-4k-2}|\hat{{v}_{k}}(\xi ){|}^{2}d\xi \underset{\mathbb{R}}{\int}|{\mathrm{\partial}}_{\tau}^{m}(\beta (\tau ){\tau}^{k}){|}^{2}d\tau .\end{array}$$

Hence,
$$\underset{{\mathbb{R}}^{n}}{\int}|{\mathrm{\partial}}_{t}^{m}(\beta (\u3008\xi {\u3009}^{2}t){t}^{k})\hat{{v}_{k}}(\xi ){|}^{2}d\xi dt={c}_{1}\parallel {v}_{k}{\parallel}_{{H}^{2m-2k-1}{\mathbb{R}}^{n-1})}^{2},$$

with
$${c}_{1}:=\underset{\mathbb{R}}{\int}|{\mathrm{\partial}}_{\tau}^{m}(\beta (\tau ){\tau}^{k}){|}^{2}d\tau <\mathrm{\infty}.$$

Using the same changing of the variable *t* in the second integral, we obtain the following:
$$\begin{array}{ll}\underset{{\mathbb{R}}^{n}}{\int}|{\xi}_{j}^{2m}\beta ({\u3008\xi \u3009}^{2}t)\hat{{v}_{k}}(\xi ){t}^{k}{|}^{2}d\xi dt& =\underset{{\mathbb{R}}^{n-1}}{\int}|{\xi}_{j}{|}^{4m}|\hat{{v}_{k}}(\xi ){|}^{2}d\xi \underset{\mathbb{R}}{\int}|{t}^{k}\beta ({\{\xi \}}^{2}t){|}^{2}dt\\ & =\underset{{\mathbb{R}}^{n-1}}{\int}|{\xi}_{j}{|}^{4m}{\u3008\xi \u3009}^{-4k-2}|\hat{{v}_{k}}(\xi ){|}^{2}d\xi \underset{\mathbb{R}}{\int}|{\tau}^{k}\beta (\tau ){|}^{2}d\tau \\ & \u2a7d\underset{{\mathbb{R}}^{n-1}}{\int}{\u3008\xi \u3009}^{4m-4k-2}|\hat{{v}_{k}}(\xi ){|}^{2}d\xi \underset{\mathbb{R}}{\int}|{\tau}^{k}\beta (\tau ){|}^{2}d\tau ,\end{array}$$

Hence,
$$\underset{{\mathbb{R}}^{n}}{\int}|{\xi}_{j}^{2m}\beta (\u3008\xi {\u3009}^{2}t)\hat{{v}_{k}}(\xi ){t}^{k}{|}^{2}d\xi dt\le {c}_{2}\parallel {v}_{k}{\parallel}_{{H}^{2m-2k-1}({\mathbb{R}}^{n-1})}^{2}.$$

with
$${c}_{2}:=\underset{\mathbb{R}}{\int}|{\tau}^{k}\beta (\tau ){|}^{2}d\tau <\mathrm{\infty}.$$

Finally, replacing the symbol ξ_{j}with 1 in the previous reasoning, we obtain the following estimate for the first integral:
$$\underset{{\mathbb{R}}^{n}}{\int}|\beta (\u3008\xi {\u3009}^{2}t)\hat{{v}_{k}}(\xi ){t}^{k}{|}^{2}d\xi dt\le {c}_{2}\parallel {v}_{k}{\parallel}_{{H}^{-2k-1}({\mathbb{R}}^{n-1})}^{2}\le {c}_{2}\parallel {v}_{k}{\parallel}_{{H}^{2m-2k-1}({\mathbb{R}}^{n-1})}^{2}.$$

Thus, we conclude that
$$\parallel {T}_{0}v{\parallel}_{{H}^{2m,m}({\mathbb{R}}^{n})}^{2}\le c\sum _{k=0}^{r-1}\parallel {v}_{k}{\parallel}_{{H}^{2m-2k-1}({\mathbb{R}}^{n-1})}^{2}=c\parallel v{\parallel}_{{\mathrm{H}}^{2m}({\mathbb{R}}^{n-1})}^{2}$$

for any *v*∈ (*S*(ℝ^{n}^{-1}))^{r}, with the number *c >*0 being independent of υ. Since the set (*S*(ℝ^{n}^{-1}))^{r} is dense in ℍ^{2m} (ℝ^{n}^{-1}), it follows from the latter estimate that the mapping (45) sets a bounded linear operator
$${T}_{0}:{\mathbb{H}}^{2m}({\mathbb{R}}^{n-1})\to {H}^{2m,m}({\mathbb{R}}^{n})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{whenever}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}0\le m\in \mathbb{Z}.$$

Let us deduce from this fact that the mapping (45) acts continuously between the spaces ℍ^{s}^{;φ}(ℝ^{n}^{-1}) and *H*^{s,s/}^{2;φ}(ℝ^{n}^{-1}) *s* > 2r — 1 and *φ* ∈ 𝓜 . Put *s*_{0}= 0, choose an even integer *s*_{1}*> s*, and consider the linear bounded operators
$${T}_{0}:{\mathbb{H}}^{{s}_{j}}({\mathbb{R}}^{n-1})\to {H}^{{s}_{j},{s}_{j}/2}({\mathbb{R}}^{n}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\in \{0,1\}.$$(48)

Let, as above, *ψ* be the interpolation parameter (32). Then the restriction of the mapping (48) with *j* = 0 to the space
$$[{\mathbb{H}}^{{s}_{0}}({\mathbb{R}}^{n-1}),{\mathbb{H}}^{{s}_{1}}({\mathbb{R}}^{n-1}){]}_{\psi}={\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1})$$

is a bounded operator
$${T}_{0}:{\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1})\to {H}^{s,s/2;\phi}({\mathbb{R}}^{n}).$$(49)

Here, we have used formulas (41) and (43), which remain true for the considered *s*_{0}and *s*_{1}.

Now the equality (46) extends by continuity over all vectors *v*∈ℍ^{s}^{;φ}(ℝ^{n}^{-1}) . Hence, the operator (49) is right inverse to (39). Thus, the required mapping (44) is built.

We need to introduce analogs of the operators (39) and (49) for the strip
$$\mathrm{\Pi}=\{(x,t):x\in {\mathbb{R}}^{n-1},0<t<\tau \}.$$

Let *s > 2r* — 1 and *φ* ∈ 𝓜. Given *u* ∈*H*^{s,s/}^{2;φ}(Π), we put *R*_{1}*u := R*_{0}*w*, where a function *w* ∈*H*^{s,s/}^{2;φ}(ℝ^{n}) satisfies the condition *w* ↾Π = *u*. Evidently, this definition does not depend on the choice of *w*. The linear mapping *u* →* R*_{1}*u* is a bounded operator
$${R}_{1}:{H}^{s,s/2;\phi}(\mathrm{\Pi})\to {\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1}).$$(50)

This follows immediately from the boundedness of the operator (39) and from the definition of the norm in *H*^{s,s/}^{2;φ}(Π)

Let us introduce a right-inverse of (50) on the base of the mapping (45). We put *T*_{1}* v :=* (*Τ*_{0}*v*) ↾Π for arbitrary *v*∈(*L*_{2}(ℝ^{n}^{-1}))^{r}. The restriction of the linear mapping *v*↦* T*_{1}*v* over vectors *v*∈ ℍ^{s}^{;φ}(ℝ^{n}^{-1})is a bounded operator
$${T}_{1}:{\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1})\to {H}^{s,s/2;\phi}(\mathrm{\Pi}).$$(51)

This follows directly from the boundedness of the operator (49). Observe that
$$R{}_{1}{T}_{1}v={R}_{1}(({T}_{0}v)\upharpoonright \mathrm{\Pi})={R}_{0}{T}_{0}v=v\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for every}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v\in {\mathbb{H}}^{s;\phi}({\mathbb{R}}^{n-1}).$$

Thus, the operator (51) is right inverse to (50).

Using operators (50) and (51), we can now prove our lemma with the help of the special local charts (10) on *S*. As above, let*s*>*2r —* 1 and *φ* ∈ 𝓜. Choosing *k* ∈{0,..., *r—* 1} and *g*∈ C^{∞}(*S*̅) arbitrarily, we get the following:
$$\begin{array}{rl}\parallel {\mathrm{\partial}}_{t}^{k}g\upharpoonright \mathrm{\Gamma}{\parallel}_{{H}^{s-2k-1;\phi}(\mathrm{\Gamma})}^{2}& =\sum _{j=1}^{\lambda}\parallel ({\chi}_{j}({\mathrm{\partial}}_{t}^{k}g\upharpoonright \mathrm{\Gamma}))\circ {\theta}_{j}{\parallel}_{{H}^{s-2k-1;\phi}({\mathbb{R}}^{n-1})}^{2}\\ & =\sum _{j=1}^{\lambda}\parallel {\mathrm{\partial}}_{t}^{k}(({\chi}_{j}g)\circ {\theta}_{j}^{\ast})({\mathbb{R}}^{n-1}){\parallel}_{{H}^{s-2k-1;\phi}({\mathbb{R}}^{n-1})}^{2}\\ & \le {c}^{2}\sum _{j=1}^{\lambda}\parallel (xjg)\circ {\theta}_{j}^{\ast}{\parallel}_{{H}^{s,s/2;\phi}\cdot \cdots )}^{2}={c}^{2}\parallel g{\parallel}_{{H}^{s,s/2;\phi}\cdot S)}^{2}.\end{array}$$

Here, *c* denotes the norm of the bounded operator (50), and, as usual, symbol "o" designates a composition of functions. Recall that {θ_{j}·} is a collection of local charts on Γ and that {*x*_{j}} is an infinitely smooth partition of unity on Γ. Thus,
$$\parallel Rg{\parallel}_{{\mathrm{H}}^{s;\phi}}\le c\sqrt{r}\parallel g{\parallel}_{{H}^{s,s/2;\phi )}(S)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{for every}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}g\in {C}^{\mathrm{\infty}}(\overline{S}).$$

This implies that the mapping (35) extends by continuity to the bounded linear operator (36).

Let us build the linear mapping *T*: (*L*_{2}(Γ))^{r}→* L*_{2}(*S*)whose restriction to ℍ^{s}^{;φ}(Γ) is a right-inverse of (36). Consider the linear mapping of flattening of Γ
$$L:v\mapsto (({\chi}_{1}v)\circ {\theta}_{1,\cdots ,}({\chi}_{\lambda}v)\circ {\theta}_{\lambda}),\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v\in {L}_{2}(\mathrm{\Gamma}).$$

Its restriction to *Η*^{σ;φ}(Γ)is an isometric operator
$$L:{H}^{\sigma ;\phi}(\mathrm{\Gamma})\to ({H}^{\sigma ;\phi}({\mathbb{R}}^{n-1}){)}^{\lambda}\phantom{\rule{2em}{0ex}}\text{whenever}\phantom{\rule{1em}{0ex}}\sigma >0.$$(52)

Besides, consider the linear mapping of sewing of Γ
$$K:({h}_{1,\cdots ,}{h}_{\lambda}){\displaystyle \mapsto \sum _{j=1}^{\lambda}{O}_{j}(({\eta}_{j}{h}_{j})\circ {\theta}_{j}^{-1}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{h}_{1,\cdots ,}{h}_{\lambda}\in {L}_{2}({\mathbb{R}}^{n-1}).}$$

Here, each function ${\eta}_{j}\in {C}_{0}^{\mathrm{\infty}}({\mathbb{R}}^{n-1})$is chosen so that η_{j}= 1 on the set ${\theta}_{j}^{-1}$ (*supp X*_{j}) whereas *O*_{j} denotes the operator of the extension by zero to Γ of a function given on Γ_{j}·. The restriction of this mapping to (*Η*^{σ;φ}(ℝ^{n}^{-1}))^{λ} is a bounded operator
$$({H}^{\sigma ;\phi}({\mathbb{R}}^{n-1}){)}^{\lambda}\to {H}^{\sigma ;\phi}({\mathrm{r}}^{\tau})\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{whenever}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\sigma >0,$$

and this operator is left inverse to (52) (see [18, the proof of Theorem 2.2]).

The mapping *Κ* induces the operator *K*_{1}of the sewing of the manifold *S =* Γ x(0, τ) by the formula
$$({K}_{1}({g}_{1},\dots ,{g}_{\lambda}))(x,t):=(K({g}_{1}(\cdot ,t),\dots ,{g}_{\lambda}(,t)))(x)$$

for arbitrary functions *g*_{1}*, . . . ,g*_{λ}∈*L*_{2}(Π) and almost all *x*∈Γ and *t* ∈(0, τ). The restriction of the mapping *K*_{1} to (*Η*^{σ,σ/2;φ}Π)^{λ}is a bounded operator
$${K}_{1}:({H}^{\sigma ,\sigma /2;\phi}(\mathrm{\Pi}){)}^{\lambda}\to {H}^{\sigma ,\sigma /2;\phi}(S)\phantom{\rule{1em}{0ex}}\text{whenever}\phantom{\rule{1em}{0ex}}\sigma >0$$(53)

(see [36, the proof of Theorem 2]).

Given *v*:= (*v*_{0}*, v*_{1}*, ..., v*_{r}_{-1})*∈*(*L*_{2}(Γ))^{r}, we set
$$Tv:={K}_{1}({T}_{1}({v}_{0,1,\cdots ,}{v}_{r-1,1}{)}_{,\cdots ,}{T}_{1}({v}_{0,\lambda ,\cdots ,}{v}_{r-1,\lambda})),$$

where
$$({v}_{k,1,\cdots ,}{v}_{k,\lambda}):=L{v}_{k}\in ({L}_{2}({\mathbb{R}}^{n-1}){)}^{\lambda}$$

for each integer *k* ∈ {0,..., *r —* 1}. The linear mapping *v*↦*Tv* acts continuously between (*L*_{2}(Γ))^{r}and *L*_{2}(*S*), which follows directly from the definitions of *L*, *T*_{1}, and *K*_{1}. The restriction of this mapping to *Η*^{s}^{;φ}(Γ)is the bounded operator (37). This follows immediately from the boundedness of the operators (51), (52), and (53). The operator (37) is right inverse to (36). Indeed, choosing a vector *v*= (*v*_{0}, *v*_{1},..., *v*_{r}_{-1}) ∈*Η*^{s}^{;φ}(Γ)arbitrarily, we obtain the following equalities:
$$\begin{array}{ll}{(RTv)}_{k}& ={(R{K}_{1}({T}_{1}{({v}_{0,1,\cdots ,}{v}_{r-1,1})}_{,\cdots ,}{T}_{1}({v}_{0,\lambda ,\cdots ,}{v}_{r-1,\lambda})))}_{k}\\ & =K({({R}_{1}{T}_{1}({v}_{0,1,}\dots ,{v}_{r-1,1}))}_{k},\dots ,\phantom{\rule{thickmathspace}{0ex}}{({R}_{1}{T}_{1}({v}_{0,\lambda ,}\dots ,{v}_{r-1,\lambda}))}_{k})\\ & =K({v}_{k,1},\dots ,{v}_{k,\lambda})=KL{v}_{k}={v}_{k}.\end{array}$$

Here, the index *k* runs over the set {0,..., r — 1} and denotes the *k*-th component of a vector. Hence, *RTv = v*.

Using this lemma, we will now prove a version of Proposition 5.5 for the target spaces of isomorphisms (24) and (31). Note that the number of the compatibility conditions (19) and (28) are constant respectively on the intervals
$${J}_{0,1}:=(2,7/2),{J}_{0,r}:=(2r-1/2,2r+3/2),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}2\le r\in \mathbb{Z},$$

and
$${J}_{1,0}:=(2,5/2),{J}_{1,r}:=(2r+1/2,2r+5/2),\text{with}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le r\in \mathbb{Z},$$

of the varying of *s*. Namely, if *s* ranges over some *J*_{l},_{ r},then this number equals *r*.

#### Proof

Recall that ${\mathcal{Q}}_{l}^{s-2,s/2-1;\phi}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{Q}}_{l}^{{s}_{j}-2,{s}_{j}/2-1},$ with *j* ∈ {0,1} are subspaces of the Hilbert spaces ${\mathcal{H}}_{l}^{s-2,s/2-1;\phi}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{H}}_{l}^{{s}_{j}-2,{s}_{j}/2-1}$respectively. According to Propositions 5.2, 5.4, and 5.5 we obtain the following:
$$\begin{array}{l}[{\mathcal{H}}_{l}^{{s}_{0}-2,{s}_{0}/2-1,}{\mathcal{H}}_{l}^{{s}_{1}-2,{s}_{1}/2-1}{]}_{\psi}\\ \phantom{\rule{thickmathspace}{0ex}}=[{H}^{{s}_{0}-2,{s}_{0}/2-1}(\mathrm{\Omega})\oplus {H}^{{s}_{0}-\cdot 2l+1)/2,{s}_{0}/2-\cdot 2l+1)/4}(S)\oplus {H}^{{s}_{0}-1}(G),\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{H}^{{s}_{1}-2,{S}_{1/2-1}}(\mathrm{\Omega})\oplus {H}^{{s}_{1}-\cdot 2l+1)/2s};1/2-\cdot 2l+1)/4(S)\oplus {H}^{{s}_{1}-1}(G){]}_{\psi}\\ =[{H}^{{s}_{0}-2,{s}_{0}/2-1}(\mathrm{\Omega}),{H}^{{s}_{1}-2,{s}_{1}/2-1}(\mathrm{\Omega}){]}_{\psi}\\ \phantom{\rule{2em}{0ex}}\oplus [{H}^{{s}_{0}-\cdot 2l+1)/2,{s}_{0}/2-\cdot 2l+1)/4}(S),{H}^{{s}_{1}-\cdot 2l+1)/2,{s}_{1}/2-\cdot 2l+1)/4}(S){]}_{\psi}\\ \phantom{\rule{2em}{0ex}}\oplus [{H}^{{s}_{0}-1}(G),{H}^{{s}_{1}-1}(G){]}_{\psi}\\ \phantom{\rule{thickmathspace}{0ex}}={H}^{s-2,s/2-1;\phi}(\mathrm{\Omega})\oplus {H}^{s-\cdot 2l+1)/2,s/2-\cdot 2l+1)/4;\phi}(S)\oplus {H}^{s-1;\phi}(G)={\mathcal{H}}_{l}^{s-2,s/2-1;\phi}.\end{array}$$

Thus,
$$[{\mathcal{H}}_{l}^{{s}_{0}-2,{s}_{0}/2-1,}{\mathcal{H}}_{l}^{{s}_{1}-2,{s}_{1}/2-1}{]}_{\psi}={\mathcal{H}}_{l}^{s-2,s/2-1;\phi}$$(55)

up to equivalence of norms.

We will deduce the required formula (54) from (55) with the help of Proposition 5.1. To this end, we need to present a linear mapping *Ρ* on ${\mathcal{H}}_{l}^{{s}_{0}-2,{s}_{0}/2-1}$suchthat *Ρ* is a projector of the space ${\mathcal{H}}_{l}^{{s}_{j}-2,{s}_{j}/2-1}$ onto its subspace ${\mathcal{Q}}_{l}^{{s}_{j}-2,{s}_{j}/2-1}$ for each *j* ∈{0, 1}. If we have this mapping, we will get
$$\begin{array}{ll}{[{\mathcal{Q}}_{l}^{{s}_{0}-2,{s}_{0}/2-1,}{\mathcal{Q}}_{l}^{{s}_{1}-2,{s}_{1}/2-1}]}_{\psi}& ={[{\mathcal{H}}_{l}^{{s}_{0}-2,{s}_{0}/2-1,}{\mathcal{H}}_{l}^{{s}_{1}-2,{s}_{1}/2-1}]}_{\psi}\cap {\mathcal{Q}}_{l}^{{s}_{0}-2,{s}_{0}/2-1}\\ & ={\mathcal{H}}_{l}^{s-2,s/2-1;\phi}\cap {\mathcal{Q}}_{l}^{{s}_{0}-2,{s}_{0}/2-1}\\ & ={\mathcal{Q}}_{l}^{s-2,s/2-1;\phi}\end{array}$$

due to Proposition 5.1, formula (55), and the conditions *s*_{0}*,s* ∈*J*_{ι}, _{r}and *s*_{0}*< s*. Note that these conditions imply the last equality because the elements of the spaces ${\mathcal{Q}}_{l}^{{s}_{0}-2,{s}_{0}/2-1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathcal{Q}}_{l}^{s-2,s/2-1;\phi}$ satisfy the same compatibility conditions and because ${\mathcal{H}}_{l}^{s-2,s/2-1;\phi}$ is embedded continuously in ${\mathcal{H}}_{l}^{{s}_{0}-2,{s}_{0}/2-1}.$

We will build the above-mentioned mapping *Ρ* with the help of Lemma 6.1. Consider first the case of / = 0. Given $(f,g,h)\in {\mathcal{H}}_{0}^{{s}_{0}-2,{s}_{0}/2,-1}$ we put
$${g}^{\ast}:=g+T({v}_{0}\upharpoonright \mathrm{\Gamma}-g\upharpoonright \mathrm{\Gamma},\dots ,{v}_{r-1}\upharpoonright \mathrm{\Gamma}-{\mathrm{\partial}}_{t}^{r-1}g\upharpoonright \mathrm{\Gamma}).$$

Here, the functions ${v}_{k}\in {H}^{{s}_{0}-1-2k}(G),$ with *k =* 0, . . ., r — 1, are defined by the recurrent formula (20), and the mapping *Τ* is taken from Lemma 6.1. The linear mapping *Ρ* : (*f, g, h*)↦(*f,g*,h*)defined on all vectors $(f,g,h)\in {\mathcal{H}}_{0}^{{s}_{0}-2,{s}_{0}/2,-1}$ is required. Indeed, its restriction to each space ${\mathcal{H}}_{0}^{{s}_{\dot{j}}-2,{s}_{j}/2-1},$ with *j* ∈{0, 1}, is a bounded operator on this space. This follows directly from Lemma 6.1 in which we take *s := S*_{j} — 1/2. Moreover, if $(f,g,h)\in {\mathcal{Q}}_{0}^{{s}_{j}-2,{j}_{0}/2,-1}$ then *P*(*f,g,h*)* =* (ƒ, *g, h*) due to the compatibility conditions (19).

Consider now the case of *l* = 1. Given $(f,g,h)\in {\mathcal{H}}_{1}^{{s}_{0}-2,{s}_{0}/2-1}$, we put
$${g}^{\ast}:=g+T({B}_{0}[{v}_{0}]\upharpoonright \mathrm{\Gamma}-g\upharpoonright \mathrm{\Gamma},\cdots ,{B}_{r-1}[{v}_{0},\cdots ,{v}_{r-1}]\upharpoonright \mathrm{\Gamma}-{\mathrm{\partial}}_{t}^{r-1}g\upharpoonright \mathrm{\Gamma}).$$

Here, the functions *v*_{0},..., *v*_{r-}_{1} and mapping *Τ* are the same as in the *l = 0* case. The linear mapping *Ρ* : (ƒ, *g, h*) ↦(*f, g*, h*) defined on all vectors $(f,g,h)\in {\mathcal{H}}_{0}^{{s}_{0}-2,{0}_{0}/2,-1}$is required. Indeed, its restriction to each space ${\mathcal{H}}_{0}^{{s}_{j}-2,{s}_{j}/1-2},$ with *j*∈{0, 1}, is a bounded operator on this space due to Lemma 6.1 in which *s* := *s*_{j}—3/2. Moreover, if $(f,g,h)\in {\mathcal{Q}}_{0}^{{s}_{j}-2,{s}_{j}/2-1},$ then *P*(*f, g, h*)* =* (ƒ, *g, h*)by the compatibility conditions (28).

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