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anach

ournal of

athematical

### A

nalysis ISSN: 1735-8787 (electronic)

www.emis.de/journals/BJMA/

THE REFINED SOBOLEV SCALE,

INTERPOLATION, AND ELLIPTIC PROBLEMS

VLADIMIR A. MIKHAILETS1 AND ALEKSANDR A. MURACH2∗

Abstract. The paper gives a detailed survey of recent results on elliptic prob- lems in Hilbert spaces of generalized smoothness. The latter are the isotropic ormander spacesHs,ϕ:=B2,µ, withµ(ξ) =hξisϕ(hξi) forξRn. They are parametrized by both the real numbers and the positive function ϕvarying slowly at +∞ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale{Hs} ≡ {Hs,1} and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continu- ous derivatives are found. Some applications to the spectral theory of elliptic operators are given.

1. Introduction

In the theory of partial differential equations, the questions concerning the exis- tence, uniqueness, and regularity of solutions are in the focus of investigations.

Note that the regularity properties are usually formulated in terms of the belong- ing of solutions to some standard classes of function spaces. Thus, the finer a used scale of spaces is calibrated, the sharper and more informative results will be.

In contrast to the ordinary differential equations with smooth coefficients, the above questions are complicated enough. Indeed, some linear partial differential

Date: Received: 8 May 2012; Accepted: 7 June 2012.

Corresponding author.

2010Mathematics Subject Classification. Primary 46E35; Secondary 35J40.

Key words and phrases. Sobolev scale, H¨ormander spaces, interpolation with function pa- rameter, elliptic operator, elliptic boundary-value problem.

211

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equations with smooth coefficients and right-hand sides are known to have no solutions in a neighbourhood of a given point, even in the class of distributions , [43, Sec. 6.0 and 7.3], [45, Sec. 13.3]. Next, certain homogeneous equations (specifically, of elliptic type) with smooth but not analytic coefficients have non- trivial solutions supported on a compact set , [45, Theorem 13.6.15]. Hence, the nontrivial null-space of this equation cannot be removed by any homogeneous boundary-value conditions; i.e., the operator of an arbitrary boundary-value prob- lem is not injective. Finally, the question about the regularity of solutions is not simple either. For example, it is known [32, Ch. 4, Notes] that

4u=f ∈ C(Ω); u∈C2(Ω),

with4 being the Laplace operator, and Ω being an arbitrary Euclidean domain.

These questions have been investigated most completely for the elliptic equa- tions, systems, and boundary-value problems. This was done in the 1950s and 1960s by S. Agmon, A. Douglis, L. Nirenberg, M.S. Agranovich, A.C. Dynin, Yu.M. Berezansky, S.G. Krein, Ya.A. Roitberg, F. Browder, L. Hermander, J.- L. Lions, E. Magenes, M. Schechter, L.N. Slobodetsky, V.A. Solonnikov, L.R. Vo- levich and some others (see, e.g., M.S. Agranovich’s surveys [7, 8] and the ref- erences given therein). Note that the elliptic equations and problems have been investigated in the classical scales of H¨older spaces (of noninteger order) and Sobolev spaces (both of positive and negative orders).

The fundamental result of the theory of elliptic equations consists in that they generate bounded and Fredholm operators (i.e., the operators with finite index) between appropriate function spaces. For instance, let Au = f be an elliptic linear differential equation of order m given a closed smooth manifold Γ. Then the operator

A: Hs+m(Γ)→Hs(Γ), s∈R,

is bounded and Fredholm. Moreover, the finite-dimensional spaces formed by solutions to homogeneous equations Au = 0 and A+v = 0 both lie in C(Γ).

HereA+ is the formally adjoint operator to A, whereas Hs+m(Γ) and Hs(Γ) are inner product Sobolev spaces over Γ and of the orders s+m and s respectively.

It follows from this that the solution uhave an important regularity property on the Sobolev scale, namely

(f ∈Hs(Γ) for some s ∈R) ⇒ u∈Hs+m(Γ). (1.1) If the manifold has a boundary, then the Fredholm operator is generated by an elliptic boundary-value problem for the equation Au =f, specifically, by the Dirichlet problem.

Some of these results were extended by H. Triebel [133, 134] and the sec- ond author [92, 93] of the survey to finer scales of function spaces, namely the Nikolsky–Besov, Zygmund, and Lizorkin–Triebel scales.

The results mentioned above have various applications in the theory of differen- tial equations, mathematical physics, the spectral theory of differential operators;

see M.S. Agranovich’s surveys [7, 8] and the references therein.

As for applications, especially to the spectral theory, the case of Hilbert spaces is of the most interest. Until recently, the Sobolev scale had been a unique scale

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of Hilbert spaces in which the properties of elliptic operators were investigated systematically. However, it turns out that this scale is not fine enough for a number of important problems.

We will give two representative examples. The first of them concerns with the smoothness properties of solutions to the elliptic equation Au = f on the manifold Γ. According to Sobolev’s Imbedding Theorem, we have

Hσ(Γ)⊂Cr(Γ) ⇔ σ > r+n/2, (1.2) where the integer r ≥ 0 and n := dim Γ. This result and property (1.1) allow us to investigate the classical regularity of the solution u. Indeed, if f ∈ Hs(Γ) for some s > r−m +n/2, then u ∈ Hs+m(Γ) ⊂ Cr(Γ). However, this is not true for s = r − m + n/2; i.e., the Sobolev scale cannot be used to express unimprovable sufficient conditions for belonging of the solution u to the class Cr(Γ). An analogous situation occurs in the theory of elliptic boundary-value problems too.

The second demonstrative example is related to the spectral theory. Suppose that the differential operator A is of order m >0, elliptic on Γ, and self-adjoint on the space L2(Γ). Given a functionf ∈L2(Γ), consider the spectral expansion

f =

X

j=1

cj(f)hj, (1.3)

where (hj)j=1 is a complete orthonormal system of eigenfunctions ofA, andcj(f) is the Fourier coefficient off with respect to hj. The eigenfunctions are enumer- ated so that the absolute values of the corresponding eigenvalues form a (non- strictly) increasing sequence. According to the Menshov–Rademacher theorem, which are valid for the general orthonormal series too, the expansion (1.3) con- verges almost everywhere on Γ provided that

X

j=1

|cj(f)|2 log2(j+ 1)<∞. (1.4) This hypotheses cannot be reformulated in equivalent manner in terms of the belonging off to Sobolev spaces because

kfk2Hs(Γ)

X

j=1

|cj(f)|2j2s

for every s >0. We may state only that the conditionf ∈Hs(Γ) for some s >0 implies convergence of the series (1.3) almost everywhere on Γ. This condition does not adequately express the hypotheses (1.4) of the Menshov–Rademacher theorem.

In 1963 L. H¨ormander [43, Sec. 2.2] proposed a broad and informative gen- eralization of the Sobolev spaces in the category of Hilbert spaces (also see [45, Sec. 10.1]). He introduced spaces that are parametrized by a general enough weight function, which serves as an analog of the differentiation order or smooth- ness index used for the Sobolev spaces. In particular, H¨ormander considered the

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following Hilbert spaces

B2,µ(Rn) :=

u: µFu∈L2(Rn) , (1.5) kukB2,µ(Rn) :=kµFukL2(Rn).

HereFu is the Fourier transform of a tempered distribution ugiven on Rn, and µis a weight function of n arguments.

In the case where

µ(ξ) =hξis, hξi:= (1 +|ξ|2)1/2, ξ∈Rn, s∈R, we have the Sobolev space B2,µ(Rn) = Hs(Rn) of differentiation orders.

The H¨ormander spaces occupy a central position among the spaces of general- ized smoothness, which is characterized by a function parameter, rather than a number. These spaces are under various and profound investigations; a good deal of the work was done in the last decades. We refer to G.A. Kalyabin and P.I. Li- zorkin’s survey , H. Triebel’s monograph [135, Sec. 22], the recent papers by A.M. Caetano and H.-G. Leopold , W. Farkas, N. Jacob, and R.L. Schilling , W. Farkas and H.-G. Leopold , P. Gurka and B. Opic , D.D. Haroske and S.D. Moura [38, 39], S.D. Moura , B. Opic and W. Trebels , and references given therein. Various classes of spaces of generalized smoothness ap- pear naturally in embedding theorems for function spaces, the theory of inter- polation of function spaces, approximation theory, the theory of differential and pseudodifferential operators, theory of stochastic processes; see the monographs by D.D. Haroske , N. Jacob , V.G. Maz’ya and T.O. Shaposhnikova [66, Sec. 16], F. Nicola and L. Rodino , B.P. Paneah , A.I. Stepanets [130, Part I, Ch. 3, Sec. 7.1], and also the papers by F. Cobos and D.L. Fernandez , C. Merucci , M. Schechter  devoted to the interpolation of function spaces, and the papers by D.E. Edmunds and H. Triebel [23,24], V.A. Mikhailets and V.M. Molyboga [72, 73, 74] on spectral theory of some elliptic operators appearing in mathematical physics.

Already in 1963 L. H¨ormander applied the spaces (1.5) and more general Ba- nach spacesBp,µ(Rn), with 1≤p≤ ∞, to an investigation of regularity properties of solutions to the partial differential equations with constant coefficients and to some classes of equations with varying coefficients. However, as distinct from the Sobolev spaces, the H¨ormander spaces have not got a broad application to the general elliptic equations on manifolds and to the elliptic boundary-value prob- lems. This is due to the lack of a reasonable definition of the H¨ormander spaces on smooth manifolds (the definition should be independent of a choice of local charts covering the manifold) an on the absence of analytic tools fit to use these spaces effectively.

Such a tool exists in the Sobolev spaces case; this is the interpolation of spaces.

Namely, an arbitrary fractional order Sobolev space can be obtained by the inter- polation of a certain couple of integer order Sobolev spaces. This fact essentially facilitates both the investigation of these spaces and proofs of various theorems of the theory of elliptic equations because the boundedness and the Fredholm property (if the defect is invariant) preserve for linear operators under the inter- polation.

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Therefore it seems reasonable to distinguish the H¨ormander spaces that are obtained by the interpolation (with a function parameter) of couples of Sobolev spaces; we will consider only inner product spaces. For this purpose we introduce the following class of isotropic spaces

Hs,ϕ(Rn) := B2,µ(Rn) for µ(ξ) = hξisϕ(hξi). (1.6) Here the number parameters is real, whereas the positive function parameter ϕ varies slowly at +∞ in the Karamata sense [15, 126]. (We may assume thatϕ is constant outside of a neighbourhood of +∞.) For example, ϕ is admitted to be the logarithmic function, its arbitrary iteration, their real power, and a product of these functions.

The class of spaces (1.6) contains the Sobolev Hilbert scale {Hs} ≡ {Hs,1}, is attached to it by the number parameter, but is calibrated much finer than the Sobolev scale. Indeed,

Hs+ε(Rn)⊂Hs,ϕ(Rn)⊂Hs−ε(Rn) for every ε >0.

Therefore the number parametersdefines the main (power) smoothness, whereas the function parameterϕdetermines an additional (subpower) smoothness on the class of spaces (1.6). Specifically, if ϕ(t) → ∞ (or ϕ(t) → 0) as t → ∞, then ϕ determines an additional positive (or negative) smoothness. Thus, the parameter ϕrefines the main smoothness s. Therefore the class of spaces (1.6) is naturally called the refined Sobolev scale.

This scale possesses the following important property: every space Hs,ϕ(Rn) is a result of the interpolation, with an appropriate function parameter, of the couple of Sobolev spaces Hs−ε(Rn) andHs+δ(Rn), with ε, δ >0. The parameter of the interpolation is a function that varies regularly (in the Karamata sense) of index θ ∈ (0, 1) at +∞; namely θ := ε/(ε+δ). Moreover, the refined Sobolev scale proves to be closed with respect to this interpolation.

Thus, every H¨ormander space Hs,ϕ(Rn) possesses the interpolation property with respect to the Sobolev Hilbert scale. This means that each linear operator bounded on both the spacesHs−ε(Rn) andHs+δ(Rn) is also bounded onHs,ϕ(Rn).

The interpolation property plays a decisive role here; namely, it permits us to establish some important properties of the refined Sobolev scale. They enable this scale to be applied in the theory of elliptic equations. Thus, we can prove with the help of the interpolation that each spaceHs,ϕ(Rn), as the Sobolev spaces, is invariant with respect to diffeomorphic transformations of Rn. This permits the spaceHs,ϕ(Γ) to be well defined over a smooth closed manifold Γ because the set of distributions and the topology in this space does not depend on a choice of local charts covering Γ. The spacesHs,ϕ(Rn) andHs,ϕ(Γ) are useful in the theory of elliptic operators on manifolds and in the theory of elliptic boundary-value problems; these spaces are present implicitly in a number of problems appearing in calculus.

Let us dwell on some results that demonstrate advantages of the introduced scale as compared with the Sobolev scale. These results deal with the examples considered above. As before, let Abe an elliptic differential operator given on Γ,

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with m:= ordA. Then A sets the bounded and Fredholm operators A : Hs+m,ϕ(Γ)→Hs,ϕ(Γ) for all s∈R, ϕ∈ M.

HereM is the class of slowly varying function parameters ϕused in (1.6). Note that the differential operator Aleaves invariant the function parameter ϕ, which refines the main smoothnesss. Besides, we have the following regularity property of a solution to the elliptic equation Au=f:

(f ∈Hs,ϕ(Γ) for some s ∈R, ϕ∈ M) ⇒ u∈Hs+m,ϕ(Γ).

For the refined Sobolev scale, we have the following sharpening of Sobolev’s Imbedding Theorem: given an integerr ≥0 and functionϕ∈ M, the embedding Hr+n/2,ϕ(Γ)⊂Cr(Γ) is equivalent to that

Z

1

dt

t ϕ2(t) <∞. (1.7)

Therefore, iff ∈Hr−m+n/2,ϕ(Γ) for some parameterϕ∈ M satisfying (1.7), then the solutionu∈Cr(Γ).

Similar results are also valid for the elliptic systems and elliptic boundary-value problems.

Now let us pass to the analysis of the spectral expansion (1.3) convergence.

We additionally suppose that the operatorA is of orderm >0 and is unbounded and self-adjoint on the spaceL2(Γ). Condition (1.4) for the convergence of (1.3) almost everywhere on Γ is equivalent to the inclusion

f ∈H0,ϕ(Γ), with ϕ(t) := max{1,logt}.

The latter is much wider than the condition f ∈Hs(Γ) for some s >0. We can also similarly represent conditions for unconditional convergence almost every- where or convergence in the H¨older space Cr(Γ), with integral r≥0.

The above and some other results show that the refined Sobolev scale is helpful and convenient. This scale can be used in different topics of the modern analysis as well; see, e.g., the articles by M. Hegland [40,41], P. Math´e and U. Tautenhahn .

This paper is a detailed survey of our recent articles [75–87, 94–100], which are summed up in the monograph  published in Russian in 2010. In them, we have built a theory of general elliptic (both scalar and matrix) operators and elliptic boundary-value problems on the refined Sobolev scales of function spaces.

Let us describe the survey contents in greater detail. The paper consists of 13 sections.

Section1 is Introduction, which we are presenting now.

Section2 is preliminary and contains a necessary information about regularly varying functions and about the interpolation with a function parameter. Here we distinguish important Theorem2.12, which gives a description of all interpolation parameters for the category of separable Hilbert spaces.

In Section3, we consider the H¨ormander spaces, give a definition of the refined Sobolev scale, and study its properties. Among them, we especially note the

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interpolation properties of this scale, formulated as Theorems3.8 and 3.9. They are very important for applications.

Section 4 deals with uniformly elliptic pseudodifferential operators that are studied on the refined Sobolev scale over Rn. We get an a priory estimate for a solution of the elliptic equation and investigate an interior smoothness of the solution. As an application, we obtain a sufficient condition for the existence of continuous bounded derivatives of the solution.

Next in Section5, we define a class of H¨ormander spaces, the refined Sobolev scale, over a smooth closed manifold. We give three equivalent definitions of these spaces: local (in terms of local properties of distributions), interpolational (by means of the interpolation of Sobolev spaces with an appropriate function param- eter), and operational (via the completion of the set of infinitely smooth functions with respect to the norm generated by a certain function of the Beltrami–Laplace operator). These definitions are similar to those used for the Sobolev spaces. We study properties of the refined Sobolev scale over the closed manifold. Important applications of these results are given in Sections6 and 7.

Section6 deals with elliptic pseudodifferential operators on a closed manifold.

We show that they are Fredholm (i.e. have a finite index) on appropriate cou- ples of H¨ormander spaces. As in Section 4, a priory estimates for solutions of the elliptic equations are obtained, and the solutions regularity is investigated.

Using elliptic operators, we give equivalent norms on H¨ormander spaces over the manifold.

In Section 7, we investigate a convergence of spectral expansions correspond- ing to elliptic normal operators given on the closed manifold. We find sufficient conditions for the following types of the convergence: almost everywhere, uncon- ditionally almost everywhere, and in the space Ck, with integral k ≥ 0. These conditions are formulated in constructive terms of the convergence on some func- tion classes, which are H¨ormander spaces.

Section8deals with the classes of H¨ormander spaces that relate to the refined Sobolev scale and are given over Euclidean domains being open or closed. For these classes, we study interpolation properties, embeddings, traces, and riggings of the space of square integrable functions with H¨ormander spaces. The results of this section are applied in next Sections9–12, where a regular elliptic boundary- value problem is investigated in appropriate H¨ormander spaces.

In Section9, this problem is studied on the one-sided refined Sobolev scale. We show that the problem generates a Fredholm operator on this scale. We investi- gate some properties of the problem; namely, a priory estimates for solutions and local regularity are given. Moreover, a sufficient condition for the weak solution to be classical is found in terms of H¨ormander spaces.

Section10deals with semihomogeneous elliptic boundary-value problems. They are considered on H¨ormander spaces which form an appropriate two-sided re- fined Sobolev scale. We show that the operator corresponding to the problem is bounded and Fredholm on this scale.

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In Sections11–12, we give various theorems about a solvability of nonhomoge- neous regular elliptic boundary-value problems in H¨ormander spaces of an arbi- trary real main smoothness. Developing the methods suggested by Ya.A. Roit- berg  and J.-L. Lions, E. Magenes , we establish a certain generic theorem and a wide class of individual theorems on the solvability. The generic theorem is featured by that the domain of the elliptic operator does not depend on the co- efficients of the elliptic equation and is common for all boundary-value problems of the same order. Conversely, the individual theorems are characterized by that the domain depends essentially on the coefficients, even of the lower order deriva- tives. In Section 11, we elaborate on Roitberg’s approach in connection with H¨ormander spaces and then deduce the generic theorem about the solvability of elliptic boundary-value problems on the two-sided refined Sobolev scale modified in the Roitberg sense.

Section 12 is devoted to J.-L. Lions and E. Magenes’ approach, which we de- velop for various Hilbert scales consisting of Sobolev or H¨ormander spaces. For the space of right-hand sides of an elliptic equation, we find a sufficiently general condition under which the operator of the problem is bounded and Fredholm (see key Theorems12.6 and12.16). As a consequence, we obtain new various individ- ual theorems on the solvability of elliptic boundary-value problems considered in Sobolev or H¨ormander spaces, both nonweighted and weighted.

In final Section13, we indicate application of H¨ormander spaces to other impor- tant classes of elliptic problems. They are nonregular boundary-value problems, parameter-elliptic problems, certain mixed elliptic problems, elliptic systems and corresponding boundary-value problems.

It is necessary to note that some results given in the survey are new even for the Sobolev spaces. These results are Theorem 10.1 in the case of half-integer s and individual Theorems12.6,12.10, and 12.14.

In addition, note that we have also investigated a certain class of H¨ormander spaces, which is wider than the refined Sobolev scale. Interpolation properties of this class are studied and then applied to elliptic operators [85, 87, 99, 140]. It is remarkable that this class consists of all the Hilbert spaces which possess the interpolation property with respect to the Sobolev Hilbert scale. These results fall beyond the limits of our survey.

2. Preliminaries

In this section we recall some important results concerning the regularly varying functions and the interpolation with a function parameter of couples of Hilbert spaces. These results will be necessary for us in the sequel.

2.1. Regularly varying functions. We recall the following notion.

Definition 2.1. A positive function ψ defined on a semiaxis [b,+∞) is said to be regularly varying of indexθ ∈R at +∞ if ψ is Borel measurable on [b0,+∞) for some number b0 ≥b and

t→+∞lim

ψ(λ t)

ψ(t) =λθ for each λ >0.

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A function regularly varying of the index θ = 0 at +∞ is called slowly varying at +∞.

The theory of regularly varying functions was founded by Jovan Karamata [49, 50] in the 1930s. These functions are closely related to the power functions and have numerous applications, mainly due to their special role in Tauberian- type theorems (see the monographs [15, 64, 113, 126] and references therein).

Example 2.2. The well-known standard case of functions regularly varying of the indexθ at +∞ is

ψ(t) :=tθ(logt)r1(log logt)r2. . .(log. . .logt)rk for t 1 (2.1) with arbitrary parameters k ∈ Z+ and r1, r2, . . . , rk ∈ R. In the case where θ = 0 these functions form the logarithmic multiscale, which has a number of applications in the theory of function spaces.

We denote by SV the set of all functions slowly varying at +∞. It is evident thatψis a function regularly varying at +∞of indexθif and only ifψ(t) = tθϕ(t), t 1, for some function ϕ ∈ SV. Thus, the investigation of regularly varying functions is reduced to the case of slowly varying functions.

The study and application of regularly varying functions are based on two fundamental theorems: the Uniform Convergence Theorem and Representation Theorem. They were proved by Karamata  in the case of continuous functions and, in general, by a number of authors later (see the monographs cited above).

Theorem 2.3 (Uniform Convergence Theorem). Suppose that ϕ ∈ SV; then ϕ(λt)/ϕ(t)→1 as t →+∞ uniformly on each compact λ-set in (0,∞).

Theorem 2.4 (Representation Theorem). A function ϕ belongs to SV if and only if it can be written in the form

ϕ(t) = exp β(t) +

t

Z

b

α(τ) τ dτ

!

, t≥b, (2.2)

for some number b > 0, continuous function α : [b,∞) → R approaching zero at ∞, and Borel measurable bounded function β : [b,∞) →R that has the finite limit at ∞.

The Representation Theorem implies the following sufficient condition for a function to be slowly varying at infinity [126, Sec. 1.2].

Theorem 2.5. Suppose that a function ϕ : (b,∞) → (0,∞) has a continuous derivative and satisfies the condition tϕ0(t)/ϕ(t)→0 as t→ ∞. Then ϕ∈SV.

Using Theorem 2.5 one can give many interesting examples of slowly varying functions. Among them we mention the following.

Example 2.6. Let ϕ(t) := expψ(t), with ψ being defined according to (2.1), where θ= 0 and r1 <1. Then ϕ∈SV.

Example 2.7. Let α, β, γ ∈ R, β 6= 0, and 0 < γ < 1. We set ω(t) := α + βsin logγt and ϕ(t) := (logt)ω(t) for t >1. Then ϕ∈SV.

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Example 2.8. Letα, β, γ ∈R, α6= 0, 0< γ < β <1, and ϕ(t) := exp(α(logt)1−β sin logγt) for t >1.

Then ϕ∈SV.

The last two examples show that a function ϕ varying slowly at +∞ may exhibit infinite oscillation, that is

lim inf

t→+∞ ϕ(t) = 0 and lim sup

t→+∞

ϕ(t) = +∞.

We will use regularly varying functions as parameters when we define certain Hilbert spaces. If the function parameters are equivalent in a neighbourhood of +∞, we get the same space up to equivalence of norms. Therefore it is useful to introduced the following notion [81, p. 90].

Definition 2.9. We say that a positive function ψ defined on a semiaxis [b,+∞) is quasiregularly varying of indexθ∈Rat +∞if there exist a numberb1 ≥band a function ψ1 : [b1,+∞) → (0,+∞) regularly varying of the same index θ ∈ R at +∞ such that ψ ψ1 on [b1,+∞). A function quasiregularly varying of the index θ = 0 at +∞ is called quasislowly varying at +∞.

As usual, the notationψ ψ1 on [b1,+∞) means that the functionsψ and ψ1 are equivalent there, that is both the functions ψ/ψ1 and ψ1/ψ are bounded on [b1,+∞).

We denote by QSV the set of all functions varying quasislowly at +∞. It is evident that ψ is quasiregularly varying of the index θ at +∞ if and only if ψ(t) =tθϕ(t),t1, for some function ϕ∈QSV.

We note the following properties of the class QSV.

Theorem 2.10. Let ϕ, χ∈QSV. The next assertions are true:

i) There is a function ϕ1 ∈ C((0; +∞)) ∩SV such that ϕ ϕ1 in a neighbourhood of +∞.

ii) If θ >0, then both t−θϕ(t)→0 and tθϕ(t)→+∞ as t→+∞.

iii) All the functions ϕ+χ, ϕ χ, ϕ/χ and ϕσ, with σ∈R, belong to QSV.

iv) Let θ ≥ 0, and in the case where θ = 0 suppose that ϕ(t) → +∞ as t →+∞. Then the composite function χ(tθϕ(t)) of t belongs to QSV.

Theorem 2.10 are known for slowly varying functions, even with the strong equivalenceϕ(t)∼ϕ1(t) as t →+∞ being in assertion i); see, e.g., [15, Sec. 1.3]

and [126, Sec. 1.5]. This implies the case when ϕ, χ∈QSV [81, p. 91].

2.2. The interpolation with a function parameter of Hilbert spaces. It is a natural generalization of the classical interpolation method by J.-L. Lions and S.G. Krein (see, e.g., [30, Ch. IV, § 9] and [61, Ch. 1, Sec. 2 and 5]) to the case when a general enough function is used as an interpolation parameter instead of a number parameter. The generalization appeared in the paper by C. Foia¸s and J.-L. Lions [29, p. 278] and then was studied by W.F. Donoghue , E.I. Pustyl‘nik , V.I. Ovchinnikov [105, Sec. 11.4], and the authors .

We recall the definition of this interpolation. For our purposes, it is sufficient to restrict ourselves to the case of separable Hilbert spaces.

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Let an ordered coupleX := [X0, X1] of complex Hilbert spacesX0 and X1 be such that these spaces are separable and that the continuous dense embedding X1 ,→ X0 holds true. We call this couple admissible. For the couple X there exists an isometric isomorphismJ :X1 ↔X0such thatJ is a self-adjoint positive operator on the space X0 with the domain X1 (see [61, Ch. 1, Sec. 2.1] and [30, Ch. IV, Sec. 9.1]). The operatorJ is said to be generating for the coupleX and is uniquely determined by X.

We denote by B the set of all functions ψ : (0,∞)→(0,∞) such that:

a) ψ is Borel measurable on the semiaxis (0,+∞);

b) ψ is bounded on each compact interval [a, b] with 0< a < b <+∞;

c) 1/ψ is bounded on each set [r,+∞) with r >0.

Let ψ ∈ B. Generally, the unbounded operator ψ(J) is defined in the space X0 as a function of J. We denote by [X0, X1]ψ or simply by Xψ the domain of the operator ψ(J) endowed with the inner product (u, v)Xψ := (ψ(J)u, ψ(J)v)X0 and the corresponding norm kukXψ := (u, u)1/2X

ψ. The space Xψ is Hilbert and separable.

Definition 2.11. We say that a function ψ ∈ B is an interpolation parameter if the following property is fulfilled for all admissible couples X = [X0, X1],Y = [Y0, Y1] of Hilbert spaces and an arbitrary linear mapping T given on X0. If the restriction of the mapping T to the space Xj is a bounded operator T :Xj →Yj for each j = 0, 1, then the restriction of the mapping T to the space Xψ is also a bounded operator T :Xψ →Yψ.

Otherwise speaking,ψ is an interpolation parameter if and only if the mapping X 7→Xψ is an interpolation functor given on the category of all admissible couples X of Hilbert spaces. (For the notion of interpolation functor, see, e.g., [14, Sec.

2.4] and [133, Sec. 1.2.2]) In the case where ψ is an interpolation parameter, we say that the space Xψ is obtained by the interpolation with the function parameterψ of the admissible coupleX. Then the continuous dense embeddings X1 ,→Xψ ,→X0 are fulfilled.

The classical result by J.-L. Lions and S.G. Krein consists in the fact that the power function ψ(t) :=tθ is an interpolation parameter whenever 0< θ < 1; see [30, Ch. IV, § 9, Sec. 3] and [61, Ch. 1, Sec. 5.1].

We have the following criterion for a function to be an interpolation parameter.

Theorem 2.12. A function ψ ∈ B is an interpolation parameter if and only if ψ is pseudoconcave in a neighbourhood of +∞, i.e. ψ ψ1 for some concave positive function ψ1

This theorem follows from Peetre’s results  on interpolations functions (see also the monograph [14, Sec. 5.4]). The corresponding proof is given in [81, Sec.

2.7].

For us, it is important the next consequence of Theorem 2.12.

Corollary 2.13. Suppose a function ψ ∈ B to be quasiregularly varying of index θ at +∞, with 0< θ <1. Then ψ is an interpolation parameter.

The direct proof of this assertion is given in [76, Sec. 2].

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3. H¨ormander spaces

In 1963 Lars H¨ormander [43, Sec. 2.2] introduced the spaces Bp,µ(Rn), which consist of distributions inRn and are parametrized by a numberp∈[1,∞] and a general enough weight functionµof argumentξ ∈Rn; see also [45, Sec. 10.1]. The number parameter p characterizes integrability properties of the distributions, whereas the function parameterµ describes their smoothness properties. In this section, we recall the definition of the spacesBp,µ(Rn), some their properties, and an application to constant-coefficient partial differential equations. Further we consider the important case where the H¨ormander space Bp,µ(Rn) is Hilbert, i.e.

p= 2, and µis a quasiregularly varying function of (1 +|ξ|2)1/2 at infinity.

3.1. The spaces Bp,µ(Rn). Let an integer n ≥ 1 and a parameter p ∈ [1,∞].

We use the following conventional designations, where Ω is an nonempty open set in Rn, in particular Ω =Rn:

a) Lp(Ω) :=Lp(Ω, dξ) is the Banach space of complex-valued functions f(ξ) of ξ∈Ω such that |f|p is integrable over Ω (if p=∞, then f is essentially bounded in Ω);

b) Cbk(Ω) is the Banach space of functions u: Ω→Chaving continuous and bounded derivatives of order ≤k on Ω;

c) C0(Ω) is the linear topological space of infinitely differentiable functions u : Rn → C such that their supports are compact and belong to Ω; we will identify functions from C0(Ω) with their restrictions to Ω;

d) D0(Ω) is the linear topological space of all distributions given in Ω; we always suppose that distributions are antilinear complex-valued function- als;

e) S0(Rn) is the linear topological Schwartz space of tempered distributions given in Rn;

f) ub := Fu is the Fourier transform of a distribution u ∈ S0(Rn); F−1f is the inverse Fourier transform of f ∈ S0(Rn);

g) hξi:= (1 +|ξ|2)1/2 is a smoothed modulus ofξ ∈Rn.

Suppose a continuous function µ : Rn → (0,∞) to be such that, for some numbers c≥1 and l >0, we have

µ(ξ)

µ(η) ≤c(1 +|ξ−η|)l for all ξ, η ∈Rn. (3.1) The function µis called a weight function.

Definition 3.1. The H¨ormander space Bp,µ(Rn) is a linear space of the distribu- tionsu∈ S0(Rn) such that the Fourier transform buis locally Lebesgue integrable on Rn and, moreover, µbu ∈ Lp(Rn). The space Bp,µ(Rn) is endowed with the norm kukBp,µ(Rn) :=kµbukLp(Rn).

The space Bp,µ(Rn) is complete and continuously embedded in S0(Rn). If 1≤ p < ∞, then this space is separable, and the set C0(Rn) is complete in it [43, Sec. 2.2]. Of special interest is thep= 2 case, when Bp,µ(Rn) becomes a Hilbert space.

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Remark 3.2. H¨ormander assumes initially that µ satisfies a stronger condition than (3.1); namely, there exist some positive numbers c and l such that

µ(ξ)

µ(η) ≤(1 +c|ξ−η|)l for all ξ, η ∈Rn. (3.2) But he notices that two sets of functions satisfying either (3.1) or (3.2) lead to the same class of spaces Bp,µ(Rn) [43, the remark at the end of Sec. 2.1].

The term ‘H¨ormander space’ was suggested by H. Triebel in [133, Sec. 4.11.4].

The following H¨ormander’s theorem establishes an important relation between the spaces Bp,µ(Rn) and Cbk(Rn) [43, Sec. 2.2, Theorem 2.2.7].

Theorem 3.3 (H¨ormander’s Embedding Theorem). Let p, q ∈ [1,∞], 1/p + 1/q= 1, and an integer k ≥0. Then the condition

hξikµ−1(ξ)∈Lq(Rn, dξ) (3.3) entails the continuous embedding Bp,µ(Rn),→Cbk(Rn). Conversely, if

{u∈Bp,µ(Rn) : suppu⊂V} ⊂Ck(Rn) for some nonempty open set V ⊆Rn, then (3.3) is valid.

The spacesBp,µ(Rn) were applied by H¨ormander to investigation of regularity properties of solutions to some partial differential equations (see [43, Ch. IV, VII]

and [45, Ch. 11, 13]). We state one of his results relating to elliptic equations [43, Sec 7.4].

Let Ω be a nonempty open set inRn. In Ω, consider a partial differential equa- tionP(x, D)u=f of an orderr with coefficients belonging to C(Ω). Introduce the local H¨ormander space over Ω:

Bp,µloc(Ω) :={f ∈ D0(Ω) : χf ∈Bp,µ(Ω) ∀ χ∈C0(Ω)}.

HereBp,µ(Ω) is the space of restrictions of all the distributionsu∈Bp,µ(Rn) to Ω.

Theorem 3.4 (H¨ormander’s Regularity Theorem). Let the operator P(x, D) be elliptic in Ω, and u ∈ D0(Ω). If P(x, D)u ∈ Bp,µloc(Ω) for some p ∈ [1,∞] and weight function µ, then u∈Blocp,µ

r(Ω) with µr(ξ) :=hξirµ(ξ).

For applications of the spaces Bp,µ(Rn), the Hilbert case of p = 2 is the most interesting. This case was investigated by B. Malgrange  and L.R. Volevich, B.P. Paneah  (see also Paneah’s monograph [106, Sec. 1.4]). Specifically, if µ(ξ) =hξis for all ξ ∈ Rn with some s ∈R, then B2,µ(Rn) becomes the Sobolev inner product spaceHs(Rn) of order s.

In what follows we consider the isotropic H¨ormander inner product spaces B2,µ(Rn), withµ(ξ) being a radial function, i.e. depending only on hξi.

3.2. The refined Sobolev scale. It useful to have a class of the H¨ormander inner product spaces B2,µ(Rn) that are close to the Sobolev spaces Hs(Rn) with s∈R. For this purpose we chooseµ(ξ) := hξisϕ(hξi) for some functionsϕ∈QSV;

then µ is a quasiregularly varying function of hξi at infinity of index s. In this case it is naturally to rename the H¨ormander space B2,µ(Rn) by Hs,ϕ(Rn). Let

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us formulate the corresponding definitions. First we introduce the following set M ⊂QSV of function parameters ϕ.

ByM we denote the set of all functions ϕ: [1; +∞)→(0; +∞) such that:

a) ϕ is Borel measurable on [1; +∞);

b) ϕ and 1/ϕ are bounded on every compact interval [1;b], where 1 < b <

+∞;

c) ϕ∈QSV.

It follows from Theorem 2.4 thatϕ∈ M if and only ifϕcan be written in the form (2.2) with b = 1 for some continuous function α : [1,∞)→ R approaching zero at +∞ and Borel measurable bounded function β : [1,∞)→R.

Lets∈R and ϕ∈ M.

Definition 3.5. The space Hs,ϕ(Rn) is the H¨ormander inner product space B2,µ(Rn) withµ(ξ) := hξisϕ(ξ) forξ ∈Rn.

Thus Hs,ϕ(Rn) consists of the distributions u ∈ S0(Rn) such that the Fourier transformubis a function locally Lebesgue integrable on Rn and

Z

Rn

hξi2sϕ2(hξi)|u(ξ)|b 2dξ <∞.

The inner product in the spaceHs,ϕ(Rn) is defined by the formula (u1, u2)Hs,ϕ(Rn):=

Z

Rn

hξi2sϕ2(hξi)ub1(ξ)ub2(ξ)dξ

and induces the norm in the usual way,Hs,ϕ(Rn) being a Hilbert space.

The function µ used in Definition 3.5 is a weight function that follows from the integral representation of the set M given above. We consider the Borel measurable weight functions µ, rather than continuous as H¨ormander does. By Theorem 2.10 i) we do not obtain the spaces different from those considered by H¨ormander.

In the simplest case where ϕ(·)≡ 1, the space Hs,ϕ(Rn) =Hs,1(Rn) coincides with the Sobolev spaceHs(Rn).

By Theorem2.10 (ii), for each ε >0 there exist a numbercε ≥1 such that c−1ε t−ε≤ϕ(t)≤cεtε for all t ≥1.

This implies the inclusions [

ε>0

Hs+ε(Rn) =:Hs+(Rn)⊂Hs,ϕ(Rn)⊂Hs−(Rn) := \

ε>0

Hs−ε(Rn). (3.4) They show that in the class of spaces

Hs,ϕ(Rn) : s∈R, ϕ∈ M (3.5) the functional parameter ϕ defines a supplementary (subpower) smoothness to the basic (power) s-smoothness. If ϕ(t) → ∞ [ϕ(t) → 0] as t → ∞, then ϕ defines a positive [negative] supplementary smoothness. Otherwise speaking, ϕ refines the power smoothness s. Therefore, it is naturally to give

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Definition 3.6. The class of spaces (3.5) is called the refined Sobolev scale overRn.

Obviously, the scale (3.5) is much finer than the Hilbert scale of Sobolev spaces.

The scale (3.5) was considered by the authors in [75, 77, 81]. Let us formulate some important properties of it.

Theorem 3.7. Let s ∈R and ϕ, ϕ1 ∈ M. The following assertions are true:

i) The dense continuous embedding Hs+ε,ϕ1(Rn) ,→ Hs,ϕ(Rn) is valid for each ε >0.

ii) The function ϕ/ϕ1 is bounded in a neighbourhood of +∞ if and only if Hs,ϕ1(Rn),→Hs,ϕ(Rn). This embedding is continuous and dense.

iii) Let an integer k ≥0 be given. The inequality

Z

1

dt

t ϕ2(t) <∞ (3.6)

is equivalent to the embedding

Hk+n/2,ϕ(Rn),→Cbk(Rn). (3.7) The embedding is continuous.

iv) The spaces Hs,ϕ(Rn) and H−s,1/ϕ(Rn) are mutually dual with respect to the inner product in L2(Rn).

Assertion i) of this theorem follows from (3.4), whereas assertions ii) – iv) are inherited from the H¨ormander spaces properties [43, Sec. 2.2], in particular, iii) from Theorem 3.3. Note thatϕ ∈ M ⇔1/ϕ∈ M, so the spaceH−s,1/ϕ(Rn) in assertion iv) is defined as an element of the refined Sobolev scale.

The refined Sobolev scale possesses the interpolation property with respect to the Sobolev scale because every spaceHs,ϕ(Rn) is obtained by the interpolation, with an appropriate function parameter, of a couple of inner product Sobolev spaces.

Theorem 3.8. Let a functionϕ∈ M and positive numbers ε, δ be given. We set ψ(t) :=

(tε/(ε+δ)ϕ(t1/(ε+δ)) for t≥1,

ϕ(1) for 0< t <1. (3.8)

Then the following assertions are true:

i) The function ψ belongs to the set B and is an interpolation parameter.

ii) For an arbitrary s∈R, we have

[Hs−ε(Rn), Hs+δ(Rn)]ψ =Hs,ϕ(Rn) (3.9) with equality of norms in the spaces.

Assertion i) holds true by Corollary2.13 because the function (3.8) is quasireg- ularly varying of index θ := ε/(ε+δ) ∈ (0, 1) at +∞. Assertion ii) is directly verified if we note that the operatorJ :u7→ F−1(hξiε+δu(ξ)) is generating for theb couple on the left of (3.9). Then the operator ψ(J) : u 7→ F−1(hξiεϕ(hξi)u(ξ))b

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maps Hs,ϕ(Rn) onto Hs−ε(Rn) that means (3.9); for details, see [77, Sec. 3] or [81, Sec. 3.2].

The refined Sobolev scale is closed with respect to the interpolation with the functions parameters that are quasiregularly varying at +∞.

Theorem 3.9. Let s0, s1 ∈ R, s0 ≤ s1, and ϕ0, ϕ1 ∈ M. In the case where s0 =s1 we suppose that the function ϕ01 is bounded in a neighbourhood of ∞.

Let ψ ∈ B be a quasiregularly varying function of an index θ ∈ (0, 1) at ∞. We represent ψ(t) = tθχ(t) with χ∈QSV and set s:= (1−θ)s0+θs1,

ϕ(t) :=ϕ1−θ0 (t)ϕθ1(t)χ

ts1−s0 ϕ1(t) ϕ0(t)

for t≥1.

Then ϕ∈ M, and

[Hs00(Rn), Hs11(Rn)]ψ =Hs,ϕ(Rn) (3.10) with equality of norms in the spaces.

This theorem can be proved by means of the repeated application of Theorem 3.8if we employ the reiteration formula [Xf, Xg]ψ =Xω, whereXis an admissible couple of Hilbert spaces, f, g, ψ ∈ B, f /g is bounded in a neighbourhood of ∞, and ω(t) := f(t)ψ(g(t)/f(t)) for t > 0; see [81, Sec. 2.3]. Besides, it is possible to give the direct proof, which is similar to that used for Theorem3.8.

Remark 3.10. The interpolation of the H¨ormander spacesBp,µ(Rn), with 1≤p≤

∞, was studied by M. Schechter  with the help of the complex method of interpolation. C. Merucci  and F. Cobos, D.L. Fernandez  considered the interpolation of various Banach spaces of generalized smoothness by means of the real method involving a function parameter.

4. Elliptic operators in Rn

In this section we consider an arbitrary uniformly elliptic classical pseudodif- ferential operator (PsDO)Aon the scale (3.5). We establish an a priory estimate for a solution to the equationAu=f and investigate the solution smoothness in this scale. Our results refine the classical theorems on elliptic operators on the Sobolev scale; see, e.g., [7, Sec. 1.8] or [46, Sec. 18.1].

Following [7, Sec. 1.1], we denote by Ψr(Rn) with r ∈ R the class of all the PsDOs A in Rn (generally, not classical) such that their symbolsa(x, ξ) are complex-valued infinitely smooth functions satisfying the following condition. For arbitrary multi-indexes α and β, there exist a number cα,β >0 such that

|∂xαξβa(x, ξ)| ≤ cα,βhξir−|β| for every x, ξ ∈Rn.

Lemma 4.1. Let A ∈ Ψr(Rn) with r ∈ R. Then the restriction of the mapping u7→Au, u∈ S0(Rn), to the space Hs,ϕ(Rn) is a bounded linear operator

A:Hs,ϕ(Rn)→Hs−r, ϕ(Rn) for each s ∈R and ϕ∈ M.

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This lemma follows from the Sobolev ϕ ≡1 case [7, Sec. 1.1, Theorem 1.1.2]

by the interpolation formula (3.9).

By Ψrph(Rn) we denote the subset in Ψr(Rn) that consists of all the classical (polyhomogeneous) PsDOs of the orderr; see [7, Sec. 1.5]. An important example of PsDO from Ψrph(Rn) is given by a partial differential operator of order r with coefficients belonging to Cb(Rn).

Definition 4.2. A PsDO A∈Ψrph(Rn) is called uniformly elliptic in Rn if there exists a number c > 0 such that |a0(x, ξ)| ≥ c for each x, ξ ∈ Rn with |ξ| = 1.

Herea0(x, ξ) is the principal symbol of A.

Letr ∈R. Suppose a PsDO A ∈Ψrph(Rn) to be uniformly elliptic in Rn. Theorem 4.3. Let s ∈ R, ϕ∈ M, and σ < s. The following a priori estimate holds true:

kukHs,ϕ(Rn) ≤c kAukHs−r,ϕ(Rn)+kukHσ,ϕ(Rn)

for all u∈Hs,ϕ(Rn). (4.1) Here c=c(s, ϕ, σ) is a positive number not depending on u.

We prove this theorem with the help of the left parametrix of A if we apply Lemma 4.1. As knows [7, Sec. 1.8, Theorem 1.8.3] there exists a PsDO B ∈ Ψ−rph(Rn) such that BA =I +T, where I is identical operator and T ∈Ψ−∞ :=

T

m∈R Ψm(Rn). The operator B is called the left parametrix of A. Writing u=BAu−T u, we easily get (4.1) by Lemma 4.1.

Let Ω be an arbitrary nonempty open subset in Rn. We study an interior smoothness of a solution to the equation Au=f in Ω.

Let us introduce some relevant spaces. By H−∞(Rn) we denote the union of all the spaces Hs,ϕ(Rn) with s ∈ R and ϕ ∈ M. The linear space H−∞(Rn) is endowed with the inductive limit topology. We set

Hints,ϕ(Ω) :=

f ∈H−∞(Rn) : χ f ∈Hs,ϕ(Rn)

for all χ∈Cb(Rn), suppχ⊂Ω, dist(suppχ, ∂Ω)>0 . (4.2) A topology in Hints,ϕ(Ω) is defined by the seminorms f 7→ kχ fkHs,ϕ(Rn) with χ being the same as in (4.2).

Theorem 4.4. Letu∈H−∞(Rn)be a solution to the equation Au=f inΩ with f ∈Hints,ϕ(Ω) for some s∈R and ϕ∈ M. Then u∈Hints+r,ϕ(Ω).

The special case when Ω = Rn (global smoothness) follows at once from the equality u = Bf −T u, with B being the left parametrix, and Lemma 4.1. In general, we deduce Theorem4.4from this case if we rearrangeAand the operator of multiplication by a functionχ satisfying (4.2). Then we write

Aχu=Aχηu=χ Aηu+A0ηu=χf +χ A(η−1)u+A0ηu, (4.3) where A0 ∈ Ψr−1(Rn), and the function η has the same properties as χ and is equal to 1 in a neighbourhood of suppχ. Now, if u ∈ Hints+r−k,ϕ(Ω) for some integer k ≥ 1, then the right-hand side of (4.3) belongs to Hs−k+1,ϕ(Rn) that impliesχu∈Hs+r−k+1,ϕ(Rn), i.e. u∈Hints+r−k+1,ϕ(Ω). By induction in k we have u∈Hints+r,ϕ(Ω).

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It is useful to compare Theorem 4.4 with H¨ormander’s Regularity Theorem.

If A is a partial differential operator, and Ω is bounded, then Theorem 4.4 is a consequence of the H¨ormander theorem.

Applying Theorems 4.4 and 3.7 iii) we get the following sufficient condition for the solution u to have continuous and bounded derivatives of the prescribed order.

Theorem 4.5. Let u ∈ H−∞(Rn) be a solution to the equation Au = f in Ω, with f ∈ Hintk−r+n/2,ϕ(Ω) for some integer k ≥0 and function parameter ϕ∈ M.

Suppose thatϕsatisfies (3.6). Thenuhas the continuous partial derivatives onΩ up to the orderk, and they are bounded on every setΩ0 ⊂Ωwithdist(Ω0, ∂Ω)>0.

In particular, if Ω = Rn, then u∈Cbk(Rn).

This theorem shows an advantage of the refined Sobolev scale over the Sobolev scale when a classical smoothness of a solution is under investigation. Indeed, if we restrict ourselves to the Sobolev case of ϕ≡ 1, then we have to replace the condition f ∈ Hintk−r+n/2,ϕ(Ω) with the condition f ∈ Hk−r+ε+n/2,1

int (Ω) for some ε >0. The last condition is far stronger than previous one.

Note that the condition (3.6) not only is sufficient in Theorem 3.3 but also is necessary on the class of all the considered solutionsu. Namely, (3.6) is equivalent to the implication

u∈H−∞(Rn), f :=Au∈Hintk−r+n/2,ϕ(Ω)

⇒ u∈Ck(Ω). (4.4) Indeed, if u∈ Hintk+n/2,ϕ(Ω), then f =Au ∈ Hintk−r+n/2,ϕ(Ω), whence u∈ Ck(Ω) if (4.4) holds. Thus (4.4) entails (3.6) in view of H¨ormander’s Theorem 3.3.

The analogs of Theorems 4.3–4.5 were proved in  for uniformly elliptic matrix PsDOs.

5. H¨ormander spaces over a closed manifold

In this section we introduce a certain class of H¨ormander spaces over a closed (compact) smooth manifold. Namely, using the spacesHs,ϕ(Rn) with s∈R and ϕ∈ M we construct their analogs for the manifold. We give three equivalent def- initions of the analogs; these definitions are similar to those used for the Sobolev spaces (see, e.g., [132, Ch. 1, Sec. 5]).

5.1. The equivalent definitions. In what follows except Subsection 7.1, Γ is a closed (i.e. compact and without a boundary) infinitely smooth oriented man- ifold of an arbitrary dimension n ≥ 1. We suppose that a certain C-density dx is defined on Γ. As usual, D0(Γ) denotes the linear topological space of all distributions on Γ. The spaceD0(Γ) is antidual to the space C(Γ) with respect to the natural extension of the scalar product inL2(Γ) := L2(Γ, dx) by continuity.

This extension is denoted by (f, w)Γ forf ∈ D0(Γ) and w∈C(Γ).

Let s ∈ R and ϕ ∈ M. We give the following three equivalent definitions of the H¨ormander space Hs,ϕ(Γ).

The first definition exhibits the local properties of those distributionsf ∈ D0(Γ) that form Hs,ϕ(Γ). From the C-structure on Γ, we arbitrarily choose a finite collection of the local chartsαj :Rn ↔Γj, j = 1, . . . ,κ, such that the open sets

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Γj form the finite covering of Γ. Let functions χj ∈C(Γ),j = 1, . . . ,κ, form a partition of unity on Γ satisfying the condition suppχj ⊂Γj.

Definition 5.1. The linear space Hs,ϕ(Γ) is defined by the formula Hs,ϕ(Γ) :=

f ∈ D0(Γ) : (χjf)◦αj ∈Hs,ϕ(Rn) ∀j = 1, . . .κ .

Here (χjf)◦αj is the representation of the distributionχjf in the local chartαj. The inner product in the spaceHs,ϕ(Γ) is introduced by the formula

(f1, f2)Hs,ϕ(Γ) :=

κ

X

j=1

((χjf1)◦αj,(χjf2)◦αj)Hs,ϕ(Rn) and induces the norm in the usual way.

In the special case where ϕ ≡ 1 the space Hs,ϕ(Γ) coincides with the inner product Sobolev space Hs(Γ) of orders. The Sobolev spaces on Γ are known to be complete and independent (up to equivalence of norms) of the choice of the local charts and the partition of unity.

The second definition connects the space Hs,ϕ(Γ) with the Sobolev scale by means of the interpolation.

Definition 5.2. Let two integers k0 and k1 be such that k0 < s < k1. We define Hs,ϕ(Γ) := [Hk0(Γ), Hk1(Γ)]ψ, (5.1) where the interpolation parameterψis given by the formula (3.8) withε:=s−k0 and δ:=k1−s.

It is useful in the spectral theory to have the third definition of Hs,ϕ(Γ) that connects the norm inHs,ϕ(Γ) with a certain function of 1−∆Γ. As usual, ∆Γ is the Beltrami-Laplace operator on the manifold Γ endowed with the Riemannian metric that induces the density dx; see, e.g., [127, Sec. 22.1].

Definition 5.3. The space Hs,ϕ(Γ) is defined to be the completion of C(Γ) with respect to the Hilbert norm

f 7→ k(1−∆Γ)s/2ϕ((1−∆Γ)1/2)fkL2(Γ), f ∈C(Γ). (5.2) Theorem 5.4. Definitions 5.1, 5.2, and 5.3 are mutually equivalent, that is they define the same Hilbert space Hs,ϕ(Γ) up to equivalence of norms.

Let us explain how to prove this fundamental theorem.

The equivalence of Definitions5.1 and 5.2. We use Definition5.1 as a starting point and show that the equality (5.1) holds true up to equivalence of norms.

We apply the Rn-analog of (5.1), due to Theorem 3.8, and pass to local coordi- nates on Γ. Namely, let the mapping T take each f ∈ D0(Γ) to the vector with components (χjf)◦αj, j = 1, . . . ,κ. We get the bounded linear operator

T : Hs,ϕ(Γ)→(Hs,ϕ(Rn))κ. (5.3) It has the right inverse bounded linear operator

K : (Hs,ϕ(Rn))κ →Hs,ϕ(Γ), (5.4)

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Amount of Remuneration, etc. The Company does not pay to Directors who concurrently serve as Executive Officer the remuneration paid to Directors. Therefore, “Number of Persons”

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