An isomorphism theorem for parabolic problems in H\"ormander spaces and its applications

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic H\"ormander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate H\"ormander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

To formulate Condition 2.2, we arbitrarily choose a point x ∈ Γ, real number t ∈ [0, τ ], vector ξ ∈ R n tangent to the boundary Γ at x, and number p ∈ C with Re p ≥ 0 such that |ξ| + |p| = 0. Let ν(x) is the unit vector of the inward normal to Γ at x. It follows from Condition 2.1 and the inequality n ≥ 2 that the polynomial A • (x, t, ξ + ζν(x), p) in ζ ∈ C has m roots ζ + j (x, t, ξ, p), j = 1, . . . , m, with positive imaginary part and m roots with negative imaginary part provided that each root is taken the number of times equal to its multiplicity.
Note that Condition 2.1 is the condition for the partial differential equation Au = f to be 2b-parabolic in Ω in the sense of I. G. Petrovskii [40], whereas Condition 2.2 claims that the system of boundary partial differential expressions {B 1 , . . . , B m } covers A on S.
We associate the linear mapping In the paper, all functions and distributions are supposed to be complex-valued.
The main purpose of the paper is to prove that the mapping (2.4) extends uniquely (by continuity) to an isomorphism between appropriate Hörmander inner product spaces.

Hörmander spaces
Here, we will define the Hörmander inner product spaces being used in the paper. They are built on the base of the anisotropic function spaces H s,s/(2b);ϕ (R k+1 ) given over R k+1 , with k ≥ 1. These spaces are parametrized with the pair of the real numbers s and s/(2b) and with the function ϕ ∈ M.
Note that the theory of slowly varying functions is set forth, e.g., in monographs [6,42]. An important example of a function ϕ ∈ M is given by a continuous function ϕ : with k ∈ Z, k ≥ 1, and q 1 , q 2 , . . . , q k ∈ R.
By definition, the complex linear space H s,sγ;ϕ (R k+1 ) consists of all tempered distributions w on R k+1 whose (complete) Fourier transform w is locally Lebesgue integrable over R k+1 and meets the condition Here and below, we use the notation This space is equipped with the inner product The inner product naturally induces the norm . We note that H s,sγ;ϕ (R k+1 ) is a special case of the spaces B p,k introduced by L. Hörmander [13,Sect. 2.2]. Namely, H s,sγ;ϕ (R k+1 ) = B p,k provided that p = 2 and the function parameter k(ξ, η) = r s γ (ξ, η) ϕ(r γ (ξ, η)) for all ξ ∈ R k and η ∈ R. The spaces B p,k were systematically investigated by L. Hörmander [13,Sect. 2.2], [14,Sect. 10.1] and in the p = 2 case by L. R. Volevich and B. P. Paneah [45]. If γ = 1/(2b), then we say that H s,sγ;ϕ (R k+1 ) is a 2b-anisotropic Hörmander space.
In the ϕ(r) ≡ 1 case, the space H s,sγ;ϕ (R k+1 ) becomes an anisotropic Sobolev space and is denoted by H s,sγ (R k+1 ). Generally, we have the continuous and dense embeddings They result directly from the following property of ϕ ∈ M: for every ε > 0 there exists The embeddings (3.1) clarify the role of the function parameter ϕ ∈ M in the class of Hilbert function spaces H s,sγ;ϕ (R k+1 ) : s ∈ R, ϕ ∈ M .
(3.2) We see that ϕ defines a supplementary (subpower) regularity of distributions with respect to the basic (power) regularity given by the pair of numbers (s, sγ). Specifically, if ϕ(r) → ∞ [or ϕ(r) → 0] as r → ∞, then ϕ defines a positive [or negative] supplementary regularity. So, we can briefly say that ϕ refines the power anisotropic regularity (s, sγ).
Note that in the γ = 1 case the space H s,sγ;ϕ (R k+1 ) becomes the isotropic Hörmander space denoted by H s;ϕ (R k+1 ). The spaces H s;ϕ (R k+1 ), with s ∈ R and ϕ ∈ M, form the refined Sobolev scale introduced and investigated by V. A. Mikhailets and A. A. Murach [25,27]. This scale has various applications in the theory of elliptic partial differential equations [32,34].
Using the scale (3.2), we now introduce the function spaces relating to the problem (2.1)- 3) The norm in the linear space (3.3) is defined by the formula with u ∈ H s,sγ;ϕ and is a (closed) subspace of the Hilbert space H s,sγ;ϕ (R k+1 ). According to [45,Lemma 3.3], the set The norm (3.4) is induced by the inner product Here, w j ∈ H s,sγ;ϕ + (R k+1 ), w j = u j in V for every j ∈ {1, 2}, and Υ is the orthogonal projector of the space H s,sγ;ϕ + (R k+1 ) onto its subspace (3.5). In the Sobolev case of ϕ(r) ≡ 1, we will omit the index ϕ in the designations of H s,sγ;ϕ + (V ) and similar spaces. We note the continuous and dense embeddings They result from (3.1) and the density of the set {w ↾ V : w ∈ C ∞ 0 (R k × (0, ∞))} in the spaces appearing in (3.6).
As to the problem (2.1)-(2.3), we need the space H s,sγ;ϕ + (V ) in the case where k = n and V = Ω. Note that the set C ∞ + (Ω) is dense in H s,sγ;ϕ + (Ω). We also need to introduce an analog of the space H s,sγ;ϕ + (V ) for the lateral area S of the cylinder Ω. It is sufficient for our purposes to restrict ourselves to the s > 0 case. Let Π := R n−1 × (0, τ ), and consider the Hilbert space H s,sγ;ϕ + (V ) in the case where k = n − 1 and V = Π. We will define the space H s,sγ;ϕ + (S) on the base of the space H s,sγ;ϕ + (Π) with the help of special local charts on S.
We arbitrarily choose a finite atlas from the C ∞ -structure on the closed manifold Γ. Let this atlas be formed by the local charts θ j : R n−1 ↔ Γ j , with j = 1, . . . , λ.
This atlas of Γ induces the collection of the special local charts They form a C ∞ -partition of unity on S which is subordinate to the covering {Γ j × [0, τ ] : j = 1, . . . , λ} of S. Now, we put H s,sγ;ϕ Here, recall, s > 0, and, L 2 (S) is the Hilbert space of all square integrable functions v : S → C with respect to the Lebesgue measure on the smooth surface S. As usual, • is the sign of composition of functions or mappings, so that for all x ∈ R n−1 and t ∈ (0, τ ). The inner product in the linear space (3.7) is defined by the formula (ii) The set C ∞ + (S) is dense in this space. We will prove this lemma at the end of Section 7. Ending the present section, we note that statement (3.6) about continuous and dense embeddings also holds true in the case where V = S and s 0 > 0. This follows directly from the validity of this statement for the open set V = Π ⊂ R n and from Lemma 3.1(ii).

Main results
Here, we formulate an isomorphism theorem for the parabolic problem (2.1)-(2.3) in Hörmander spaces introduced above and then consider applications of this theorem to the investigation of the regularity of the generalized solutions to the problem.
Note, if m j ≤ 2m − 1 for each j ∈ {1, . . . , m}, then σ 0 = 2m. Isomorphism Theorem is formulated as follows. where H σ−2m,(σ−2m)/(2b);ϕ + (Ω, S) In the Sobolev case of ϕ(r) ≡ 1 and σ/(2b) ∈ Z, this theorem follows from the result by M. S. Agranovich and M. I. Vishik [1, Theorem 12.1], the limiting case of σ = σ 0 being included. This will be demonstrated in Section 5. In the general situation, we will deduce Theorem 4.1 from the Sobolev case with the help of the interpolation with a function parameter between Hilbert spaces. This will be done in Section 8 after we investigate the necessary interpolation properties of the Hörmander spaces appearing in (4.1) and (4.2).
As has just been mentioned, the mapping (2.4) extends by continuity to an isomorphism which acts between some anisotropic Sobolev spaces. All the isomorphisms (4.1), with σ > σ 0 and ϕ ∈ M, are restrictions of (4.3). This results from the embeddings (3.6) being valid for has a unique preimage u ∈ H σ 0 ,σ 0 /(2b) + (Ω) relative to the one-to-one mapping (4.3). The function u is said to be a (strong) generalized solution to the parabolic problem (2.1)-(2.3) with the right-hand sides (4.4).
Let us now discuss the regularity properties of this solution in Hörmander spaces. The next result follows immediately from Theorem 4.1.
(Ω) is a generalized solution to the problem (2.1)-(2.3) whose right-hand sides satisfy the condition We observe that the supplementary regularity ϕ of the right-hand sides is inherited by the solution.
In the special case where ω = Ω and π 1 = ∂Ω (then π 2 = S), Theorem 4.3 says about the global increasing in smoothness, so that we arrive at Corollary 4.2. If π 1 = ∅, then this theorem asserts that the regularity increases in neighbourhoods of interior points of the closed domain Ω.
Using Hörmander spaces, we can obtain fine sufficient conditions under which the generalized solution u and its generalized derivatives of a prescribed order are continuous on ω ∪ π 1 .
3) whose right-hand sides satisfy conditions (4.5), (4.6) for σ := p+b+n/2 and given ϕ ∈ M complies with the conclusion of Theorem 4.4, then ϕ meets condition (4.7). Remark 4.6. If we formulate an analog of Theorem 4.4 for the Sobolev case of ϕ ≡ 1, we have to change the condition of this theorem for a stronger one. Namely, we have to claim that the right-hand sides of the problem (2.1)-(2.3) satisfy conditions (4.5), (4.6) for certain σ > p + b + n/2.
In Section 8, we will deduce Theorem 4.3 from Isomorphism Theorem 4.1 and will show that Theorem 4.4 is a consequence of Theorem 4.3 and a version of Hörmander's embedding theorem [13,Theorem 2.2.7]. We will also justify Remark 4.5.

Isomorphism Theorem in the Sobolev case
The goal of this section is to show that Theorem 4.1 follows from the above-mentioned result by M. S. Agranovich and M. I. Vishik [1,Theorem 12.1] in the Sobolev case where ϕ(r) ≡ 1 and σ/(2b) ∈ Z. Beforehand, we will prove a lemma about a description of the spaces H s,sγ + (Ω) and H s,sγ + (S) in terms of the Hilbert spaces H s,sγ (Ω) and H s,sγ (S) used in this result. The latter two are defined quite similarly to the first two, with no restrictions on support of distributions being imposed. For the reader's convenience, we give the relevant definitions.
Let real s > 0 and . This space is Hilbert with respect to the norm (5.2). We are interested in the case where V = Ω, with k = n, and also in the case where V = Π, with Π := R n−1 × (0, τ ) and k = n − 1. Using H s,sγ (Π), we now define the Hilbert space H s,sγ (S) by formulas (3.7), (3.8) in which we omit the subscript +. The anisotropic Sobolev spaces just defined are well known in the theory of parabolic equations [1,10,20,43].
We have the continuous embeddings They follow immediately from the definitions of the spaces appearing in (5.3). Besides, we let H θ (U) denote the isotropic Sobolev inner product space of order θ over a Euclidean domain U; specifically, H 0 (U) = L 2 (U). We Moreover, the norm in H s,sγ + (Ω) is equivalent to the norm in H s,sγ (Ω). This lemma remains valid if we replace Ω by S and G by Γ. They found necessary and sufficient conditions under which the extension of every function u ∈ H s,sγ (G × (0, θ)) by zero belongs to H s,sγ (G × (−∞, θ)) with 0 < θ ≤ ∞, they restricting themselves to the case where s ∈ Z and γ = 1/(2b). These conditions are tantamount to (5.4) and imply equivalence of the norms of u and its extension by zero. M. S. Agranovich and M. I. Vishik also considered the case of functions given on Γ × (0, θ) Proof of Lemma 5.1. We note first that condition (5.4) is well posed by virtue of the trace theorem for anisotropic Sobolev spaces (see, e.g., [43,Part II,Theorem 4]). Let Υ s,sγ (Ω) denote the linear manifold of all functions u ∈ H s,sγ (Ω) that satisfy (5.4). According to this trace theorem, we may and will consider Υ s,sγ (Ω) as a (closed) subspace of H s,sγ (Ω). It follows directly from (5.3) that we have the continuous embedding H s,sγ + (Ω) ֒→ Υ s,sγ (Ω). Therefore (in view of the Banach theorem on inverse operator) it remains to prove the converse inclusion Υ s,sγ (Ω) ⊂ H s,sγ + (Ω). Let u ∈ Υ s,sγ (Ω). We must prove that u = w on Ω for a certain function w ∈ H s,sγ + (R n+1 ). To this end, we use the following three extension operators O, T τ , and T G acting between some isotropic Sobolev spaces.
Given a function v ∈ L 2 ((0, ∞)), we define the function Ov ∈ L 2 (R) by the formulas . It follows directly from [44, Theorems 2.9.3(a) and 2.10.3(b)] and the condition sγ −1/2 / ∈ Z that the restriction of (5.5) to H sγ ((0, ∞)) defines an isomorphism We consider a bounded linear operator τ )) and that the restriction of the mapping T τ to the space H sγ ((0, τ )) is a bounded operator We also consider a bounded linear operator such that T G h = h on G for every function h ∈ L 2 (G) and that the restriction of the mapping T G to the space H s (G) is a bounded operator and 12) up to equivalence of norms; see, e.g., [1, § 8, Subection 1]. (As usual, E ⊗F denotes the tensor product of arbitrary Hilbert spaces E and F . Besides, their intersection E ∩F is considered as a Hilbert space endowed with the inner product It follows directly from (5.7), (5.8), (5.11) and the inclusion u ∈ Υ s,sγ (Ω) that Here, I is the identity operator on L 2 (G). Then in view of formulas (5.5), (5.6), (5.9), (5.10), and (5.12). Besides, w = u on Ω. Thus, u ∈ H s,sγ + (Ω). The same reasoning shows that Lemma 5.1 remains valid if we replace Ω by Π := R n−1 × (0, τ ) and G by R n−1 . (Of course, we take R n instead of R n+1 and need not use the extension operator T G in this case.) It follows directly from this fact and the definitions of H s,sγ + (S) and H s,sγ (S) that Lemma 5.1 also remains valid if we replace Ω by S and G by Γ. Let us show that u ∈ H σ,σ/(2b) + (Ω). To this end, we use Lemma 5.1 with s := σ and γ := 1/(2b). According to (2.3), the function u satisfies condition (5.4) provided that 0 ≤ k ≤ κ−1. Here, The fulfilment of (5.4) for the rest values of the integer k (when κ − 1 < k < σ/(2b) − 1/2) is proved (if these values exist) in the following way. Let the number of these values is l ≥ 1; then The parabolicity Condition 2.1 in the case of ξ = 0 and p = 1 means that the coefficient a (0,...,0),κ (x, t) = 0 for all x ∈ G and t ∈ [0, τ ]. Therefore we can resolve the parabolic equation (2.1) with respect to ∂ κ t u(x, t); namely, we can write for some functions a α,β 0 ∈ C ∞ (Ω). If l ≥ 2, then we differentiate the equality (5.17) l − 1 times with respect to t and obtain l − 1 equalities with j = 1, . . . , l − 1.  Here, c 3 and c 4 are some positive numbers that do not depend on (5.14) and u. Thus, we conclude that for an arbitrary vector (5.14) there exists a unique solution u ∈ H Completing this section, we will show that the mapping (2.4) is a bijection such that (A, B)u = (f, g 1 , . . . , g m ). To deduce the desired inclusion u ∈ C ∞ + (Ω) from (5.21), we use the extension operators O, T τ , and T G from the proof of Lemma 5.1. We can suppose that the mappings T τ and T G are independent of s > 0. The extension operators of this kind are constructed, e.g., in [41]. Then, according to (5.13) (with s = 2bl and γ = 1/(2b)) and (5.21), we can write Here, we note that u ∈ Υ 2bl,l (Ω) by Lemma 5.1 for l indicated in (5.22), the space Υ 2bl,l (Ω) being defined in the proof of this lemma. We also remark that the letter inclusion in (5.22) holds true by the Sobolev embedding theorem (see also [43, Part II, Theorem 13]). Hence, u = w ↾ Ω ∈ C ∞ + (Ω). Thus, the mapping (2.4) is surjective. We have justified the bijection (5.20).

Interpolation with a function parameter
In this section we discuss the method of the interpolation with a function parameter between Hilbert spaces, which was introduced by C. Foiaş and J.-L. Lions [11, p. 278]. This interpolation is a natural generalization of the classical interpolation method by S. G. Krein and J.-L. Lions to the case where a sufficiently general function is used instead of a number as an interpolation parameter (see, e.g., monographs [17, Chapter IV, Section 1, Subsection 10] and [19, Chapter 1, Sections 2 and 5]). Interpolation with a function parameter will play a key role in the proof of Isomorphism Theorem. It is sufficient for our purposes to restrict the discussion to the case of separable complex Hilbert spaces. We follow the monograph [34, Section 1.1], which systematically sets forth this interpolation (see also [30,Section 2]).
Let X := [X 0 , X 1 ] be an ordered pair of separable complex Hilbert spaces such that X 1 ⊆ X 0 and this embedding is continuous and dense. This pair is said to be admissible. There exists a positive-definite self-adjoint operator J on X 0 that has the domain X 1 and that satisfies the condition Jv X 0 = v X 1 for every v ∈ X 1 . This operator is uniquely determined by the pair X and is called a generating operator for X (see, e.g., [17, Chapter IV, Theorem 1.12]). It defines an isometric isomorphism J : X 1 ↔ X 0 .
Given a function ψ ∈ B, we consider the (generally, unbounded) operator ψ(J), which is defined on X 0 as the Borel function ψ of J. This operator is built with the help of Spectral Theorem applied to the self-adjoint operator J. Let [X 0 , X 1 ] ψ or, simply, X ψ denote the domain of the operator ψ(J) endowed with the inner product The linear space X ψ is Hilbert and separable with respect to this inner product. The latter induces the norm v X ψ := ψ(J)v X 0 .
A function ψ ∈ B is called an interpolation parameter if the following condition is fulfilled for all admissible pairs X = [X 0 , X 1 ] and Y = [Y 0 , Y 1 ] of Hilbert spaces and for an arbitrary linear mapping T given on X 0 : if the restriction of T to X j is a bounded operator T : X j → Y j for each j ∈ {0, 1}, then the restriction of T to X ψ is also a bounded operator T : X ψ → Y ψ .
If ψ is an interpolation parameter, then we say that the Hilbert space X ψ is obtained by the interpolation with the function parameter ψ of the pair X = [X 0 , X 1 ] (or, otherwise speaking, between the spaces X 0 and X 1 ). In this case, we have the dense and continuous embeddings X 1 ֒→ X ψ ֒→ X 0 .
It is known that a function ψ ∈ B is an interpolation parameter if and only if ψ is pseudoconcave in a neighbourhood of infinity, i.e. there exists a concave positive function ψ 1 (r) of r ≫ 1 such that both the functions ψ/ψ 1 and ψ 1 /ψ are bounded in some neighbourhood of infinity. This criterion follows from J. Peetre's [38,39] description of all interpolation functions for the weighted L p (R n )-type spaces (this result of J. Peetre is also set forth in the monograph [5,Theorem 5.4.4]). The proof of the criterion is given in [34, Section 1.1.9].
We will use the next consequence of this criterion [34,Theorem 1.11].
Then ψ is an interpolation parameter.
Note that, in the case of power functions, Proposition 6.1 leads us to the above-mentioned classical result by J.-L. Lions and S. G. Krein. Namely, they proved that the function ψ(r) ≡ r θ is an interpolation parameter whenever 0 < θ < 1. In this case, the exponent θ is regarded as a number parameter of the interpolation.
We end this section with two properties of the interpolation, which will be used in our proofs. The first of them enables us to reduce the interpolation of subspaces or factor spaces to the interpolation of initial spaces (see [34,Sec. 1.1.6] or [44,Sec. 1.17]). Note that subspaces (of Hilbert spaces) are assumed to be closed and that we generally consider nonorthogonal projectors onto subspaces. Proposition 6.2. Let X = [X 0 , X 1 ] be an admissible pair of Hilbert spaces, and let Y 0 be a subspace of X 0 . Then Y 1 := X 1 ∩ Y 0 is a subspace of X 1 . Suppose that there exists a linear mapping P : X 0 → X 0 such that P is a projector of the space X j onto its subspace Y j for every j ∈ {0, 1}. Then the pairs [Y 0 , Y 1 ] and [X 0 /Y 0 , X 1 /Y 1 ] are admissible, and with equivalence of norms for an arbitrary interpolation parameter ψ ∈ B. Here, The second property reduces the interpolation of ortogonal sums of Hilbert spaces to the interpolation of their summands.

The interpolation between anisotropic Sobolev spaces
The purpose of this section is to prove that the Hörmander spaces appearing in Theorem 4.1 can be obtained by means of the interpolation with a function parameter between their Sobolev analogs.
Beforehand, we will derive the necessary interpolation formulas for the basic Hörmander spaces H s,sγ;ϕ (R k+1 ) and H s,sγ;ϕ + (R k+1 ), with the integer k ≥ 1. All the results of this section are formulated as lemmas.
Lemma 7.1. On the assumption (7.1) we have with equality of norms.
The proof of this lemma is analogous to the proof of [21, Lemma 5.1], where the k = 1 case was examined. Nevertheless, we give this proof for the reader's convenience.
Proof. According to (3.1), the pair of anisotropic Sobolev spaces is admissible. It follows immediately from their definition that the generating operator for X is given by the formula Here, F [F −1 respectively] denotes the operator of the direct [inverse] Fourier transform in all variables of tempered distributions given in R k+1 .
The generating operator J is reduced to the operator of multiplication by r s 1 −s 0 γ by means of the Fourier transform which sets an isometric isomorphism Here, of course, the second Hilbert space consists of all functions (of ξ ∈ R k and η ∈ R) that are square integrable over R k+1 with respect to the Radon measure r 2s 0 γ (ξ, η)dξdη. Hence, F reduces ψ(J) to the operator of multiplication by the function ψ(r s 1 −s 0 γ (ξ, η)) ≡ r s−s 0 γ (ξ, η) ϕ(r γ (ξ, η)), the identity being due to (7.2). Therefore, given w ∈ C ∞ 0 (R k+1 ), we can write . This implies the equality of spaces (7.3) as the set C ∞ 0 (R k+1 ) is dense in both of them. Here, we remark that this set is dense in the second space denoted by X ψ because C ∞ 0 (R k+1 ) is dense in the space H s 1 ,s 1 γ (R k+1 ) embedded continuously and densely in X ψ . Lemma 7.2. Assume in addition to (7.1) that s 0 ≥ 0. Then H s,sγ;ϕ with equivalence of norms.
Proof. We will deduce this lemma from Lemma 7.1 with the help of Proposition 6.2. To this end, we need to present a linear mapping P on L 2 (R k+1 ) that the restriction of P to each space

Let us consider a bounded linear operator
T : L 2 ((−∞, 0)) → L 2 (R) (7.5) such that T h = h on (−∞, 0) for every function h ∈ L 2 ((−∞, 0)) and that the restriction of the mapping T to the Sobolev space H s j γ ((−∞, 0)) is a bounded operator The operator T exists [44, Lemma 2.9.3]. Using the tensor product of bounded operators on Hilbert spaces, we obtain the bounded linear operator such that (I ⊗ T )v = v on R k × (−∞, 0) for every function v ∈ L 2 (R k × (−∞, 0)). Here, I is the identity operator on L 2 (R k ). Given j ∈ {0, 1}, we can write and (7.9) These equalities of spaces hold true up to equivalence of norms. Being based on (7.5), (7.6), (7.8), and (7.9), we conclude that the restriction of (7.7) to the space H s j ,s j γ (R k × (−∞, 0)) is a bounded operator (7.10) Consider the linear mapping It is easy to see that supp P w ⊆ R k × [0, ∞) and that the inclusion supp w ⊆ R k × [0, ∞) implies the equality P w = w on R k+1 . Using these properties of P and the boundedness of the operator (7.10), we conclude that the mapping P is required. Now, by virtue of Proposition 6.2 (formula (6.1)) and Lemma 7.1, we can write These equalities of spaces hold true up to equivalence of norms. (Note that the first pair is admissible by Proposition 6.2.) Thus, we have proved (7.4).
Lemma 7.3. Assume in addition to (7.1) that s 0 ≥ 0 and with equivalence of norms.
Proof. We will first prove (7.12). Recall that, by definition, and H s j ,s j γ + for each j ∈ {0, 1}. Here, the denominators are defined by (3.5). We will deduce formula (7.12) from Lemma 7.2 with the help of Proposition 6.2, the interpolation of factor spaces being used. For this purpose, we need to present a linear mapping P on H s 0 ,s 0 γ + (R n+1 ) that the restriction of P to each H s j ,s j γ + (R n+1 ), with j ∈ {0, 1}, is a projector of the space H s j ,s j γ + (R n+1 ) onto its subspace H s j ,s j γ + (R n+1 , Ω). Let us make use of the reasoning and notation given in the proof of Lemma 5.1. The justification of (5.13) presented in this proof shows also that we have the bounded operator for each j ∈ {0, 1}. Here, the operators T G and T τ respectively are restrictions of the mappings (5.7) and (5.9), which do not depend on j (see [44,Theorems 4.2.2 and 4.2.3]). Therefore the operator (7.16) with j = 1 is a restriction of its counterpart with j = 0. Besides, as we have mentioned just after (5.13), the equality T + u = u holds on Ω for every u ∈ Υ s j ,s j γ (Ω). Note also that Υ s j ,s j γ (Ω) = H s j ,s j γ + (Ω) due to Lemma 5.1 and the condition (7.11). Let us consider the linear mapping Note that P w = 0 on Ω and that the condition w = 0 on Ω implies the equality P w = w on R n+1 . Taking into account these properties of P and the boundedness of the operator (7.16), we conclude that the mapping P is required. Now, using successively (7.15), Proposition 6.2 (formula (6.2)), Lemma 7.2, and (7.14), we write the following: Here, These equalities of spaces hold true up to equivalence of norms. (Remark also that the first pair is admissible by Proposition 6.2.) Thus, we have proved (7.12). Let us prove the interpolation formula (7.13). The pair of spaces on the right of (7.13) is admissible due to Lemma 3.1. (Its proof given at the end of this section does not use this lemma in the case where ϕ(r) ≡ 1 and sγ − 1/2 / ∈ Z.) We will deduce formula (7.13) from its analog H s,sγ;ϕ for Π := R n−1 × (0, τ ). The proof of (7.17) is the same as that of (7.12), but with Π, R n , and the identity operator instead of Ω, R n+1 , and T G respectively. Using the definition of anisotropic Hörmander spaces over S given in (3.7) and (3.8), we will deduce (7.13) from (7.17) with the help of certain operators of flattening and sewing of the manifold S. We define the flattening operator by the formula We define the sewing operator by the formula Here, each function η k ∈ C ∞ 0 (R n−1 ) is chosen so that η k = 1 on the set θ −1 k (supp χ k ); next η * k (x, t) := η k (x) for all x ∈ R n−1 and t ∈ (0, τ ). Besides, O k denotes the operator of the extension (of functions) by zero from Γ k × (0, τ ) to S; thus, for every y ∈ Γ and t ∈ (0, τ ), we have The mapping K is left inverse to L. Indeed,  Here, we define (Q k,l w)(x, t) := η k,l (β k,l (x)) w(β k,l (x), t) (7.25) for all w ∈ L 2 (Π), x ∈ R n−1 , and t ∈ (0, τ ), where η k,l := (χ l • θ k )η k ∈ C ∞ 0 (R n−1 ) and, moreover, β k,l : R n−1 ↔ R n−1 is an infinitely smooth diffeomorphism such that β k,l = θ −1 k • θ l in a neighbourhood of supp η k,l . As is known [15, Theorem B.1.8], the operator ω → (η k,l ω) • β k,l is bounded on every Sobolev space H σ (R n−1 ) with σ ∈ R. Therefore the operator w → Q k,l w defined by formula (7.25) for all w ∈ L 2 (R n ), x ∈ R n−1 , and t ∈ R is bounded on each space with j ∈ {0, 1}. Hence, the restriction of the mapping w → Q k,l w, with w ∈ L 2 (Π), to each space H s j ,s j γ + (Π) is a bounded operator on this space. Then, owing to the interpolation formula (7.17), the restriction of this mapping to the space H s,sγ;ϕ + (Π) is a bounded operator on this space. Combining the latter conclusion with (7.24), we can write for some number c > 0 that does not depend on h. Thus, we have proved the boundedness of the operator (7.23).
The same reasoning also proves that the restriction of the mapping K to the space (H s j ,s j γ + (Π)) λ is a bounded operator Interpolating between the spaces in (7.26) with the function parameter ψ and using Proposition 6.3 and formula (7.17), we obtain a bounded operator K : H s,sγ;ϕ Now it follows directly from (7.18), (7.27), and (7.22) that the identity operator KL realizes a continuous embedding of the space H s,sγ,ϕ + (S) in the interpolation space Moreover, it follows immediately from (7.21) and (7.23) that the identity operator KL establishes the inverse continuous embedding. Thus, we have proved that the equality (7.13) is true up to equivalence of norms.
At the end of this section, we will prove Lemma 3.1.
Proof of Lemma 3.1. We first examine the case where ϕ(r) ≡ 1 and sγ − 1/2 / ∈ Z. Lemma 5.1 implies that H s,sγ + (S) is equal up to equivalence of norms to a certain subspace of H s,sγ (S). Therefore, assertion (i) is a known property of the anisotropic Sobolev space H s,sγ (S); see [43,Chapter I,§ 5].
Let us prove assertion (ii) with the help of the flattening operator L and sewing operator K used in the proof of Lemma 7.3. As has been stated in Section 3, the set . Given a function v ∈ H s,sγ + (S), we approximate the vector Lv by the sequence of vectors h (j) ∈ (Υ ∞ 0 (Π)) λ in the norm of the space (H s,sγ + (Π) λ . Then the sequence of functions Kh (j) ∈ C ∞ + (S) approximates the function KLv = v in the norm of the space H s,sγ + (S). Assertion (ii) is proved in the case examined.
In the general situation, Lemma 3.1 follows from this case with the help of Lemma 7.3. Indeed, the space H s,sγ;ϕ + (S) is complete and separable due to properties of the interpolation used in formula (7.13). Moreover, let A 1 and A 2 be two pairs each of which consists of an atlas on Γ and a partition of unity used in the definitions (3.7) and (3.8). Let H s,sγ;ϕ + (S, A k ) and H s j ,s j γ + (S, A k ), with j ∈ {0, 1}, denote the spaces H s,sγ;ϕ + (S) and H s j ,s j γ + (S) corresponding to the pair A k with k ∈ {0, 1}. As has been stated in the previous paragraph, the identity mapping I is an isomorphism between the spaces H s j ,s j γ + (S, A 0 ) and H s j ,s j γ + (S, A 1 ) for each j ∈ {0, 1}. Therefore, by virtue of the interpolation formula (7.13), we have the isomorphism This means that the space H s,sγ;ϕ + (S) does not depend on the choice of an atlas on Γ and a partition of unity. Finally, Assertion (ii) in the general situation results from the density of H s 1 ,s 1 γ These equalities of spaces hold up to equivalence of norms. Thus, the isomorphism (8.2) becomes (4.1). This isomorphism is an extension by continuity of the mapping (2.4) because the set C ∞ + (Ω) is dense in the space H σ,σ/(2b);ϕ + (Ω).