Elliptic operators on refined Sobolev scales on vector bundles

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


Introduction
It is well known [1,2] that elliptic differential and pseudodifferential operators on a closed infinitely smooth manifold are Fredholm between appropriate Sobolev spaces. This fundamental property is used in the theory of elliptic differential equations and elliptic boundary-value problems. However, the Sobolev scale is not sufficiently finely calibrated for some mathematical problems (see monographs [3][4][5][6][7][8][9][10]). In this connection, Hörmander [3,4] introduced and investigated a broad class of normed function spaces B p; D˚w 2 S 0 .R n / W b w 2 L p .R n / « ; kwk B p; WD k b wk L p .R n / ; where 1 Ä p Ä 1, W R n ! .0; 1/ is a weight function, and b w is the Fourier transform of a tempered distribution w. Hörmander applied these spaces to investigation of solvability of partial differential equations given in Euclidean domains and to study of regularity of solutions to these equations.
Nevertheless, the class of all spaces B p; is too general for applications to differential equations on manifolds and boundary-value problems. Among these spaces, Mikhailets and Murach [11][12][13] selected the class of inner product spaces H s;' WD B 2; parametrized with the function . / D h i s '.h i/, where s 2 R, the function ' W OE1; 1/ ! .0; 1/ varies slowly at infinity in the sense of Karamata [14,15], and h i D .1 C j j 2 / 1=2 . This class is called the refined Sobolev scale. It contains the inner product Sobolev spaces H s D H s;1 and is obtained by the interpolation with a function parameter between these spaces. This interpolation property allowed Mikhailets and Murach [11][12][13][16][17][18][19][20][21] to build the theory of solvability of general elliptic systems and elliptic boundary-value problems on the refined Sobolev scale. Their theory [7] is supplemented in [22][23][24][25][26][27][28][29] for a more extensive class of Hörmander inner product spaces. The refined Sobolev scale and other classes of Hörmander spaces are applied to the spectral theory of elliptic differential operators on manifolds [7,Section 2.3], theory of interpolation of normed spaces [23,30], to some differential-operator equations [31], parabolic initial-boundary value problems [32][33][34][35], and in mathematical physics [36,37].
However, elliptic operators on vector bundles have not been covered by this theory. These operators have important applications to elliptic boundary problems on vector bundles, elliptic complexes, spectral theory of elliptic differential operators, and others (see, e.g., [1,2,38]).
The goal of this paper is to introduce and investigate the refined Sobolev scale on an arbitrary vector bundle over infinitely smooth closed manifold and to give applications of this scale to general elliptic pseudodifferential operators on vector bundles.
The paper consists of eight sections. Section 1 is Introduction. In Section 2, we introduce the refined Sobolev scale on the vector bundle. Section 3 is devoted to the method of interpolation with a function parameter between Hilbert spaces. This method plays a key role in the paper. Section 4 contains main results concerning properties of the refined Sobolev scale introduced. Section 5 presents main results regarding the properties of elliptic pseudodifferential operators on this scale. Section 6 contains some auxiliary facts. The main results of the paper formulated in Sections 4 and 5 are proved in Sections 7 and 8 respectively.

The refined Sobolev scale on a vector bundle
The refined Sobolev scale on R n and smooth manifolds was introduced and investigated by Mikhalets and Murach [13,39]. This scale consists of the inner product Hörmander spaces H s;' with s 2 R and ' 2 M. Let us give the definition of the function class M and the space H s;' .R n /. The latter will be a base for our definition of the refined Sobolev scale on vector bundles.
The class M consists of all Borel measurable functions ' W OE1; 1/ ! .0; 1/ that satisfy the following two conditions: (i) both functions ' and 1=' are bounded on each compact interval OE1; b with 1 < b < 1; (ii) the function ' varies slowly at infinity in the sense of Karamata [14], i.e. Slowly varying functions are well investigated and play an important role in mathematical analysis and its applications (see monographs [40][41][42]). A standard example of a function ' 2 M is given by a continuous function ' W OE1; 1/ ! .0; 1/ such that '.t / WD .log t / r 1 .log log t / r 2 : : : .log : : : log where 0 Ä k 2 Z and r 1 ; : : : ; r k 2 R. The class M admits the following description (see, e.g., [42, Section 1.2]): Here,˛is a continuous function on OE1; 1/ such that˛. / ! 0 as ! 1, andˇis a Borel measurable function on OE1; 1/ such thatˇ.t / ! l as t ! 1 for some l 2 R. Let s 2 R and ' 2 M. By definition, the complex linear space H s;' .R n /, with 1 Ä n 2 Z, consists of all distributions w 2 S 0 .R n / such that their Fourier transform b w is locally Lebesgue integrable over R n and satisfies the condition Z R n h i 2s ' 2 .h i/ jb w. /j 2 d < 1: Here, S 0 .R n / is the complex linear topological space of all tempered distributions on R n , and h i D .1 C j j 2 / 1=2 . An inner product in H s;' .R n / is defined by the formula with w 1 ; w 2 2 H s;' .R n /. This inner product endows H s;' .R n / with the Hilbert spaces structure and induces the norm kwk s;'IR n WD .w; w/ 1=2 s;'IR n : The space H s;' .R n / is separable with respect to this norm, and the set C 1 0 .R n / is dense in this space. Here, as usual, C 1 0 .R n / stands for the set of all infinitely differentiable compactly supported functions w W R n ! C. In this paper, we consider complex-valued functions and distributions; hence, all function spaces are supposed to be complex. Besides, we interpret distributions as antilinear functionals on corresponding spaces of test functions.
If ' D 1, then the space H s;' .R n / coincides with the inner product Sobolev space H s .R n / of order s. Generally, we have the continuous embeddings H sC" .R n / ,! H s;' .R n / ,! H s " .R n / for any " > 0: (2) They show that the function parameter ' defines a supplementary regularity with respect to the main (power) regularity s. Briefly saying, ' refines the main regularity s. Following [7,23], we call the class of function spaces the refined Sobolev scale on R n . Let be a closed (i. e. compact and without boundary) infinitely smooth real manifold of dimension n 1. We suppose that a certain C 1 -density dx is given on . Let W V ! be an infinitely smooth complex vector bundle of rank p 1 on . Here, V is the total space of the bundle, is the base space, and is the projector (see, e.g., [2, Chapter I, Section 2]). Let C 1 .; V / denote the complex linear space of all infinitely differentiable sections u W ! V . Note that u.x/ 2 1 .x/ for every x 2 and that 1 .x/ is a complex vector space of dimension p (this space is called the fiber over x). Let us introduce the Hörmander space H s;' .; V / on this vector bundle. From the C 1 -structure on , we choose a finite atlas consisting of local charts˛j W R n $ j with j D 1; : : : ;~. Here, the open sets j form a finite covering of . We choose these sets so that the local trivializationˇj W 1 . j / $ j C p is defined. We also choose real-valued functions j 2 C 1 ./, j D 1; : : : ;~, that satisfy the condition supp j j and form a partition of unity on .
Let s 2 R and ' 2 M. We introduce the norm on C 1 .; V / by the formula Here, u 2 C 1 .; V /, and the projector … k is defined as follows: … k W .x; a/ 7 ! a k for all x 2 and a D .a 1 ; : : : ; a p / 2 C p . We put for arbitrary j 2 f1; : : :~g and k 2 f1; : : : ; pg. Note that if u 2 C 1 .; V /, then each u j;k 2 C 1 0 .R n /; hence, the norms on the right-hand side of (4) are well defined. The norm (4) is Hilbert because it is induced by the inner product .u j;k ; v j;k / s;'IR n (6) of sections u; v 2 C 1 .; V /. Let H s;' .; V / be the completion of the linear space C 1 .; V / with respect to the norm (4) (and the corresponding inner product (6)). Thus, we have the Hilbert space H s;' .; V /. This space does not depend up to equivalent of norms on our choice of the atlas f˛j g, partition of unity f j g, and local trivializations fˇj g. This will be proved below as Theorem 4.2. By analogy with (3) we call the class of Hilbert function spaces the refined Sobolev scale on the bundle W V ! .
If ' D 1, then H s;' .; V / becomes the inner product Sobolev space H s .; V / of order s 2 R (see e.g. [2, Chapter IV, Section 1]). In the Sobolev case of ' D 1, we will omit the index ' in our designations concerning the Hörmander spaces H s;' . /. Specifically, k k sI;V denotes the norm in the Sobolev space H s .; V /.
In the case of trivial vector bundle of rank p D 1, the space H s;' .; V / consists of distributions on and is denoted by H s;' ./. The space H s;' ./ was introduced and investigated by Mikhailets and Murach [13,39].

Interpolation with function parameter between Hilbert spaces
The refined Sobolev scale on the vector bundle W V ! possesses an important interpolation property. Namely, every space H s;' .; V /, with s 2 R and ' 2 M, is the result of the interpolation with an appropriate function parameter between the Sobolev spaces H s " .; V / and H sCı .; V /, where "; ı > 0. We will systematically use this property in the paper. Therefore we recall the definition of interpolation with function parameter between Hilbert spaces and discuss some of its properties. We restrict ourselves to the case of separable complex Hilbert spaces and mainly follow monograph [7, Section 1.1]. Note that the interpolation with function parameter between normed spaces was introduced by Foiaş and Lions [43], who separately considered the case of Hilbert spaces.
Let X WD OEX 0 ; X 1 be an ordered pair of separable complex Hilbert spaces X 0 and X 1 such that X 1 X 0 with the continuous and dense embedding. This pair is said to be admissible. For X there exists an isometric isomorphism J W X 1 $ X 0 that J is a self-adjoint positive-definite operator in X 0 with the domain X 1 . The operator J is uniquely determined by the pair X and is called a generating operator for this pair.
Let B denote the set of all Borel measurable functions W .0; 1/ ! .0; 1/ that is bounded on every compact interval OEa; b, with 0 < a < b < 1, and that 1= is bounded on every set OEr; 1/, with r > 0. Given 2 B, consider the operator .J / defined as the Borel function of the self-adjoint operator J with the help of Spectral Theorem. The operator .J / is (generally) unbounded and positive-definite in X 0 . Let OEX 0 ; X 1 or, simply, X denote the domain of .J / endowed with the inner product .u 1 ; u 2 / X WD . .J /u 1 ; .J /u 2 / X 0 and the corresponding norm kuk X D k .J /uk X 0 . The space X is Hilbert and separable.
A function 2 B is said to be an interpolation parameter if the following condition is fulfilled for each 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 between X 0 and X 1 (or of the pair X ). In this case, the continuous and dense embeddings X 1 ,! X ,! X 0 hold true. The function 2 B is an interpolation parameter if and only if is pseudoconcave in a neighborhood of C1. The latter property means that there exists a concave function 1 W .b; 1/ ! .0; 1/, with b 1, that both functions = 1 and 1 = are bounded on .b; 1/. This criterion follows from Peetre's [44,45] description of all interpolation functions for the weighted Lebesgue spaces (see [7,Theorem 1.9]). Specifically, every function 2 B of the form .t / Á t Â 0 .t /, where 0 < Â < 1 and 0 varies slowly at infinity, is an interpolation parameter.
Let us formulate the above-mentioned interpolation property of the refined Sobolev scale on R n [7, Theorem 1.14].  '.1/ if 0 < t < 1: Then 2 B is an interpolation parameter, and with equality of norms.

Properties of the refined Sobolev scale on vector bundle
Let us formulate the main results of the paper concerning properties of the refined Sobolev scale (7) on the vector bundle W V ! .   Suppose now that the vector bundle W V ! is Hermitian. Thus, for every x 2 , a certain inner product h ; i x is defined in the fiber 1 .x/ so that the scalar function 3 x 7 ! hu.x/; v.x/i x is infinitely smooth on for arbitrary sections u; v 2 C 1 .; V /. Using the C 1 -density dx on , we define the inner product of these sections by the formula In view of this theorem note that ' 2 M , 1=' 2 M; hence, the space H s;1=' .; V / is well defined.
Theorem 4.5. Let 0 Ä q 2 Z and ' 2 M. Then the condition is equivalent to that the identity mapping u 7 ! u, with u 2 C 1 .; V /, extends uniquely to a continuous embedding H qCn=2;' .; V / ,! C q .; V /. Moreover, this embedding is compact.
Here, of course, C q .; V / denotes the Banach space of all q times continuously differentiable sections u W ! V . The norm in this space is defined by the formula where each u j;k 2 C q b .R n / is given by (5). Here, C q b .R n / denotes the Banach space of all q times continuously differentiable functions on R n whose partial derivatives up to the q-th order are bounded on R n . This space is endowed with the norm kwk .q/IR n WD X 1 C C n Äq sup t 2R nˇ@ 1 C:::C n w.t / @t 1 1 ; : : : ; @t n nǒ f a function w.
We will prove Theorems 4.1-4.5 in Section 7. In the case where W V ! is a trivial vector bundle of rank p D 1, they are established by Mikhailets and Murach [39] (see also their monograph [7, Section 2.1.2]).

Elliptic operators on the refined Sobolev scale on a vector bundle
Consider elliptic PsDOs on a pair of vector bundles on . Let 1 W V 1 ! and 2 W V 2 ! be two infinitely smooth complex vector bundles of the same rank p 1 on . We choose the atlas f˛j W R n ! j g so that both the local trivializationsˇ1 ;j W ; 2g is Hermitian. Let hu; vi ;V k denote the corresponding inner product of sections u; v 2 C 1 .; V k / and its extension by continuity indicated in Theorem 4.4.
Given m 2 R, we let ‰ m ph .I V 1 ; V 2 / denote the class of all polyhomogeneous (classical) PsDOs A W C 1 .; V 1 / ! C 1 .; V 2 / of order m (see, e.g., [2, Chapter IV, Section 3]). Recall that if is an open nonempty subset of with j for some j 2 f1; : : : ;~g, then a PsDO for every section u 2 C 1 .; V 1 / and arbitrary scalar functions '; 2 C 1 ./ with supp ' and supp . Here, A is a certain p p-matrix whose entries are polyhomogeneous PsDOs on R n of order m, and … is the projector defined by the formula … W .x; a/ 7 ! a for arbitrary x 2 and a 2 C p .
Hereafter we let m 2 R and suppose that A is an arbitrary elliptic PsDO from the class ‰ m ph .I V 1 ; V 2 /. The ellipticity of A is equivalent to that each operator A D .A l;r / p l;rD1 from (15) is elliptic on the set˛ 1 j . /, i.e. det.a l;r ;0 .x; // p l;rD1 ¤ 0 for all x 2˛ 1 j . / and 2 R n n f0g, with a l;r ;0 .x; / being the principal symbol of the scalar PsDO A l;r on R n . Put Since the PsDOs A and A C are elliptic, the spaces N and N C are finite-dimensional (see, e.g., [2, Theorem 4.8]).
This operator is Fredholm. Its kernel is N, and its domain The index of operator (18) is equal to dim N dim N C and does not dependent on s and '.
Recall that a bounded linear operator T W E 1 ! E 2 between Banach spaces E 1 and E 2 is called Fredholm if its kernel ker T and co-kernel coker T WD E 2 =T .X / are finite-dimensional. If the operator T is Fredholm, then its domain T .X/ is closed in E 2 and its index ind T WD dim ker T dim coker T is finite (see, e.g. [1, Lemma 19.1.1]). If N D f0g and N C D f0g, then operator (18) is an isomorphism between the spaces H sCm;' .; V 1 / and H s;' .; V 2 / by virtue of the Banach theorem on inverse operator. In the general situation, this operator induces an isomorphism between their certain subspaces of finite codimension. In this connection consider the following decompositions of these spaces into direct sums of their subspaces: These decompositions are well defined because the summands in them have the trivial intersection and the finite dimension of the first summand is equal to the codimension of the second one. This equality is due to the following fact: if we consider N as a subspace of H s m;1=' .; V 1 /, then the dual of N with respect to the form h ; i ;V 1 coincides with the second summand in (20) according to Theorem 4.4, analogous reasoning being valid for (21). Let P and P C denote the oblique projectors of the spaces H sCm;' .; V 1 / and H s;' .; V 2 / onto the second summands parallel to the first summands in (20) and (21) respectively. These projectors are independent of s and '.
The solutions u 2 H sCm;' .; V 1 / to the elliptic equation Au D f satisfy the following a priori estimate.
If s C m 1 < < s C m, then we can take Á WD in (23); this follows in view of Theorem 4.3 from the estimate k uk sCm;'I;V 1 Ä c k f k s;'I;V 2 C kuk sCm 1;'I;V 1 : Consider the local regularity of the solutions to the elliptic equation Au D f . Given j 2 f1; 2g, we put  As an application of the refine Sobolev scale, we give the following result: Theorem 5.6. Let 0 Ä q 2 Z. Suppose that a section u 2 H 1 .; V 1 / is a solution to the elliptic equation Au D f on where f 2 H q mCn=2;' loc . 0 ; V 2 / for a certain function ' 2 M subject to condition (13). Then u 2 C q loc . 0 ; V 1 /.
Here, C q loc . 0 ; V 1 / denotes the linear space of all sections u 2 H 1 .; V 1 / such that u 2 C q .; V 1 / for arbitrary 2 C 1 ./ with supp 0 .
Remark 5.7. Let ' 2 M. Condition (13) is sharp in Theorem 5.6. Namely, this condition is equivalent to the implication We will prove Theorems 5.

Auxiliary results
We will use three properties of the interpolation with a function parameter. The first of them reduces the interpolation between orthogonal sums of Hilbert spaces to the interpolation between the summands (see, e.g., [7, Theorem 1.5]).
, with j D 1; : : : ; r, be a finite collection of admissible couples of Hilbert spaces. Then for every function 2 B we have with equality of norms.
The second property shows that this interpolation preserves the Fredholm property of the bounded operators that have the same defect (see, e.g., [7, Theorem 1.7]).
Proposition 6.2. Let X D OEX 0 ; X 1 and Y D OEY 0 ; Y 1 be admissible pairs of Hilbert spaces, and let 2 B be an interpolation parameter. Suppose that a linear mapping T is given on X 0 and satisfies the following property: the restrictions of T to the spaces X j , where j D 0; 1, are Fredholm bounded operators T W X j ! Y j that have a common kernel and the same index. Then the restriction of T to the space X is a Fredholm bounded operator T W X ! Y with the same kernel and index and, besides, T .X / D Y \ T .X 0 /.
The third property reduces the interpolation between the dual or antidual spaces of given Hilbert spaces to the interpolation between these given spaces (see [7,Theorem 1.4]). We need this property in the case of antidual spaces. If H is a Hilbert space, then H 0 stands for the antidual of H ; namely, H 0 consists of all antilinear continuous functionals l W H ! C. The linear space H 0 is Hilbert with respect to the inner product .l 1 ; l 2 / H 0 WD .v 1 ; v 2 / H of functionals l 1 ; l 2 2 H 0 ; here v j , with j 2 f1; 2g, is a unique vector from H such that l j .w/ D .v j ; w/ H for every w 2 H . Note that we do not identify H and H 0 on the base of the Riesz theorem (according to which v j exists). Proposition 6.3. Let a function 2 B be such that the function .t /=t is bounded in a neighbourhood of infinity. Then for every admissible pair OEX 0 ; X 1 of Hilbert spaces we have the equality OEX 0 1 ; X 0 0 D OEX 0 ; X 1 0 with equality of norms. Here, the function 2 B is defined by the formula .t / WD t = .t / for t > 0. If is an interpolation parameter, then is an interpolation parameter as well.
In view of this theorem we note that if OEX 0 ; X 1 is an admissible pair of Hilbert spaces, then the dual pair OEX 0 1 ; X 0 0 is also admissible provided that we identify functions from X 0 0 with their restrictions on X 1 .

Proofs of properties of the refined Sobolev scale
In this section we will prove Theorems 4.1-4.5.
in which the function Á j 2 C 1 0 .R n / is chosen so that Á j D 1 on the set˛ 1 j .supp j /. We have the linear mapping K W .C 1 0 .R n // p~! C 1 .; V /: It is left inverse to the flattening mapping (27). Indeed, given u 2 C 1 .; V /, we write K T u D K.u 1;1 ; : : : ; u 1;p ; : : : ; u~; 1 ; : : : ; u~; p / DX j D1 u j ; where each section u j 2 C 1 .; V / is defined by formula (35) with u instead of w. In this formula, for arbitrary k 2 f1; : : : ; pg and x 2 j , we have the equalities .Á j u j;k /.˛ 1 j .x// D Á j .˛ 1 j .x// … k ˇj .. j u/.x// D … k ˇj .. j u/.x// due to our choice of Á j . Therefore Here, Á j;l WD . l ı˛l /Á j 2 C 1 0 .R n /, whereas˛j ;l W R n $ R n is an infinitely smooth diffeomorphism such that˛j ;l WD˛ 1 j ı˛l in a neighbourhood of supp Á j;l and that˛j ;l .t / D t whenever jtj 1. Then, given k 2 f1; : : : ; pg, we have the equalities Here, eachˇk ;r l;j is a certain complex-valued function from C 1 ./ such that the matrix-valued function .ˇk ;r l;j .x// p k;rD1 of x 2 supp l \ supp.Á j ı˛ 1 j / corresponds to the transition mappingˇl ıˇ 1 j . Thus, ;r l;j .˛l .t // ..Á j;l w j;r / ı˛j ;l /.t / for arbitrary t 2 R (if˛l .t / 6 2 j , then this equality becomes 0 D 0). Owing to (38) and (39) Taking here 2 fs "; s C ıg and using the interpolation with the function parameter , we conclude that the restriction of the operator (43) It follows from equality (37) and from the boundedness of operators (44) and (28) that kuk X D kK T uk X Ä c 1 kuk s;'I;V for every u 2 C 1 .; V /; where c 1 is the norm of the product of these operators, and X WD OEH s " .; V /; H sCı .; V / : Besides, the boundedness of operators (42) and (32)  with c 2 being the norm of the product of the last two operators. Thus, the norms in the spaces H s;' .; V / and X are equivalent on the linear manifold C 1 .; V /. Since this manifold is dense in these spaces, they coincide up to equivalence of norms (the set C 1 .; V / is dense in X due to (8)). for each 2 R. Considering this isomorphism for WD fs "; s C ıg and using the interpolation with the function parameter defined by formula (9), we conclude that the identity mapping is an isomorphism Our proof of the next Theorem 4.5 is based on the following result. 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 2 C 1 .; V / we have the inequality Here, the first equality is due to (14), and c is the norm of the continuous embedding operator H qCn=2;' .R n / ,! C q b .R n /, which holds due to (13) and Proposition 7.1. Hence, the identity mapping I W u 7 ! u, with u 2 C 1 .; V /, extends uniquely (by continuity) to a linear bounded operator If this operator is injective, then it sets the continuous embedding of H qCn=2;' .; V / in C q .; V /. Let us prove the injectivity of (47). Consider the isometric flattening operator (28) with s D q C n=2. It is an extension by continuity of mapping (27). This mapping is well defined on functions u 2 C q .; V / and sets an isometric operator Let us now prove that this embedding is compact. Without loss of generality we may consider ' 2 M as a continuous function on OE1; 1/. Indeed, as is known [42, Section 1.4], there exists a continuous function ' 1 2 M that both functions '=' 1 and ' 1 =' are bounded on OE1; 1/. Therefore the spaces H qCn=2;' .; V / and H qCn=2;' 1 .; V / are equal up to equivalence of norms. Then we may use the second space instead of the first in our reasoning. We put for arbitrary t 1: Owing to [7,Lemma 1.4], the function ' 0 belongs to M and has the following two properties: It follows from the first property that we have the compact embedding 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 1 6 . 2 [ [ ~/. Therefore, there exists a nonempty open set U 1 such that 1 .x/ D 1 for every x 2 U . We arbitrarily choose a function w 2 H qCn=2;' .R n / such that supp w ˛ 1 1 .U /. Turn to the operator K defined by formulas (33)- (35). According to (42) with s D q C n=2 and owing to our assumption, we have the inclusion K.w; 0; : : : ; 0 Let us deduce from this inclusion that w 2 C q .R n /.
To this end we introduce the linear mapping It acts continuously from C q .; V / to C q b .R n /. Besides, the operator K acts continuously from .H qCn=2;' .R n // p~t o C q .; V / according to (42) with s D q C n=2 and our assumption. Hence, T 1 K.w; 0; : : : ; 0/ D w; this equality is evident if additionally w 2 C 1 0 .R n / and then extends by closure over each function w chosen above. Now w D T 1 K.w; 0; : : : ; 0/ 2 C q .R n / Thus, we obtain embedding (46) with G WD˛ 1 1 .U /. It implies condition (13) due to Proposition 7.1.

Proofs of properties of elliptic operators on the refined Sobolev scale
Beforehand we will prove the following result: Proof of Theorem 5:1. According to Lemma 8.1 the mapping u 7 ! Au, with u 2 C 1 .; V 1 /, extends by continuity to the bounded linear operator (18). Let us prove that this operator is Fredholm.
Using the interpolation with the function parameter defined by formula (9) with " D ı D 1, we conclude by Proposition 6.2 that the bounded operator is also Fredholm. According to Proposition 3.1 this operator coincides with (18). Moreover, owing to Proposition 6.2, the kernel of the Fredholm operator (18) equals N, the index equals dim N dim N C , and the range is in view of (53).
Proof of Theorem 5:2. Owing to Theorem 4.1, N is the kernel and P C .H s;' .; V 2 // is the range of the operator (18). Hence, the restriction of (18) to the subspace P .H sCm;' .; V 1 // is the bijective linear bounded operator (22). This operator is an isomorphism by the Banach theorem on inverse operator.  (24) we may take not only < s C m but also arbitrary 2 R.) We will deduce (23) from (24). Beforehand, let us prove the following result: for each integer r 1 and for arbitrary functions ; Á from Theorem 5.3 there exists a number c > 0 such that k uk sCm;'I;V 1 Ä c kÁAuk s;'I;V 2 C kÁuk sCm r;'I;V 1 C kuk I;V 1 (54) for every u 2 H sCm;' .; V 1 /. According to (24) there exists a number c 0 > 0 such that k uk sCm;'I;V 1 Ä c 0 kA. u/k s;'I;V 2 C k uk I;V 1 (55) for arbitrary u 2 H sCm;' .; V 1 /. Rearranging the PsDO A and the operator of the multiplication by , we arrived at the formula Here, A 0 is a certain PsDO from ‰ m 1 ph .I V 1 ; V 2 / (see, e.g., [38, p. 13]), and the PsDO u 7 ! A..Á 1/u/ belongs to each class ‰ ph .I V 1 ; V 2 / with 2 R because supp \ supp.Á 1/ D ;. Therefore, owing to Lemma 8.1, we obtain the inequalities kA. u/k s;'I;V 2 Ä k Auk s;'I;V 2 C k A..Á 1/u/k s;'I;V 2 C kA 0 .Áu/k s;'I;V 2 Ä k Auk s;'I;V 2 C c 1 kuk 1;'I;V 1 C c 2 kÁuk sCm 1;'I;V 1 Ä k Auk s;'I;V 2 C c 1 c 3 kuk I;V 1 C c 2 kÁuk sCm 1;'I;V 1 : Here, c 1 is the norm of the operator u 7 ! A..Á 1/u/ that acts continuously from H 1;' .; V 1 / to H s;' .; V 2 /, and c 2 is the norm of the bounded operator with e c being the norm of the bounded operator v 7 ! v on the space H s;' .; V 2 /. Thus, we have proved (54) for r D 1.
Choose an integer k 1 arbitrarily and assume that (54) is true for r D k. Let us prove that (54) is also true for r D k C 1. We choose a function Á 1 2 C 1 ./ such that Á 1 D 1 in a neighbourhood of supp and that Á D 1 in a neighbourhood of supp Á 1 . According to our assumption, there exists a number c 5 > 0 such that k uk sCm;'I;V 1 Ä c 5 kÁ 1 Auk s;'I;V 2 C kÁ 1 uk sCm k;'I;V 1 C kuk I;V 1 (59) for arbitrary u 2 H sCm;' .; V 1 /. Owing to (24) we write kÁ 1 uk sCm k;'I;V 1 Ä c 6 kA.Á 1 u/k s k;'I;V 2 C kÁ 1 uk I;V 1 I here, c 6 is a certain positive number that does not depend on u. Rearranging the PsDO A and the operator of the multiplication by Á 1 , we obtain Here, A 0 1 is a certain PsDO from ‰ m 1 ph .I V 1 ; V 2 /, and the PsDO u 7 ! Á 1 A..Á 1/u/ belongs to each class ‰ ph .I V 1 ; V 2 / with 2 R because supp Á 1 \ supp.Á 1/ D ;. Therefore, owing to Lemma 8.1, we obtain the inequalities kA.Á 1 u/k s k;'I;V 2 Ä kÁ 1 Auk s k;'I;V 2 C kÁ 1 A..
Here, c 9 is the norm of the bounded operator v 7 ! Á 1 v on the space H s;' .; V 2 /. Now formulas (63) and (64) give the inequality (54) with r D k C 1. Owing to the principle of mathematical induction, this inequality is true for each integer r 1.
The required estimate (23) follows from the inequality (54), where r 2 Z such that s C m r < , in view of kÁuk sCm r;'I;V 1 Ä c 10 kÁuk I;V 1 Ä c 10 c 11 kuk I;V 1 : Here, c 10 is the norm of the embedding operator H .; V 1 / ,! H sCm r;' .; V 1 /, and c 11 is the norm of the operator u 7 ! Áu on the space H .; V 1 / As to Remark 5.4 note that inequality (25) follows from (58) with < s C m 1 in view of Theorem 4.3. Proof of Theorem 5:6. Owing to Theorem 5.5 where s WD q m C n=2 we have the inclusion u 2 H qCn=2;' loc . 0 ; V 1 /. We arbitrarily choose a function 2 C 1 ./ such that supp 0 . Then u 2 H qCn=2;' .; V 1 / ,! C q .; V 1 / due to condition (13) and Theorem 4.5. Therefore u 2 C q . 0 ; V 1 /.