Isomorphism theorems for some parabolic initial-boundary value problems in H\"ormander spaces

In H\"ormander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the operators corresponding to these problems are isomorphisms between appropriate H\"ormander spaces. The regularity of the functions which form these spaces is characterized by a pair of number parameters and a function parameter varying regularly at infinity in the sense of Karamata. Owing to this function parameter, the H\"ormander spaces describe the regularity of functions more finely than the anisotropic Sobolev spaces.


Introduction
The modern theory of general parabolic initial-boundary problems has been developed for the classical scales of Hölder-Zygmund and Sobolev function spaces [1][2][3][4][5][6][7][8][9]. The central result of this theory are the theorems on wellposedness by Hadamard of these problems on appropriate pairs of these spaces. For applications, especially to the spectral theory of differential operators, inner product Sobolev spaces play a special role.
In 1963 Hörmander [10] proposed a broad and meaningful generalization of the Sobolev spaces in the framework of Hilbert spaces. He introduced the spaces for which a general Borel measurable weight function W R k ! .0; 1/ serves as an index of regularity of a distribution w. (Here, b w denotes the Fourier transform of w.) These spaces and their versions within the category of normed spaces (so called spaces of generalized smoothness) have found various applications to analysis and partial differential equations [11][12][13][14][15][16][17][18][19].
Recently Mikhailets and Murach [20][21][22][23][24] have built a theory of solvability of general elliptic systems and elliptic boundary-value problems on Hilbert scales of spaces H sI' WD B 2; for which the index of regularity is of the form . / WD .1 C j j 2 / s=2 '..1 C j j 2 / 1=2 /: *Corresponding Author: Valerii Los: National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Prospect Peremohy 37, 03056, Kyiv-56, Ukraine, E-mail: v_los@yahoo.com Aleksandr Murach: Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01004, Ukraine and Chernihiv National Pedagogical University Het'mana Polubotka str. 53, 14013 Chernihiv, Ukraine, E-mail: murach@imath.kiev.ua Here, s is a real number, and ' is a function varying slowly at infinity in the sense of Karamata [26]. This theory is based on the method of interpolation with a function parameter between Hilbert spaces, specifically between Sobolev spaces. This allows Mikhailets and Murach to deduce theorems about solvability of elliptic systems and elliptic problems from the known results on the solvability of elliptic equations in Sobolev spaces. This theory is set force in [18,25].
Generally, the method of interpolation between normed spaces proved to be very useful in the theory of elliptic [27][28][29] and parabolic [4,8] partial differential equations. Specifically, Lions and Magenes [4] systematically used the interpolation with a number (power) parameter between Hilbert spaces in their theory of solvability of parabolic initial-boundary value problems on a complete scale of anisotropic Sobolev spaces. Using the more flexible method of interpolation with a function parameter between Hilbert spaces, Los, Mikhailets, and Murach [30,31] proved theorems on solvability of semi-homogeneous parabolic problems in 2b-anisotropic Hörmander spaces H s;s=.2b/I' , where 2b is a parabolic weight and where the parameters s and ' are the same as those in the above mentioned elliptic theory. These problems were considered in the case of homogeneous initial conditions (Cauchy data).
The purpose of this paper is to establish the well-posedness of inhomogeneous parabolic problems on appropriate pairs of the Hörmander spaces, i.e. to prove new isomorphism theorems for these problems. We consider the problems that consist of a general second order parabolic partial differential equation, the Dirichlet boundary condition or a general first order boundary condition, and the Cauchy datum. We deduce these isomorphism theorems from Lions and Magenes' result [4] with the help of the interpolation with a function parameter between anisotropic Sobolev spaces. The use of this method in the case of inhomogeneous parabolic problems meets additional difficulties connected with the necessity to take into account quite complex compatibility conditions imposed on the right-hand sides of the problem. The model case of initial boundary-value problems for heat equation is investigated in [32].

Statement of the problem
We arbitrarily choose an integer n 2 and a real number > 0. Let G be a bounded domain in R n with an infinitely smooth boundary WD @G. We put WD G .0; / and S WD .0; /; so, is an open cylinder in R nC1 , and S is its lateral boundary. Then WD G OE0; and S WD OE0; are the closures of and S respectively.
In , we consider a parabolic second order partial differential equation for all x 2 G and t 2 .0; /: Here and below, we use the following notation for partial derivatives: @ t WD @=@t and Dx WD D˛1 1 : : : D˛n n , where D j WD i @=@x j , x D .x 1 ; : : : ; x n / 2 R n , and˛WD .˛1; : : : ;˛n/ with 0 Ä˛1; :::;˛n 2 Z and j˛j WD˛1 C C˛n.
We suppose that all the coefficients a˛of A belong to the space C 1 . /. In the paper, all functions and distributions are supposed to be complex-valued, so we consider complex function spaces. We suppose that the partial differential operator A is Petrovskii parabolic on , i.e. it satisfies the following condition (see, e.g. [1, Section 9, Subsection 1]): For arbitrary x 2 G, t 2 OE0; , D . 1 ; : : : ; n / 2 R n , and p 2 C with Re p 0, the inequality a˛.x; t / ˛1 1 ˛n n ¤ 0 holds whenever j j C jpj ¤ 0: In the paper, we investigate the initial-boundary value problem that consists of the parabolic equation (1), the initial condition and the zero-order (Dirichlet) boundary condition u.x; t / D g.x; t / for all x 2 and t 2 .0; /

59
or the first order boundary condition for all x 2 and t 2 .0; /: As to (4), we assume that all the coefficients b 0 , b 1 , ..., b n of B belong to C 1 .S / and that B covers A on S [1, Section 9, Subsection 1]. The latter assumption means the fulfilment of the following: Choose arbitrarily x 2 , t 2 OE0; , vector Á D .Á 1 ; : : : ; Á n / 2 R n tangent to the boundary at the point x, and number p 2 C with Re p 0 so that jÁj C jpj ¤ 0. Let .x/ D . 1 .x/; : : : ; n .x// be the unit vector of the inward normal to at x. Then: a) the inequality is not a root of the polynomial It is useful to note that if all the coefficients b 1 ,..., b n are real-valued, then part b) of Condition 2.2 is satisfied. This follows directly from Condition 2.1. Thus, we examine both the parabolic problem (1), (2), (3) and the parabolic problem (1), (2), (4). We investigate them in appropriate Hörmander inner product spaces considered in the next section.

Hörmander spaces
Among the normed function spaces B p; introduced by Hörmander in [10, Section 2.2], we use the inner product spaces H .R k / WD B 2; defined over R k , with 1 Ä k 2 Z. Here, W R k ! .0; 1/ is an arbitrary Borel measurable function that satisfies the following condition: there exist positive numbers c and l such that By definition, the (complex) linear space H .R k / consists of all tempered distributions w 2 S 0 .R k / whose Fourier transform b w is a locally Lebesgue integrable function subject to the condition The inner product in H .R k / is defined by the formula where w 1 ; w 2 2 H .R k /. This inner product induces the norm According to [10, Section 2.2], the space H .R k / is Hilbert and separable with respect to this inner product. Besides that, this space is continuously embedded in the linear topological space S 0 .R k / of tempered distributions on R k , and the set C 1 0 .R k / of test functions on R k is dense in H .R k / (see also Hörmander's monograph [33, Section 10.1]). We will say that the function parameter is the regularity index for the space H .R k / and its versions H . /.
A version of H .R k / for an arbitrary nonempty open set V R k is introduced in the standard way. Namely, where u 2 H .V /. Here, as usual, w V stands for the restriction of the distribution w 2 H .R k / to the open set V . In other words, H .V / is the factor space of the space H .R k / by its subspace Thus, H .V / is a separable Hilbert space. The norm (5) is induced by the inner product where w j 2 H .R k /, w j D u j in V for each j 2 f1; 2g, and ‡ is the orthogonal projector of the space H .R k / onto its subspace (6). The spaces H .V / and H Q .R k / were introduced and investigated by Volevich and Paneah [11,Section 3]. It follows directly from the definition of H .V / and properties of H .R k / that the space H .V / is continuously embedded in the linear topological space D 0 .V / of all distributions on V and that the set Suppose that the integer k 2. Dealing with the above-stated parabolic problems, we need the Hörmander spaces H .R k / and their versions in the case where the regularity index takes the form for all 0 2 R k 1 and k 2 R: Here, the number parameter s is real, whereas the function parameter ' runs over a certain class M. By definition, the class M consists of all Borel measurable functions ' W OE1; 1/ ! .0; 1/ such that a) both the functions ' and 1=' are bounded on each compact interval OE1; b, with 1 < b < 1; b) the function ' varies slowly at infinity in the sense of Karamata [26], i.e. '. r/='.r/ ! 1 as r ! 1 for each > 0.
The theory of slowly varying functions (at infinity) is expounded, e.g., in [34,35]. Their standard examples are the functions '.r/ WD .log r/ Â 1 .log log r/ Â 2 : : : . log : : : log " ƒ‚ … k times r / Â k of r 1; where the parameters k 2 N and Â 1 ; Â 2 ; : : : ; Â k 2 R are arbitrary. Let s 2 R and ' 2 M. We put H s;s=2I' .R k / WD H .R k / in the case where is of the form (7). Specifically, if '.r/ Á 1, then H s;s=2I' .R k / becomes the anisotropic Sobolev inner product space H s;s=2 .R k / of order .s; s=2/. Generally, if ' 2 M is arbitrary, then the following continuous and dense embeddings hold: Indeed, let s 0 < s < s 1 ; since ' 2 M, there exist positive numbers c 0 and c 1 such that c 0 r s 0 s Ä '.r/ Ä c 1 r s 1 s for every r 1 (see e.g., [35, Section 1.5, Property 1 ı ]). Then for arbitrary 0 2 R k 1 and k 2 R. This directly entails the continuous embeddings (8). They are dense because the set C 1 0 .R k / is dense in all the spaces from (8). Consider the class of Hörmander inner product spaces The embeddings (8) show, that in (9) the function parameter ' defines additional regularity with respect to the basic anisotropic .s; s=2/-regularity. Specifically, if '.r/ ! 1 [or '.r/ ! 0] as r ! 1, then ' defines additional positive [or negative] regularity. In other words, ' refines the basic smoothness .s; s=2/. We need versions of the function spaces (9) for the cylinder D G .0; / and its lateral boundary S D .0; /. We put H s;s=2I' . / WD H . / in the case where is of the form (7) with k WD n C 1. For the function space H s;s=2I' . /, the numbers s and s=2 serve as the regularity indices of distributions u.x; t / with respect to the spatial variable x 2 G and to the time variable t 2 .0; / respectively. Following [36, Section 1], we will define the function space H s;s=2I' .S / with the help of special local charts on S . Let s > 0 and ' 2 M. We put H s;s=2I' .…/ WD H .…/ for the strip … WD R n 1 .0; / in the case where is defined by formula (7) with k WD n. Recall that, according to our assumption D @ is an infinitely smooth closed manifold of dimension n 1, the C 1 -structure on being induced by R n . From this structure we arbitrarily choose a finite atlas formed by local charts Â j W R n 1 $ j with j D 1; : : : ; . Here, the open sets 1 ; : : : ; make up a covering of . We also arbitrarily choose functions j 2 C 1 ./, with j D 1; : : : ; , so that supp j j and 1 C D 1 on . By definition, the linear space H s;s=2I' .S / consists of all square integrable functions g W S ! C that the function g j .x; t / WD j .Â j .x// g.Â j .x/; t / of x 2 R n 1 and t 2 .0; / belongs to H s;s=2I' .…/ for each number j 2 f1; : : : ; g. The inner product in H s;s=2I' .S / is defined by the formula where g; g 0 2 H s;s=2I' .S /. This inner product naturally induces the norm kgk H s;s=2I' .S / WD .g; g/ 1=2 H s;s=2I' .S/ : The space H s;s=2I' .S / is complete (i. e. Hilbert) and does not depend up to equivalence of norms on the choice of local charts and partition of unity on [36,Theorem 1]. Note that this space is actually defined with the help of the following special local charts on S: where Â j .x; t/ WD .Â j .x/; t / for all x 2 R n 1 and t 2 .0; /. We also need isotropic Hörmander spaces H sI' .V / over an arbitrary open nonempty set V Â R k with k 1. Let s 2 R and ' 2 M. We put H sI' .V / WD H .V / in the case where the regularity index takes the form Since the function (11) is radial (i.e., depends only on j j), the space H sI' .V / is isotropic. We will use the spaces Besides, we will use Hörmander spaces H sI' ./ over D @ . The are defined with the help of the abovementioned collection of local charts fÂ j g and partition of unity f j g on similarly to the spaces over S. Let s 2 R and ' 2 M. By definition, the linear space H sI' ./ consists of all distributions ! 2 D 0 ./ on that for each number j 2 f1; : : : ; g the distribution ! j .x/ WD j .Â j .x// !.Â j .x// of x 2 R n 1 belongs to H sI' .R n 1 /. The inner product in H sI' ./ is defined by the formula where !; ! 0 2 H sI' ./. It induces the norm The space H sI' ./ is Hilbert separable and does not depend up to equivalence of norms on our choice of local charts and partition of unity on [37, Theorem 3.6(i)]. Note that the classes of isotropic inner product spaces were selected, investigated, and systematically applied to elliptic differential operators and elliptic boundary-value problems by Mikhailets and Murach [18,25].
If ' Á 1, then the considered spaces H s;s=2I' . / and H sI' . / become the Sobolev spaces H s;s=2 . / and H s . / respectively. It follows directly from (8) that Analogously, In the Sobolev case of ' Á 1, we will omit the index ' in designations of function spaces that will be introduced on the base of the Hörmander spaces H s;s=2I' . / and H sI' . /.

Main results
Consider first the parabolic problem (1)-(3), which corresponds to the Dirichlet boundary condition on S. In order that a regular enough solution u to this problem exist, the right-hand sides of the problem should satisfy certain compatibility conditions (see, e.g., [1,Section 11] or [3, Chapter 4, Section 5]). These conditions consist in that the partial derivatives @ k t u.x; t /ˇt D0 , which could be found from the parabolic equation (1) and initial condition (2), should satisfy the boundary condition (3) and some relations that are obtained by means of the differentiation of the boundary condition with respect to t . To write these compatibility conditions we use Sobolev inner product spaces.
We associate the linear mapping with the problem (1) of the problem by the formula .f; g; h/ WD ƒ 0 u with the help of this bounded operator. According to [38,Chapter II,Theorem 7], the traces @ k t u. ; 0/ 2 H s 1 2k .G/ are well defined by closure for all k 2 Z such that 0 Ä k < s=2 1=2 (and only for these k). Using (1) and (2), we express these traces in terms of the functions f .x; t / and h.x/ by the recurrent formula the equalities holding for almost all x 2 G. Besides, the traces @ k t g. ; 0/ 2 H s 3=2 2k ./ are well defined by closure for all k 2 Z such that 0 Ä k < s=2 3=4 (and only for these k). Therefore, owing to the Dirichlet boundary condition (3), the equality holds for these integers k. The right-hand part of this equality is well defined because the function @ k t u. ; 0/ 2 H s 1 2k .G/ has the trace @ k t u. ; 0/ 2 H s 3=2 2k ./ in view of s 3=2 2k > 0. Now, substituting (17) into (18), we obtain the compatibility conditions Here, the functions v k are defined by the recurrent formula these relations holding for almost all due to (16), the trace v k 2 H s 3=2 2k ./ is defined by closure whenever s 3=2 2k > 0. Thus, the compatibility conditions (19) are well posed. For instance, if 2 < s Ä 7=2, then formula (19) gives one compatibility condition g D h . Next, if 7=2 < s Ä 11=2, then (19) gives two compatibility conditions g D h and and so on.
We put E 0 WD f2r C 3=2 W 1 Ä r 2 Zg. Note that E 0 is the set of all discontinuities of the function that assigns the number of compatibility conditions (19) to s 2.
Our main result on the parabolic problem (1)-(3) consists in that the linear mapping (14) extends uniquely to an isomorphism between appropriate pairs of Hörmander spaces introduced in the previous section. Let us indicate these spaces. We arbitrarily choose a real number s > 2 and function parameter ' 2 M. We take H s;s=2I' . / as the source space of this isomorphism; otherwise speaking, H s;s=2I' . / serves as a space of solutions u to the problem. To introduce the target space of the isomorphism, consider the Hilbert space In the Sobolev case of ' Á 1 this space coincides with the target space of the bounded operator (15 The latter follows directly from (12) and (13). Thus, our definition is reasonable. Here, the space Q s 2 ";s=2 1 "=2 0 is complete because the differential operators and traces operators used in the compatibility conditions are bounded on the corresponding pairs of Sobolev spaces. Therefore the right-hand side of this equality is complete with respect to the sum of the norms in the components of the intersection, this sum being equivalent to the norm in H s 2;s=2 1I' 0 due to (22). Thus, the space Q s 2;s=2 1I' 0 is complete (with respect to the latter norm).
If s 2 E 0 , then we define the Hilbert space Q s 2;s=2 1I' 0 by means of the interpolation between its analogs just introduced. Namely, we put Here, the number " 2 .0; 1=2/ is arbitrarily chosen, and the right-hand side of the equality is the result of the interpolation of the written pair of Hilbert spaces with the parameter 1=2. We will recall the definition of the interpolation between Hilbert spaces in Section 5. The Hilbert space Q s 2;s=2 1I' 0 defined by formula (23) does not depend on the choice of " up to equivalence of norms and is continuously embedded in H s 2;s=2 1I' 0 . This will be shown in Remark 6.3 at the end of Section 6. Now we can formulate our main result concerning the parabolic initial-boundary value problem (1)-(3).
Otherwise speaking, the parabolic problem (1) Note that the necessity to define the target space Q s 2;s=2 1I' 0 separately in the s 2 E 0 case is caused by the following: if we defined this space for s 2 E 0 in the way used in the s … E 0 case, then the isomorphism (24) would not hold at least for ' Á 1. This follows from a result by Solonnikov [39,Section 6].
Consider now the parabolic problem (1), (2), (4), which corresponds to the first order boundary condition on S. Let us write the compatibility conditions for the right-hand sides of this problem.
We associate the linear mapping ƒ 1 W u 7 ! Au; Bu; u. ; 0/ ; where u 2 C 1 . /; with the problem (1) . According to this theorem, the traces @ k t g. ; 0/ 2 H s 5=2 2k ./ are defined by closure for all k 2 Z such that 0 Ä k < s=2 5=4 (and only for these k). We can express these traces in terms of the function u.x; t / and its time derivatives; namely, for almost all x 2 . Here, all the functions u.x; 0/, @ t u.x; 0/,..., @ k t u.x; 0/ of x 2 G are expressed in terms of the functions f .x; t/ and h.x/ by the recurrent formula (17).
Substituting (17) in the right-hand side of formula (27), we obtain the compatibility conditions @ k t g D B k OEv 0 ; : : : ; v k ; with k 2 Z and 0 Ä k < s=2 5=4: Here, the functions v 0 , v 1 ,..., v k are defined on G by the recurrent formula (20), and we put B k OEv 0 ; : : : for all x 2 G. The right-hand side of the equality (28) is well defined because the function B k OEv 0 ; : : : ; v k belongs to H s 2 2k .G/ due to (21) and therefore the trace B k OEv 0 ; : : : ; v k 2 H s 5=2 2k ./ is defined by closure whenever s 5=2 2k > 0. Note that if s Ä 5=2, then there are no compatibility conditions. We set E 1 WD f2r C 1=2 W 1 Ä r 2 Zg. Observe that E 1 is the set of all discontinuities of the function that assigns the number of compatibility conditions (28) to s 2.
To formulate our isomorphism theorem for the parabolic problem (1), (2), (4), we introduce the source and target spaces of this isomorphism. Let s > 2 and ' 2 M. As above, we take H s;s=2I' . / as the source space.
This continuous embedding follows immediately from (12) and (13) with the number " 2 .0; 1=2/ chosen arbitrarily. This Hilbert space does not depend on the choice of " up to equivalence of norms and is embedded continuously in H s 2;s=2 1I' 1 , which will be shown in Remark 6.3. Now we can formulate our main result concerning the parabolic initial-boundary value problem (1), (2), (4).
Note that the necessity to define the target space Q s 2;s=2 1I' 1 separately in the s 2 E 1 case is stipulated by a similar cause as that indicated for the space Q s 2;s=2 1I' 0 . Namely, if we defined this space for s 2 E 1 in the way used in the s … E 1 case, then the isomorphism (31) would not hold at least when ' Á 1 and (4) is the Neumann boundary condition (see [39,Section 6] [39,Theorem 17] proved the corresponding a priory estimates for anisotropic Sobolev norms of solutions to the problem (1)-(3) and to the problem (1), (2), (4) provided that (4) is the Neumann boundary condition. Note that these results include the limiting case of s D 2.
In Section 6 we will deduce Theorems 4.1 and 4.2 from the above-mentioned results with the help of the method of interpolation with a function parameter between Hilbert spaces, specifically between Sobolev inner product spaces. Therefore we devote the next section to this method and its applications to Sobolev and Hörmander spaces.

Interpolation with a function parameter between Hilbert spaces
This method of interpolation is a natural generalization of the classical interpolation method by S. Krein and J.-L. Lions to the case when a general enough function is used instead of a number as an interpolation parameter; see, e.g., monographs [40, Chapter IV, Section 1, Subsection 10] and [28, Chapter 1, Sections 2 and 5]. For our purposes, it is sufficient to restrict the discussion of the interpolation with a function parameter to the case of separable complex Hilbert spaces. We mainly follow the monograph [18, Section 1.1], which systematically expounds this interpolation (see also [37,Section 2]).
Let X WD OEX 0 ; X 1 be an ordered pair of separable complex Hilbert spaces such that X 1 Â X 0 and this embedding is continuous and dense. This pair is said to be admissible. For X , there is a positive-definite self-adjoint operator J on X 0 with the domain X 1 such that kJ vk X 0 D kvk X 1 for every v 2 X 1 . This operator is uniquely determined by the pair X and is called a generating operator for X ; see, e.g., [40, Chapter IV, Theorem 1.12]. The operator defines an isometric isomorphism J W X 1 $ X 0 .
Let B denote the set of all Borel measurable functions W .0; 1/ ! .0; 1/ such that is bounded on each compact interval OEa; b, with 0 < a < b < 1, and that 1= is bounded on every semiaxis OEa; 1/, with a > 0.
Choosing a function 2 B arbitrarily, we consider the (generally, unbounded) operator .J / defined on X 0 as the Borel function of J . This operator is built with the help of Spectral Theorem applied to the selfadjoint operator J . Let OEX 0 ; X 1 or, simply, X denote the domain of .J / endowed with the inner product .v 1 ; v 2 / X WD . .J /v 1 ; .J /v 2 / X 0 and the corresponding norm kvk X WD k .J /vk X 0 . The linear space X is Hilbert and separable with respect to this norm.
A function 2 B is called an interpolation parameter if the following condition is satisfied for all admissible pairs X D OEX 0 ; X 1 and Y D OEY 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 W X j ! Y j for each j 2 f0; 1g, then the restriction of T to X is also a bounded operator T W X ! Y .
If is an interpolation parameter, then we say that the Hilbert space X is obtained by the interpolation with the function parameter of the pair X D OEX 0 ; X 1 or, otherwise speaking, between the spaces X 0 and X 1 . In this case, the dense and continuous embeddings X 1 ,! X ,! X 0 hold.
The class of all interpolation parameters (in the sense of the given definition) admits a constructive description. Namely, a function 2 B is an interpolation parameter if and only if is pseudoconcave in a neighbourhood of infinity. The latter property means that there exists a concave positive function 1 .r/ of r 1 that both the functions = 1 and 1 = are bounded in some neighbourhood of infinity. This criterion follows from Peetre's description of all interpolation functions for the weighted Lebesgue spaces [41,42] (this result of Peetre is set forth in the monograph [43,Theorem 5.4.4]). The proof of the criterion is given in [18,Section 1.1.9].
An application of this criterion to power functions gives the classical result by Lions and S. Krein. Namely, the function .r/ Á r Â is an interpolation parameter whenever 0 Ä Â Ä 1. In this case, the exponent Â serves as a number parameter of the interpolation, and the interpolation space X is also denoted by X Â . This interpolation was used in formulas (23) and (30) in the special case of Â D 1=2.
Let us formulate some general properties of interpolation with a function parameter; they will be used in our proofs. The first of these properties enables us to reduce the interpolation of subspaces to the interpolation of the whole spaces (see [18,Theorem 1.6] or [29, Section 1.17.1, Theorem 1]). As usual, subspaces of normed spaces are assumed to be closed. Generally, we consider nonorthogonal projectors onto subspaces of a Hilbert space.
Proposition 5.1. Let X D OEX 0 ; X 1 be an admissible pair of Hilbert spaces, and let Y 0 be a subspace of X 0 . Then Y 1 WD X 1 \ Y 0 is a subspace of X 1 . Suppose that there exists a linear mapping P W X 0 ! X 0 such that P is a projector of the space X j onto its subspace Y j for each j 2 f0; 1g. Then the pair OEY 0 ; Y 1 is admissible, and OEY 0 ; Y 1 D X \ Y 0 with equivalence of norms for an arbitrary interpolation parameter 2 B. Here, X \ Y 0 is a subspace of X .
The second property reduces the interpolation of orthogonal sums of Hilbert spaces to the interpolation of their summands (see [18,Theorem 1.8].
with equality of norms for every function 2 B.
The third property shows that the interpolation with a function parameter is stable with respect to its repeated fulfillment [18, Theorem 1.3].
Proposition 5.3. Let˛;ˇ; 2 B, and suppose that the function˛=ˇis bounded in a neighbourhood of infinity. Define the function ! 2 B by the formula !.r/ WD˛.r/ .ˇ.r/=˛.r// for r > 0. Then ! 2 B, and OEX˛; Xˇ D X ! with equality of norms for every admissible pair X of Hilbert spaces. Besides, if˛;ˇ; are interpolation parameters, then ! is also an interpolation parameter.
Our proof of Theorems 4.1 and 4.2 is based on the key fact that the interpolation with an appropriate function parameter between margin Sobolev spaces in (12) and (13)  '.1/ if 0 < r < 1: Then the function 2 B is an interpolation parameter, and the equality of spaces holds true up to equivalence of norms for arbitrary 2 R provided that W D G or W D . If W D R k with 1 Ä k 2 Z, then (33) holds true with equality of norms in spaces.
This result is due to [21, Theorems 3.1 and 3.5]; see also monograph [18, Theorems 1.14, 2.2, and 3.2] for the cases where W D R k , W D , and W D G respectively.

Proofs
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 D 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 2 H s;s=2I' .S /.
Lemma 6.1. Choose an integer r 1, and consider the linear mapping R W g 7 ! g ; @ t g ; : : : ; @ r 1 t g ; with g 2 C 1 .S /: and that RT v D v for every v 2 H sI' ./.
Proof. We first prove an analog of this lemma for Hörmander spaces defined on R n and R n 1 instead of S and .
Then we deduce the lemma with the help of the special local charts on S. Consider the linear mapping R 0 W w 7 ! w j t D0 ; @ t w j t D0 ; : : : ; @ r 1 t w j t D0 ; with w 2 C 1 0 .R n /: Here, we interpret w as a function w.x; t / of x 2 R n 1 and t 2 R so that R 0 w 2 .C 1 0 .R n 1 // r . Choose s > 2r 1 and ' 2 M arbitrarily, and prove that the mapping (38) extends uniquely (by continuity) to a bounded linear operator This fact is known in the Sobolev case of ' Á 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 ' 2 M. Namely, choose s 0 ; s 1 2 R such that 2r 1 < s 0 < s < s 1 and consider the bounded linear operators Let be the interpolation parameter (32). Then the restriction of the mapping (40) with j D 0 to the space is a bounded operator R 0 W H s;s=2I' .R n / ! H s 0 .R n 1 /; H s 1 .R n 1 / : The latter equality is due to Proposition 5.5. This operator is an extension by continuity of the mapping (38) because the set C 1 0 .R n / is dense in H s;s=2I' .R n /. Owing to Propositions 5.
Let us now build a linear mapping that its restriction to each space H sI' .R n 1 / with s > 2r 1 and ' 2 M is a bounded operator between the spaces H sI' .R n 1 / and H s;s=2I' .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 on the linear topological space of vectors v WD .v 0 ; : : : ; v r 1 / 2 S 0 .R n 1 / r : We consider T 0 v as a distribution on the Euclidean space R n of points .x; t /, with x D .x 1 ; : : : ; x n 1 / 2 R n 1 and t 2 R. In (45), the functionˇ2 C 1 0 .R/ is chosen so thatˇD 1 in a certain neighbourhood of zero. As usual, F 1 7 !x denotes the inverse Fourier transform with respect to D . 1 ; : : : ; n 1 / 2 R n 1 , and h i WD .1 C j j 2 / 1=2 . The variable is dual to x relative to the direct Fourier transform b w. / D .F w/. / of a function w.x/. Obviously, the mapping (45) is well defined and acts continuously between .S 0 .R n 1 / r and S 0 .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 /.
We assert that 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 2 .S.R n 1 / r implies T 0 v 2 S.R n 1 /, the left-hand side of the equality (46) is well defined. Let us prove this equality. Choosing j 2 f0; : : : ; r 1g and v D .v 0 ; : : : ; v r 1 / 2 .S.R n 1 // r arbitrarily, we get for every 2 R n 1 . In the fourth equality, we have used the fact thatˇD 1 in a neighbourhood of zero. Thus, the Fourier transforms of all components of the vectors R 0 T 0 v and v coincide, which is equivalent to (46). Let us now prove that the restriction of the mapping (45) to each space with 0 Ä m 2 Z is 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 (see, e.g., [44, Section 9.1]). Here and below in this proof, k k stands for the norm in the Hilbert space L 2 .R n /.
Of course, @ x j u and @ t denote the operators of generalized partial derivatives with respect to x j and t respectively.
Choosing v D .v 0 ; : : : ; v r 1 / 2 .S.R n 1 // r arbitrarily and using the Parseval equality, we obtain the following: Let us estimate each of these three integrals separately. We begin with the third integral. Changing the variable D h i 2 t in the interior integral with respect to t, we get the equalities Using the same changing of the variable t in the second integral, we obtain the following: Finally, replacing the symbol j with 1 in the previous reasoning, we obtain the following estimate for the first integral: Z R nˇˇ.