An extended Hilbert scale and its applications
aa r X i v : . [ m a t h . F A ] F e b AN EXTENDED HILBERT SCALE AND ITS APPLICATIONS
VLADIMIR MIKHAILETS, ALEKSANDR MURACH, AND TETIANA ZINCHENKO
Abstract.
We propose a new viewpoint on Hilbert scales extending them by means ofall Hilbert spaces that are interpolation ones between spaces on the scale. We prove thatthis extension admits an explicit description with the help of OR -varying functions of theoperator generating the scale. We also show that this extended Hilbert scale is obtained bythe quadratic interpolation (with function parameter) between the above spaces and is closedwith respect to the quadratic interpolation between Hilbert spaces. We give applications ofthe extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, andspectral expansions induced by abstract and elliptic operators. Introduction
Hilbert scales (above all, the Sobolev scale) play an important role in mathematical analysisand the theory of differential equations; see, e.g., the classical monographs [12, 29, 37, 39],surveys [1, 2, 17], and recent book [32]. Such scales are built with respect to an arbitrarilychosen Hilbert space H and a positive definite self-adjoint unbounded operator A actingin this space. As a result, we obtain the Hilbert scale { H sA : s ∈ R } , where H sA is thecompletion of the domain of A s in the norm k A s u k H of a vector u . This scale has thefollowing fundamental property: if < θ < , then the mapping { H s A , H s A } 7→ H sA , with s < s and s := (1 − θ ) s + θs , is an exact interpolation functor of type θ [36, Theorem 9.1].Concerning a linear operator T bounded on both spaces H s A and H s A , this means that T isalso bounded on H sA and that the norms of T on these spaces satisfy the inequality k T : H sA → H sA k ≤ k T : H s A → H s A k − θ k T : H s A → H s A k θ . (An analogous property is fulfilled for bounded linear operators that act on pairs of differentspaces belonging to two Hilbert scales.) Hence, every space H sA subject to s < s < s is aninterpolation space between H s A and H s A . However, the class of such interpolation Hilbertspaces is far broader than the section { H sA : s ≤ s ≤ s } of the Hilbert scale.It is therefore natural to consider the extension of this scale by means of all Hilbert spacesthat are interpolation ones between some spaces H s A and H s A , where the numbers s < s range over R . Such an extended Hilbert scale is an object of our investigation. We will showthat this scale admits a simple explicit description with the help of OR -varying functions of A , is obtained by the quadratic interpolation (with function parameter) between the spaces H s A and H s A , and is closed with respect to the quadratic interpolation between Hilbert spaces. Mathematics Subject Classification.
Key words and phrases.
Hilbert scale, interpolation space, interpolation with function parameter, interpo-lational inequality, generalized Sobolev space, spectral expansion.This work is supported by the European Union’s Horizon 2020 research and innovation programme underthe Marie Sk l odowska-Curie grant agreement No 873071 (SOMPATY: Spectral Optimization: From Mathe-matics to Physics and Advanced Technology). These and some other properties of the extended Hilbert scale are considered in Section 2 ofthis paper; they are proved in Section 3. Note that the above interpolation and interpolationalproperties of Hilbert scales are studied in articles [4, 5, 16, 18, 19, 35, 36, 38, 47, 59, 63] (seealso monographs [39, Chapter 1], [51, Section 1.1], and [68, Chapters 15 and 30]). Amongthem, of fundamental importance for our investigation is Ovchinnikov’s result [59, Theorem11.4.1] on an explicit description (with respect to equivalence of norms) of all Hilbert spacesthat are interpolation ones between arbitrarily chosen compatible Hilbert spaces.The next sections are devoted to various applications of the extended Hilbert scale. Sec-tion 4 considers interpolational inequalities that connect the norms in spaces on the scale toeach other, as well as the norms of linear operators acting between extended Hilbert scales.From the viewpoint of inequalities for norms of vectors, this scale can be interpreted as a vari-able Hilbert scale investigated in [23, 24, 25, 42]; the latter appears naturally in the theoryof ill-posed problems (see, e.g., [26, 30, 41, 70]). Section 5 gives applications of the extendedHilbert scale to function or distribution spaces, which are used specifically in the theory ofpseudodifferential operators. We show that the extended Hilbert scale generated by someelliptic operators consists of generalized Sobolev spaces whose regularity order is a function OR -varying at infinity. These spaces form the extended Sobolev scale considered in [50, 52]and [51, Section 2.4.2]. It has important applications to elliptic operators [54, 55, 73, 74]and elliptic boundary-value problems [6, 7, 8, 9, 31]. Among them are applications to theinvestigation of various types of convergence of spectral expansions induced by elliptic op-erators. This topic is examined in the last Section 7. Its results are based on theorems onthe convergence—in a space with two norms—of the spectral expansion induced by an ab-stract normal operator and on the degree of this convergence. These theorems are proved inSection 6. 2. Basic results
Let H be a separable infinite-dimensional complex Hilbert space, with ( · , · ) and k · k re-spectively denoting the inner product and the corresponding norm in H . Let A be a positivedefinite self-adjoint unbounded linear operator in H . The positive definiteness of A meansthat there exists a number r > such that ( Au, u ) ≥ r ( u, u ) for every u ∈ Dom A . As usual, Dom A denotes the domain of A . Without loss of generality we suppose that the lower bound r = 1 .For every s ∈ R , the self-adjoint operator A s in H is well defined with the help of thespectral decomposition of A . The domain Dom A s of A s is dense in H ; moreover, Dom A s = H whenever s ≤ . Let H sA denote the completion of Dom A s with respect to the norm k u k s := k A s u k and the corresponding inner product ( u , u ) s := ( A s u , A s u ) , with u, u , u ∈ Dom A s .The Hilbert space H sA is separable. As usual, we retain designations ( · , · ) s and k · k s for theinner product and the corresponding norm in this space. Note that the linear manifold H sA coincides with Dom A s whenever s ≥ and that H sA ⊃ H whenever s < . The set H ∞ A := T λ> H λA is dense in every space H sA , with s ∈ R .The class { H sA : s ∈ R } is called the Hilbert scale generated by A or, simply, A -scale (see.,e.g., [36, Section 9, Subsection 1]). If s , s ∈ R and s < s , then the identity mapping on Dom A s extends uniquely to a continuous embedding operator H s A ֒ → H s A , the embeddingbeing normal. Therefore, interpreting H s A as a linear manifold in H s A , we obtain the normal N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 3 pair [ H s A , H s A ] of Hilbert spaces. This means that H s A is dense in H s A and that k u k s ≤ k u k s for every u ∈ H s A . Main Definition.
The extended Hilbert scale generated by A or, simply, the extended A -scale consists of all Hilbert spaces each of which is an interpolation space for a certain pair [ H s A , H s A ] where s < s (the real numbers s and s may depend on the interpolation Hilbertspace).We will give an explicit description of this scale and prove its important interpolationproperties.Beforehand, let us recall the definition of an interpolation space in the case considered.Suppose H and H are Hilbert spaces such that H is a linear manifold in H and that theembedding H ֒ → H is continuous. A Hilbert space X is called an interpolation space forthe pair [ H , H ] (or, in other words, an interpolation space between H and H ) if X satisfiesthe following two conditions:(i) X is an intermediate space for this pair, i.e. X is a linear manifold in H and thecontinuous embeddings H ֒ → X ֒ → H hold;(ii) for every linear operator T given on H , the following implication is true: if therestriction of T to H j is a bounded operator on H j for each j ∈ { , } , then therestriction of T to X is a bounded operator on X .Property (ii) implies the following inequality for norms of operators: k T : X → X k ≤ c max (cid:8) k T : H → H k , k T : H → H k (cid:9) , where c is a certain positive number which does not depend on T (see, e.g., [11, Theorem2.4.2]). If c = 1 , the interpolation space X is called exact.Both properties (i) and (ii) are invariant with respect to the change of the norm in X foran equivalent norm. Therefore, it makes sense to describe the interpolation spaces for thepair [ H , H ] up to equivalence of norms.As is known [36, Theorem 9.1], every space H sA is an interpolation one for the pair [ H s A , H s A ] whenever s ≤ s ≤ s . To give a description of all interpolation Hilbert spaces for this pair,we need more general functions of A than power functions used in the definition of H sA .Choosing a Borel measurable function ϕ : [1 , ∞ ) → (0 , ∞ ) arbitrarily and using the spectraldecomposition of A , we define the self-adjoint operator ϕ ( A ) > which acts in H . Recall that Spec A ⊆ [1 , ∞ ) according to our assumption. Here and below, Spec A denotes the spectrumof A , and ϕ ( A ) > means that ( ϕ ( A ) u, u ) > for every u ∈ Dom ϕ ( A ) \ { } . Let H ϕA denotethe completion of the domain Dom ϕ ( A ) of ϕ ( A ) with respect to the norm k u k ϕ := k ϕ ( A ) u k of u ∈ Dom ϕ ( A ) .The space H ϕA is Hilbert and separable. Indeed, this norm is induced by the inner product ( u , u ) ϕ := ( ϕ ( A ) u , ϕ ( A ) u ) of u , u ∈ Dom ϕ ( A ) . Besides, endowing the linear space Dom ϕ ( A ) with the norm k · k ϕ and considering the isometric operator ϕ ( A ) : Dom ϕ ( A ) → H, (2.1)we infer the separability of Dom ϕ ( A ) (in the norm k·k ϕ ) from the separability of H . Therefore,the space H ϕA is separable as well. In the sequel we use the same designations ( · , · ) ϕ and k · k ϕ for the inner product and the corresponding norm in the whole Hilbert space H ϕA . V. MIKHAILETS, A. MURACH, AND T. ZINCHENKO
Operator (2.1) extends uniquely (by continuity) to an isometric isomorphism B : H ϕA ↔ H. (2.2)The equality B ( H ϕA ) = H follows from the fact that the range of ϕ ( A ) coincides with H whenever Spec ϕ ( A ) and that the range is narrower than H but is dense in H whenever ∈ Spec ϕ ( A ) . Hence, ( u , u ) ϕ = ( Bu , Bu ) for every u , u ∈ H ϕA . Besides, H ϕA =Dom ϕ ( A ) if and only if Spec ϕ ( A ) .Remark that we use the same designation H ϕA both in the case where ϕ is a function and inthe case where ϕ is a number. This will not lead to ambiguity because we will always specifywhat ϕ means, a function or number. Of course, this remark also concerns the designationsof the norm and inner product in H ϕA .We need the Hilbert spaces H ϕA such that ϕ ranges over a certain function class OR . Bydefinition, this class consists of all Borel measurable functions ϕ : [1 , ∞ ) → (0 , ∞ ) for whichthere exist numbers a > and c ≥ such that c − ≤ ϕ ( λt ) /ϕ ( t ) ≤ c for all t ≥ and λ ∈ [1 , a ] (the numbers a and c may depend on ϕ ). Such functions were introduced byV. G. Avakumovi´c [10] in 1936, are called OR-varying (or O-regularly varying) at infinity andhave been well investigated [13, 15, 66].The class OR admits the following simple description [66, Theorem A.1]: ϕ ∈ OR if andonly if ϕ ( t ) = exp β ( t ) + t Z γ ( τ ) τ dτ ! , t ≥ , for some bounded Borel measurable functions β, γ : [1 , ∞ ) → R .This class has the following important property [66, Theorem A.2(a)]: for every ϕ ∈ OR there exist real numbers s and s , with s ≤ s , and positive numbers c and c such that c λ s ≤ ϕ ( λt ) ϕ ( t ) ≤ c λ s for all t ≥ and λ ≥ . (2.3)Let ϕ ∈ OR ; considering the left-hand side of the inequality (2.3) in the t = 1 case, weconclude that ϕ ( λ ) ≥ const · e − λ whenever λ ≥ . Hence, the identity mapping on Dom ϕ ( A ) extends uniquely to a continuous embedding operator H ϕA ֒ → H / exp A . This will be shownin the first two paragraphs of the proof of Theorem 2.6, in which we put ϕ ( t ) := ϕ ( t ) and ϕ ( t ) := e − t . Here, of course, H / exp A denotes the Hilbert space H χA parametrized with thefunction χ ( t ) := e − t of t ≥ . Therefore, we will interpret H ϕA as a linear manifold in H / exp A .Thus, all the spaces H ϕA parametrized with ϕ ∈ OR and, hence, all the spaces from theextended A -scale lie in the same space H / exp A , which enables us to compare them. Theorem 2.1.
A Hilbert space X belongs to the extended A -scale if and only if X = H ϕA upto equivalence of norms for certain ϕ ∈ OR .Remark . We cannot transfer from the extended A -scale to a wider class of spaces by meansof interpolation Hilbert spaces between any spaces from this scale. Namely, suppose thatcertain Hilbert spaces H and H belong to the extended A -scale and satisfy the continuousembedding H ֒ → H . Then every Hilbert space X which is an interpolation one for thepair [ H , H ] belongs to this scale as well. Indeed, for each j ∈ { , } , the space H j is aninterpolation one for a certain pair [ H s j, A , H s j, A ] , where s j, < s j, . Besides, both H s j, A and N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 5 H s j, A are interpolation spaces for the pair [ H s A , H s A ] provided that s := min { s , , s , } and s := max { s , , s , } . Hence, the above-mentioned space X is an interpolation one for thelatter pair, which follows directly from the given definition of an interpolation space. Thus, X belongs to the extended A -scale.We will also give an explicit description (up to equivalence of norms) of all Hilbert spacesthat are interpolation ones for the given pair [ H s A , H s A ] , where s < s . Considering ϕ ∈ OR ,we put σ ( ϕ ) := sup { s ∈ R | the left-hand inequality in (2.3) holds } , (2.4) σ ( ϕ ) := inf { s ∈ R | the right-hand inequality in (2.3) holds } . (2.5)Evidently, −∞ < σ ( ϕ ) ≤ σ ( ϕ ) < ∞ . The numbers σ ( ϕ ) and σ ( ϕ ) are equal to the lowerand the upper Matuszewska indices of ϕ , respectively (see [43] and [13, Theorem 2.2.2]). Theorem 2.3.
Let s , s ∈ R and s < s . A Hilbert space X is an interpolation space forthe pair [ H s A , H s A ] if and only if X = H ϕA up to equivalence of norms for a certain functionparameter ϕ ∈ OR that satisfies condition (2.3) .Remark . Of course, we mean in Theorem 2.3 that the positive numbers c and c incondition (2.3) depend neither on t nor on λ . Evidently, this condition is equivalent to thefollowing pair of conditions: (i) s ≤ σ ( ϕ ) and, moreover, s < σ ( ϕ ) if the supremum in (2.4) is not attained; (ii) σ ( ϕ ) ≤ s and, moreover, σ ( ϕ ) < s if the infimum in (2.5) is not attained.It is important for applications that the extended A -scale can be obtained by means ofthe quadratic interpolation (with function parameter) between spaces from A -scale. Beforewe formulate a relevant theorem, we will recall the definition of the quadratic interpolationbetween Hilbert spaces. This interpolation is a natural generalization of the classical inter-polation method by J.-L. Lions [38] and S. G. Krein [35] (see also the book [39, Chapter 1,Sections 2 and 5] and survey [36, Section 9]) to the case where a general enough function isused, instead of the number θ ∈ (0 , , as an interpolation parameter. The generalization firstappeared in C. Foia¸s and J.-L. Lions’ paper [19, Section 3.4]. We mainly follow monograph[51, Section 1.1] (see also [47, Section 2.1]).Let B denote the set of all Borel measurable functions ψ : (0 , ∞ ) → (0 , ∞ ) such that ψ is bounded on each compact interval [ a, b ] , with < a < b < ∞ , and that /ψ is boundedon every set [ r, ∞ ) , with r > . We arbitrarily choose a function ψ ∈ B and a regular pair H := [ H , H ] of separable complex Hilbert spaces. The regularity of this pair means that H is a dense linear manifold in H and that the embedding H ֒ → H is continuous. For H there exists a positive definite self-adjoint linear operator J in H such that Dom J = H andthat k J u k H = k u k H for every u ∈ H . The operator J is uniquely determined by the pair H and is called the generating operator for this pair.Using the spectral decomposition of J , we define the self-adjoint operator ψ ( J ) in H .Let [ H , H ] ψ or, simply, H ψ denote the domain of ψ ( J ) endowed with the inner product ( u , u ) H ψ := ( ψ ( J ) u , ψ ( J ) u ) H and the corresponding norm k u k H ψ = k ψ ( J ) u k H , with u, u , u ∈ H ψ . The space H ψ is Hilbert and separable.A function ψ ∈ B is called an interpolation parameter if the following condition is fulfilledfor all regular pairs H = [ H , H ] and G = [ G , G ] of separable complex Hilbert spaces and V. MIKHAILETS, A. MURACH, AND T. ZINCHENKO for an arbitrary linear mapping T given on H : if the restriction of T to H j is a boundedoperator T : H j → G j for each j ∈ { , } , then the restriction of T to H ψ is also a boundedoperator T : H ψ → G ψ . If ψ is an interpolation parameter, we will say that the Hilbertspace H ψ is obtained by the quadratic interpolation with the function parameter ψ of thepair H (or, in other words, between the spaces H and H ). In this case, the dense continuousembeddings H ֒ → H ψ ֒ → H hold true.A function ψ ∈ B is an interpolation parameter if and only if ψ is pseudoconcave in aneighbourhood of infinity. The latter property means that there exists a number r > anda concave function ψ : ( r, ∞ ) → (0 , ∞ ) that both functions ψ/ψ and ψ /ψ are boundedon ( r, ∞ ) . This key fact follows from J. Peetre’s [61, 62] description of all interpolationfunctions for the weighted L p ( R n ) -type spaces (the description is also set forth in monograph[11, Theorem 5.4.4]).The above-mentioned interpolation property of the extended A -scale is formulated as fol-lows: Theorem 2.5.
Let ϕ ∈ OR , and let real numbers s < s be taken from condition (2.3) . Put ψ ( τ ) := ( τ − s / ( s − s ) ϕ ( τ / ( s − s ) ) whenever τ ≥ ,ϕ (1) whenever < τ < . (2.6) Then the function ψ belongs to B and is an interpolation parameter, and (cid:2) H s A , H s A (cid:3) ψ = H ϕA with equality of norms . (2.7)For instance, considering the function ϕ ( t ) := 1 + log t of t ≥ from the class OR , we cantake s := 0 and s := ε for every ε > and put ψ ( τ ) := 1 + ε − log τ whenever τ ≥ in theinterpolation formula (2.7).Note that, if s < σ ( ϕ ) and s > σ ( ϕ ) , the numbers s and s will satisfy the conditionof Theorem 2.5 whatever ϕ ∈ OR .The extended A -scale is closed with respect to the quadratic interpolation (with functionparameter). This follows directly from the next two results. Theorem 2.6.
Let ϕ , ϕ : [1 , ∞ ) → (0 , ∞ ) be Borel measurable functions. Suppose that thefunction ϕ /ϕ is bounded on [1 , ∞ ) . Then the pair [ H ϕ A , H ϕ A (cid:3) is regular. Let ψ ∈ B , andput ϕ ( t ) := ϕ ( t ) ψ (cid:18) ϕ ( t ) ϕ ( t ) (cid:19) whenever t ≥ . (2.8) Then (cid:2) H ϕ A , H ϕ A (cid:3) ψ = H ϕA with equality of norms . (2.9) Proposition 2.7.
Let ϕ , ϕ ∈ OR and ψ ∈ B . Suppose that the function ϕ /ϕ is boundedin a neighbourhood of infinity and that ψ is an interpolation parameter. Then the function (2.8) belongs to the class OR . This proposition is contained in [52, Theorem 5.2].As to Theorem 2.6, it is necessary to note that its hypothesis allows us to consider H ϕ A and H ϕA as linear manifolds in H ϕ A . Indeed, since the functions ϕ /ϕ and ϕ /ϕ are boundedon [1 , ∞ ) , the identity mappings on Dom ϕ ( A ) and on Dom ϕ ( A ) extend uniquely to some N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 7 continuous embedding operators H ϕ A ֒ → H ϕ A and H ϕA ֒ → H ϕ A respectively (see the first twoparagraphs of the proof of Theorem 2.6). Thus, we may say about the regularity of the pair [ H ϕ A , H ϕ A ] and compare the spaces [ H ϕ A , H ϕ A ] ψ and H ϕA in (2.9).3. Proofs of basic results
We will prove Theorems 2.1, 2.3, 2.5, and 2.6 in the reverse order, which is stipulated by aremarkable result by Ovchinnikov [59, Theorem 11.4.1]. This result explicitly describes (upto equivalence of norms) all the Hilbert spaces that are interpolation ones for an arbitrarycompatible pair of Hilbert spaces. As to our consideration, Ovchinnikov’s theorem can beformulated as follows:
Proposition 3.1.
Let H := [ H , H ] be a regular pair of separable complex Hilbert spaces. AHilbert space X is an interpolation space for H if and only if X = H ψ up to equivalence ofnorms for a certain interpolation parameter ψ ∈ B . Note that all exact interpolation Hilbert spaces for H were characterized (isometrically) byDonoghue [16].Let us turn to the proofs of the theorems formulated in Section 2. Proof of Theorem . . We first show that the pair [ H ϕ A , H ϕ A ] is regular. It follows from thehypothesis of the theorem that Dom ϕ ( A ) ⊆ Dom ϕ ( A ) and that k u k ϕ ≤ κ − k u k ϕ forevery u ∈ Dom ϕ ( A ) , with κ := inf t ≥ ϕ ( t ) ϕ ( t ) > . (3.1)Hence, the identity mapping on Dom ϕ ( A ) extends uniquely to a continuous linear operator I : H ϕ A → H ϕ A . (3.2)Let us prove that this operator is injective.Suppose that Iu = 0 for certain u ∈ H ϕ A . We must prove the equality u = 0 . Choose asequence ( u k ) ∞ k =1 ⊂ Dom ϕ ( A ) such that u k → u in H ϕ A as k → ∞ . Since operator (3.2) isbounded, we have the convergence u k = Iu k → Iu = 0 in H ϕ A . Hence, the sequence ( u k ) ∞ k =1 is a Cauchy one in Dom ϕ ( A ) , and u k → in Dom ϕ ( A ) . Here and below in the proof, thelinear space Dom ϕ j ( A ) is endowed with the norm k · k ϕ j for each j ∈ { , } . Thus, thereexists a vector v ∈ H such that ϕ ( A ) u k → v in H , and ϕ ( A ) u k → in H . Besides, ϕ ( A ) u k = ϕ ϕ ( A ) ϕ ( A ) u k → ϕ ϕ ( A ) v in H because the function ϕ /ϕ is bounded on [1 , ∞ ) . Therefore, ( ϕ /ϕ )( A ) v = 0 . Hence, v = 0 as a vector from H because the function ϕ /ϕ is positive on [1 , ∞ ) . Thus, ϕ ( A ) u k → in H . Therefore, k u k k ϕ = k ϕ ( A ) u k k → , i.e. u = lim k →∞ u k = 0 in H ϕ A . We have provedthat the operator (3.2) is injective.Hence, it realizes a continuous embedding H ϕ A ֒ → H ϕ A . The density of this embeddingfollows directly from the density of Dom ϕ ( A ) in the normed space Dom ϕ ( A ) . Let us provethe latter density. Choose a vector u ∈ Dom ϕ ( A ) arbitrarily. The domain of the operator ϕ ( A )(1 /ϕ )( A ) is dense in H because the closure of this operator coincides with the operator ( ϕ /ϕ )( A ) , whose domain is dense in H . Hence, there exists a sequence ( v k ) ∞ k =1 ⊂ Dom (cid:0) ϕ ( A )(1 /ϕ )( A ) (cid:1) V. MIKHAILETS, A. MURACH, AND T. ZINCHENKO such that v k → ϕ ( A ) u in H as k → ∞ . Putting u k := (1 /ϕ )( A ) v k ∈ Dom ϕ ( A ) for every integer k ≥ , we conclude that ϕ ( A ) u k = v k → ϕ ( A ) u in H . Therefore, Dom ϕ ( A ) ∋ u k → u in Dom ϕ ( A ) . Hence, the set Dom ϕ ( A ) is dense in Dom ϕ ( A ) .Thus, the continuous embedding H ϕ A ֒ → H ϕ A is dense, i.e. the pair [ H ϕ A , H ϕ A (cid:3) is regular.Let us build the generating operator for this pair. Choosing j ∈ { , } arbitrarily, we havethe isometric linear operator ϕ j ( A ) : Dom ϕ j ( A ) → H. This operator extends uniquely (by continuity) to an isometric isomorphism B j : H ϕ j A ↔ H, with j ∈ { , } (3.3)(see the explanation for (2.2)). Define the linear operator J in H ϕ A by the formula J u := B − B u for every u ∈ Dom J := H ϕ A . Let us prove that J is the generating operator for thepair [ H ϕ A , H ϕ A ] .Note first that J sets an isometric isomorphism J = B − B : H ϕ A ↔ H ϕ A . (3.4)Hence, the operator J is closed in H ϕ A . Besides, J is a positive definite operator in H ϕ A .Indeed, choosing u ∈ Dom ϕ ( A ) arbitrarily, we write the following: ( J u, u ) ϕ = ( B − B u, u ) ϕ = ( B u, B u ) = ( ϕ ( A ) u, ϕ ( A ) u )= (cid:16) ϕ ϕ ( A ) ϕ ( A ) u, ϕ ( A ) u (cid:17) ≥ κ ( ϕ ( A ) u, ϕ ( A ) u ) = κ ( u, u ) ϕ , the inequality being due to (3.1). Passing here to the limit and using (3.4), we conclude that ( J u, u ) ϕ ≥ κ ( u, u ) ϕ for every u ∈ H ϕ A . (3.5)Thus, J is a positive definite closed operator in H ϕ A . Moreover, since / ∈ Spec J by (3.4), thisoperator is self-adjoint on H ϕ A . Regarding (3.4) again, we conclude that J is the generatingoperator for the pair [ H ϕ A , H ϕ A ] .Let us reduce the self-adjoint operator J in H ϕ A to an operator of multiplication by function.Since the operator A is self-adjoint in H and since A ≥ , there exists a space R with a finitemeasure µ , a measurable function α : R → [1 , ∞ ) , and an isometric isomorphism I : H ↔ L ( R, dµ ) (3.6)such that Dom A = { u ∈ H : α · I u ∈ L ( R, dµ ) } and that I Au = α · I u for every u ∈ Dom A ; see, e.g, [64, Theorem VIII.4]. Otherwisespeaking, I reduces A to the operator of multiplication by α .Using (3.3) and (3.6), we introduce the isometric isomorphism I := I B : H ϕ A ↔ L ( R, dµ ) . (3.7) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 9
Let us show that I reduces J to an operator of multiplication by function. Given u ∈ Dom ϕ ( A ) , we write the following: I J u = ( I B )( B − B ) u = I B u = I ϕ ( A ) u = I (cid:16) ϕ ϕ (cid:17) ( A ) ϕ ( A ) u = (cid:16) ϕ ϕ ◦ α (cid:17) I ϕ ( A ) u = (cid:16) ϕ ϕ ◦ α (cid:17) I u. Thus, I J u = (cid:16) ϕ ϕ ◦ α (cid:17) I u for every u ∈ Dom ϕ ( A ) . (3.8)Let us prove that this equality holds true for every u ∈ H ϕ A .Choose u ∈ H ϕ A arbitrarily, and consider a sequence ( u k ) ∞ k =1 ⊂ Dom ϕ ( A ) such that u k → u in H ϕ A as k → ∞ . Owing to (3.8), we have the equality I J u k = (cid:16) ϕ ϕ ◦ α (cid:17) I u k whenever ≤ k ∈ Z . (3.9)Here, I J u k → I J u and I u k → I u in L ( R, dµ ) as k → ∞ (3.10)due to the isometric isomorphisms (3.4) and (3.7). Since convergence in L ( R, dµ ) impliesconvergence in the measure µ , it follows from (3.10) by the Riesz theorem that I J u k l → I J u and I u k l → I u µ -a.e. on R as l → ∞ (3.11)for a certain subsequence ( u k l ) ∞ l =1 of ( u k ) ∞ k =1 . The latter convergence implies that (cid:16) ϕ ϕ ◦ α (cid:17) I u k l → (cid:16) ϕ ϕ ◦ α (cid:17) I u µ -a.e. on R as l → ∞ . (3.12)Now formulas (3.9), (3.11), and (3.12) yield the required equality I J u = (cid:16) ϕ ϕ ◦ α (cid:17) I u for every u ∈ H ϕ A . (3.13)It follows from (3.7) and (3.13) that H ϕ A ⊆ n u ∈ H ϕ A : (cid:16) ϕ ϕ ◦ α (cid:17) I u ∈ L ( R, dµ ) o =: Q, (3.14)we recalling H ϕ A = Dom J . Let us prove that H ϕ A = Q in fact. We endow the linear space Q with the norm k u k Q := (cid:13)(cid:13)(cid:13)(cid:16) ϕ ϕ ◦ α (cid:17) I u (cid:13)(cid:13)(cid:13) L ( R,dµ ) . Owing to (3.13) and (3.14), we have the normal embedding H ϕ A ⊆ Q with k u k Q = k u k ϕ for every u ∈ H ϕ A . (3.15)Consider the linear mapping L : u B − I − h(cid:16) ϕ ϕ ◦ α (cid:17) · I u i where u ∈ Q. (3.16)According to the isometric isomorphisms (3.3) and (3.6), we havethe isometric operator L : Q → H ϕ A . (3.17)If Lu = u for every u ∈ H ϕ A , the required equality H ϕ A = Q will follow plainly from (3.14)and the injectivity of (3.17). Let us prove that Lu = u for every u ∈ H ϕ A . Given u ∈ Dom ϕ ( A ) , we write the following: Lu = B − I − h(cid:16) ϕ ϕ ◦ α (cid:17) I ϕ ( A ) u i = B − I − h(cid:16) ϕ ϕ ◦ α (cid:17) ( ϕ ◦ α ) I u i = B − I − (cid:2) ( ϕ ◦ α ) I u (cid:3) = B − ϕ ( A ) u = B − B u = u. Thus, Lu = u for every u ∈ Dom ϕ ( A ) . (3.18)Choose now u ∈ H ϕ A arbitrarily, and let a sequence ( u k ) ∞ k =1 ⊂ Dom ϕ ( A ) converge to u in H ϕ A . Since u k → u in Q by (3.15), we write Lu = lim k →∞ Lu k = lim k →∞ u k = u in H ϕ A in view of (3.17) and (3.18). Thus, Lu = u for every u ∈ H ϕ A , and we have proved therequired equality Dom J = n u ∈ H ϕ A : (cid:16) ϕ ϕ ◦ α (cid:17) I u ∈ L ( R, dµ ) o . (3.19)Formulas (3.13) and (3.19) mean that the operator J is reduced by the isometric isomor-phism (3.7) to the operator of multiplication by the function ( ϕ /ϕ ) ◦ α . Using this fact, wewill prove the required formula (2.9).Since ψ ∈ B , the function /ψ is bounded on [ κ , ∞ ) . Hence, the function ϕ ( t ) ϕ ( t ) = 1 ψ (cid:16) ϕ ( t ) ϕ ( t ) (cid:17) of t ≥ is bounded due to (3.1). Therefore, Dom ϕ ( A ) ⊆ Dom ϕ ( A ) .Choosing u ∈ Dom ϕ ( A ) arbitrarily and using the above-mentioned reductions of A and J to operators of multiplication by function, we write the following: L ( R, dµ ) ∋ I ϕ ( A ) u = ( ϕ ◦ α ) I u = (cid:16) ψ ◦ ϕ ϕ ◦ α (cid:17) ( ϕ ◦ α ) I u = (cid:16) ψ ◦ ϕ ϕ ◦ α (cid:17) I ϕ ( A ) u = (cid:16) ψ ◦ ϕ ϕ ◦ α (cid:17) I u = I ψ ( J ) u. Hence, k u k ϕ = k ϕ ( A ) u k = kI ϕ ( A ) u k L ( R,dµ ) = kI ψ ( J ) u k L ( R,dµ ) = k ψ ( J ) u k ϕ . Therefore,
Dom ϕ ( A ) ⊆ Dom ψ ( J ) , and k u k ϕ = k u k X for every u ∈ Dom ϕ ( A ) , where X :=[ H ϕ A , H ϕ A ] ψ = Dom ψ ( J ) . Passing here to the limit, we infer the normal embedding H ϕA ⊆ X with k u k ϕ = k u k X for every u ∈ H ϕA . (3.20)Besides, as we have just shown, I ϕ ( A ) u = I ψ ( J ) u whenever u ∈ Dom ϕ ( A ) . (3.21)Let us deduce the equality H ϕA = X from (3.20) and (3.21). Using the isometric isomor-phisms (2.2), (3.6), and (3.7), we getthe isometric operator M := B − I − I ψ ( J ) : X → H ϕA . (3.22) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 11
Owing to (3.21), we write
M u = B − I − I ϕ ( A ) u = u for every u ∈ Dom ϕ ( A ) . Therefore,choosing u ∈ H ϕA arbitrarily and considering a sequence ( u k ) ∞ k =1 ⊂ Dom ϕ ( A ) such that u k → u in H ϕA , we get M u = lim k →∞ M u k = lim k →∞ u k = u in H ϕA due to (3.20). Now the required equality H ϕA = X follows from the property M u = u whenever u ∈ H ϕA , the inclusion H ϕA ⊆ X , and the injectivity of the operator (3.22). In view of (3.20),we have proved (2.9) and, hence, Theorem 2.6. (cid:3) We will deduce Theorem 2.5 from Theorem 2.6 with the help of the following result [52,Theorem 4.2]:
Proposition 3.2.
Let s , s ∈ R , s < s , and ψ ∈ B . Put ϕ ( t ) := t s ψ ( t s − s ) for every t ≥ . Then the function ψ is an interpolation parameter if and only if the function ϕ satisfies (2.3) with some positive numbers c and c that are independent of t and λ .Proof of Theorem . . Let us show first that ψ ∈ B . Evidently, the function ψ is Borelmeasurable. Putting t := 1 in (2.3), we write c ϕ (1) λ s ≤ ϕ ( λ ) ≤ c ϕ (1) λ s for arbitrary λ ≥ . Hence, the function ϕ is bounded on every compact subset of [1 , ∞ ) , which yields theboundedness of ψ on every interval (0 , b ] with b > . Besides, ψ ( τ ) := τ − s / ( s − s ) ϕ ( τ / ( s − s ) ) ≥ c ϕ (1) whenever τ ≥ . Therefore, the function /ψ is bounded on (0 , ∞ ) . Thus, ψ ∈ B by the definition of B .It follows from the definition of ψ that ϕ ( t ) = t s ψ ( t s − s ) for every t ≥ . Hence, ψ isan interpolation parameter according to Proposition 3.2, whereas the interpolation property(2.7) is due to Theorem 2.6, in which we put ϕ ( t ) ≡ t s and ϕ ( t ) ≡ t s . (cid:3) Proof of Theorem . . Necessity. Suppose that a Hilbert space X is an interpolation space forthe pair [ H s A , H s A ] . Then, owing to Proposition 3.1, there exists an interpolation parameter ψ ∈ B such that X = [ H s A , H s A ] ψ up to equivalence of norms. According to Theorem 2.6, weget [ H s A , H s A ] ψ = H ϕA with equality of norms; here, ϕ ( t ) := t s ψ ( t s − s ) for every t ≥ . Thus, X = H ϕA up to equivalence of norms. Note that ϕ ∈ OR due to Proposition 2.7 considered inthe case of ϕ ( t ) ≡ t s and ϕ ( t ) ≡ t s . Moreover, according to Proposition 3.2, the function ϕ satisfies condition (2.3). The necessity is proved. Sufficiency.
Suppose now that a Hilbert space X coincides with H ϕA up to equivalence ofnorms for a certain function parameter ϕ ∈ OR that satisfies condition (2.3). Then, owingto Theorem 2.5, we get H ϕA = [ H s A , H s A ] ψ with equality of norms, where the interpolationparameter ψ ∈ B is defined by formula (2.6). Thus, X = [ H s A , H s A ] ψ up to equivalenceof norms. This implies that X is an interpolation space for the pair [ H s A , H s A ] in view ofthe definition of an interpolation parameter (or by Proposition 3.1). The sufficiency is alsoproved. (cid:3) Proof of Theorem . . Necessity. Suppose that a Hilbert space X belongs to the extended A -scale. Then X is an interpolation space for a certain pair [ H s A , H s A ] where s < s . Hence, weconclude by Theorem 2.3 that X = H ϕ (Ω) up to equivalence of norms for a certain functionparameter ϕ ∈ OR . The necessity is proved. Sufficiency.
Suppose now that X = H ϕA up to equivalence of norms for certain ϕ ∈ OR .The function ϕ satisfies condition (2.3) for the numbers s := σ ( ϕ ) − and s := σ ( ϕ )+1 , for example. Therefore, X is an interpolation space for the pair [ H s A , H s A ] due to Theorem 2.3;i.e., X belongs to the extended A -scale. The sufficiency is also proved. (cid:3) Interpolational inequalities
We assume in this section that functions ϕ , ϕ : [1 , ∞ ) → (0 , ∞ ) and ψ : (0 , ∞ ) → (0 , ∞ ) satisfy the hypothesis of Theorem 2.6; i.e., ϕ and ϕ are Borel measurable, and ϕ /ϕ isbounded on [1 , ∞ ) , and ψ belongs to B . Moreover, suppose that ψ is pseudoconcave in aneighbourhood of infinity; then ψ is an interpolation parameter (see, e.g., [51, Theorem 1.9]).Owing to Theorem 2.6, we have the dense continuous embedding H ϕ A ֒ → H ϕ A and the in-terpolation formula [ H ϕ A , H ϕ A ] ψ = H ϕA with equality of norms. Here, the Borel measurablefunction ϕ : [1 , ∞ ) → (0 , ∞ ) is defined by (2.8). Hence, H ϕA is an interpolation space between H ϕ A and H ϕ A .We will obtain some inequalities that estimate (from above) the norm in the interpolationspace H ϕA via the norms in the marginal spaces H ϕ A and H ϕ A with the help of the inter-polation parameter ψ . Such inequalities are naturally called interpolational. Specifically, if ϕ , ϕ ∈ OR , then ϕ ∈ OR as well, due to Proposition 2.7. In this case, these interpolationalinequalities deal with norms in spaces belonging to the extended Hilbert scale.We denote the number κ > by formula (3.1). Owing to [51, Lemma 1.1], the function ψ is pseudoconcave on ( ε, whenever ε > . Hence, according to [51, Lemma 1.2], there existsa number c ψ, κ > such that ψ ( t ) ψ ( τ ) ≤ c ψ, κ max (cid:26) , tτ (cid:27) for all t, τ ∈ [ κ , ∞ ) . (4.1) Theorem 4.1.
Let τ ≥ κ and u ∈ H ϕ A ; then k u k ϕ ≤ c ψ, κ ψ ( τ ) (cid:0) k u k ϕ + τ − k u k ϕ (cid:1) / . (4.2)Before we prove this theorem, let us comment formula (4.2). It follows from (4.1) that ψ ( t ) ≤ c ψ, κ ψ ( τ ) whenever κ ≤ t ≤ τ and that ψ ( t ) /t ≤ c ψ, κ ψ ( τ ) /τ whenever κ ≤ τ ≤ t .Hence, ψ is slowly equivalent to an increasing function on the set [ κ , ∞ ) , and the function ψ ( τ ) /τ is slowly equivalent to a decreasing function on the same set. Of the main interest isthe case where ψ ( τ ) → ∞ and ψ ( τ ) /τ → as τ → ∞ . In this case, it is useful to rewriteinequality (4.2) in the form k u k ϕ ≤ c ψ, κ (cid:18) ψ ( τ ) τ k u k ϕ + ψ ( τ ) k u k ϕ (cid:19) / . (4.3)Restricting ourselves to the Hilbert scale { H sA : s ∈ R } , we conclude that this inequalitybecomes k u k s ≤ (cid:0) τ θ − k u k s + τ θ k u k s (cid:1) / whenever u ∈ H s A ; (4.4)here, the real numbers s , s , s , θ , and τ satisfy the conditions s < s , < θ < , s = (1 − θ ) s + θs , (4.5)and τ ≥ . Indeed, we only need to take ϕ ( t ) ≡ t s , ϕ ( t ) ≡ t s , ψ ( τ ) ≡ τ θ , and ϕ ( t ) ≡ t s by (2.8) (4.6) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 13 in (4.3) and observe that c ψ, κ = 1 for ψ taken. Interpolational inequalities of type (4.4) forSobolev scales are used in the theory of partial differential operators (see, e.g., [3, Section 1,Subsection 6]). Proof of Theorem . . Let J denote the generating operator for the pair [ H ϕ A , H ϕ A ] . Recallthat J is a positive definite self-adjoint operator in the Hilbert space H ϕ A and that k J u k ϕ = k u k ϕ for every u ∈ H ϕ A = Dom J . According to (3.5), we have Spec J ⊆ [ κ , ∞ ) .Let E t , t ≥ κ , be the resolution of the identity associated with the self-adjoint operator J .Choosing τ ≥ κ and u ∈ H ϕ A arbitrarily, we get k u k ϕ = k ψ ( J ) u k ϕ = ∞ Z κ ψ ( t ) d ( E t u, u ) ϕ ≤ c ψ, κ ψ ( τ ) ∞ Z κ max (cid:26) , t τ (cid:27) d ( E t u, u ) ϕ ≤ c ψ, κ ψ ( τ ) ∞ Z κ (cid:18) t τ (cid:19) d ( E t u, u ) ϕ = c ψ, κ ψ ( τ ) (cid:0) k u k ϕ + τ − k J u k ϕ (cid:1) = c ψ, κ ψ ( τ ) (cid:0) k u k ϕ + τ − k u k ϕ (cid:1) . Here, we use the interpolation formula [ H ϕ A , H ϕ A ] ψ = H ϕA and inequality (4.1). Thus, therequired inequality (4.2) is proved. (cid:3) Let us consider an application of Theorem 4.1. We arbitrarily choose u ∈ H ϕ A such that u = 0 . Put τ := k u k ϕ / k u k ϕ in (4.2) and note that τ ≥ κ in view of (3.1). We then obtainthe interpolational inequality k u k ϕ ≤ c ψ, κ √ k u k ϕ ψ (cid:18) k u k ϕ k u k ϕ (cid:19) . (4.7)If the function χ ( τ ) := ψ ( √ τ ) is concave on [ κ , ∞ ) , (4.8)then (4.7) holds true without the factor c ψ, κ √ . Indeed, choosing v ∈ H ϕ A with k v k ϕ = 1 arbitrarily, we get k v k ϕ = ∞ Z κ ψ ( t ) d ( E t v, v ) ϕ = ∞ Z κ χ ( t ) d ( E t v, v ) ϕ ≤ χ ∞ Z κ t d ( E t v, v ) ϕ ! = χ (cid:0) k J v k ϕ (cid:1) = χ (cid:0) k v k ϕ (cid:1) = ψ (cid:0) k v k ϕ (cid:1) due to the Jensen inequality applied to the concave function χ . Here, E t is the same as thatin the proof of Theorem 4.1, and ∞ Z κ d ( E t v, v ) ϕ = k v k ϕ = 1 . Putting v := u/ k u k ϕ in the inequality k v k ϕ ≤ ψ ( k v k ϕ ) just obtained, we conclude that k u k ϕ ≤ k u k ϕ ψ (cid:18) k u k ϕ k u k ϕ (cid:19) under condition (4.8) . (4.9)This interpolational inequality is equivalent to Variable Hilbert Scale Inequality [25, The-orem 1, formula (9)] on the supplementary assumption that both functions ϕ and ϕ arecontinuous. (Note that the norm k · k ϕ used in the cited article [25] means the norm k · k √ ϕ used by us. Besides, there is no assumption in this article that the function ϕ /ϕ is bounded.)In the case of the Hilbert scale { H sA : s ∈ R } , inequality (4.9) becomes k u k s ≤ k u k − θs k u k θs (4.10)provided that the real numbers s , s , s , and θ satisfy (4.5) (we use the power functions (4.6)).The interpolational inequality (4.10) is well known [36, Section 9, Subsection 1] and meansthat the Hilbert scale is a normal scale of spaces.The interpolational inequalities just considered deal with norms of vectors. Now we focusour attention on interpolational inequalities that involve norms of linear operators actingcontinuously between appropriate Hilbert spaces H ϕA and G ηQ . Here, G (just as H ) is aseparable infinite-dimensional complex Hilbert space, and Q is a counterpart of A for G .Namely, Q is a self-adjoint unbounded linear operator in G such that Spec Q ⊆ [1 , ∞ ) .We suppose that functions η , η , η : [1 , ∞ ) → (0 , ∞ ) satisfy analogous conditions to thoseimposed on ϕ , ϕ , and ϕ at the beginning of this section. Namely, these functions are Borelmeasurable, and the function η /η is bounded, and η ( t ) = η ( t ) ψ (cid:18) η ( t ) η ( t ) (cid:19) whenever t ≥ . (4.11)We suppose that a linear mapping T is given on H ϕ A and satisfies the following condition:the restriction of T to the space H ϕ j A is a bounded operator T : H ϕ j A → G η j Q for each j ∈ { , } . (4.12)Then the restriction of T to H ϕA is a bounded operator T : H ϕA = (cid:2) H ϕ A , H ϕ A (cid:3) ψ → (cid:2) G η Q , G η Q (cid:3) ψ = G ηQ (4.13)according to Theorem 2.6 and because ψ is an interpolation parameter. Let k T k j and k T k denote the norms of operators (4.12) and (4.13) respectively. Then k T k ≤ c max {k T k , k T k } (4.14)for some number c > that does not depend on T but may depend on ψ and the spaces H ϕ j A and G η j Q (see, e.g., [11, Theorem 2.4.2]). This is an interpolational inequality for operatornorms, which means that the method of quadratic interpolation is uniform. N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 15
We will consider a more precise interpolational inequality than (4.14); it involves ψ in someway. Put ν := min (cid:26) inf t ≥ ϕ ( t ) ϕ ( t ) , inf t ≥ η ( t ) η ( t ) (cid:27) > , and let c ψ,ν denote a positive number such that inequality (4.1) holds true with ν takeninstead of κ . Without loss of generality we suppose that ψ ( t ) ψ ( τ ) ≤ c ψ,ν max (cid:26) , tτ (cid:27) for all t, τ > . (4.15)(Hence, ψ is pseudoconcave on (0 , ∞ ) according to [11, Lemma 5.4.3].) We can achieve thisby redefining ψ properly on (0 , ν ) , e.g., by the formula ψ ( t ) := ψ ( ν ) ν − t whenever < t < ν .This does not change ϕ and η in view of (2.8) and (4.11). Let e ψ denote the dilation functionfor ψ , i.e. e ψ ( λ ) := sup t> ψ ( λt ) ψ ( t ) ≤ c ψ,ν max { , λ } whenever λ > . (4.16) Theorem 4.2.
The following interpolational inequality holds true: k T k ≤ c ψ,ν √ k T k e ψ (cid:18) k T k k T k (cid:19) . (4.17) Proof.
It follows directly from Theorem 2.6 (namely, from the equalities in (4.13)) and theresult by Fan [18, formula (2.3)] that inequality (4.17) holds true with a certain number c > written instead of c ψ,ν √ . Note that Fan [18] denotes the interpolation space [ H , H ] ψ by H χ where χ ( t ) ≡ ψ ( √ t ) and that ψ is pseudoconcave on (0 , ∞ ) if and only if so is χ , withthe dilation function e χ ( t ) ≡ e ψ ( √ t ) . (As in Section 2, [ H , H ] is a regular pair of separablecomplex Hilbert spaces.) Besides, if χ is concave on (0 , ∞ ) , then c = √ . This is a directconsequence of Theorem 2.6 and the inequality [18, formula (2.2)]. Let us show that we maytake c = c ψ,ν √ in (4.17) in the general case where ψ is pseudoconcave on (0 , ∞ ) .Considering the function χ ( t ) := ψ ( √ t ) of t > and using (4.15), we have χ ( t ) χ ( τ ) ≤ c ψ,ν max (cid:26) , tτ (cid:27) for all t, τ > . It follows from this that c ψ,ν χ ( t ) ≤ χ ( t ) ≤ χ ( t ) whenever t > , with χ : (0 , ∞ ) → (0 , ∞ ) being the least concave majorant of χ (see [62, p. 91]). Hence, √ c ψ,ν ψ ∗ ( t ) ≤ ψ ( t ) ≤ ψ ∗ ( t ) whenever t > , (4.18)where ψ ∗ ( t ) := p χ ( t ) of t > . Since the function ψ ∗ ( √ t ) ≡ χ ( t ) is concave on (0 , ∞ ) , weconclude by [18, formula (2.2)] that k T k ∗ ≤ √ k T k f ψ ∗ (cid:18) k T k k T k (cid:19) . (4.19) Here, k T k ∗ denotes the norm of the bounded operator T : (cid:2) H ϕ A , H ϕ A (cid:3) ψ ∗ → (cid:2) G η Q , G η Q (cid:3) ψ ∗ , and f ψ ∗ is the dilation function for ψ ∗ . It follows from (4.13) and (4.18) that k T k ≤ √ c ψ,ν k T k ∗ . (4.20)Besides, f ψ ∗ ( λ ) = sup t> ψ ∗ ( λt ) ψ ∗ ( t ) ≤ √ c ψ,ν e ψ ( λ ) whenever λ > . (4.21)Now (4.19), (4.20), and (4.21) yield the required inequality (4.17). (cid:3) The inequality (4.17) is more precise than (4.14) in view of (4.16).
Remark . If the function ψ is concave on (0 , ∞ ) , then c ψ,ν = 1 in inequality (4.17). Besides,we may write √ instead of c ψ,ν √ in this inequality provided that the function ψ ( √ t ) of t > is concave on (0 , ∞ ) , as we have noted in the proof of Theorem 4.2.5. Applications to function spaces
In this section, we will show how the concept of the extended Hilbert scale allows us tointroduce and investigate wide classes of Hilbert function (or distribution) spaces related toSobolev spaces on manifolds.Let Γ be a separable paracompact infinitely smooth Riemannian manifold without bound-ary. Consider the separable complex Hilbert space H := L (Γ) of all functions f : Γ → C which are square integrable over Γ with respect to the Riemann measure. Let ∆ Γ be theLaplace – Beltrami operator on Γ ; it is defined on the linear manifold C ∞ (Γ) of all compactlysupported functions f ∈ C ∞ (Γ) . Assume that the closure of this operator is self-adjointin H , and denote this closure by ∆ Γ too. (Specifically, this self-adjointness follows fromthe completeness of Γ under the Riemann metric [21, p. 140]. For incomplete Riemannianmanifolds, sufficient conditions for the self-adjointness are given, e.g., in [14, 53]). Then theoperator A := (1 − ∆ Γ ) / is self-adjoint and positive definite with the lower bound r = 1 .Therefore, the separable Hilbert space H ϕA is defined for every Borel measurable function ϕ : [1 , ∞ ) → (0 , ∞ ) ; we denote this space by H ϕA (Γ) . If ϕ ( t ) ≡ t s for certain s ∈ R , then H ϕA (Γ) becomes the Sobolev space H s (Γ) of order s . According to Theorem 2.1, the extended A -scale consists (up to equivalence of norms) of all Hilbert spaces H ϕA (Γ) where ϕ ∈ OR . Inother words, the class { H ϕA (Γ) : ϕ ∈ OR } consists of all interpolation Hilbert spaces betweeninner product Sobolev spaces over Γ . Therefore, it is naturally to call this class the extendedSobolev scale over Γ .Now we focus our attention on two important cases where Γ is the Euclidean space R n and where Γ is a compact boundaryless manifold. Generalizing the above consideration, weuse some elliptic operators as A . We will prove that the extended Hilbert scales generatedby these operators consist of some distribution spaces, admit explicit description with thehelp of the Fourier transform and local charts on Γ , and do not depend on the choice ofthe elliptic operators and these charts. Since we consider complex linear spaces formed byfunctions or distributions, all functions and distributions are supposed to be complex-valuedunless otherwise stated. N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 17 R n , with n ≥ . Here, we put H := L ( R n ) and note that the Riemannmeasure on R n is the Lebesgue measure. Let ( · , · ) R n and k · k R n stand respectively for theinner product and norm in L ( R n ) .Following [1, Section 1.1], we let Ψ m ( R n ) , where m ∈ R , denote the class of all pseudo-differential operator (PsDO) A on R n whose symbols a belong to C ∞ ( R n ) and satisfy thefollowing condition: for every multi-indices α, β ∈ Z n + there exists a number c α,β > suchthat | ∂ αx ∂ βξ a ( x, ξ ) | ≤ c α,β (1 + | ξ | ) m −| β | for all x, ξ ∈ R n , with x and ξ being considered respectively as spatial and frequency variables. If A ∈ Ψ m ( R n ) ,we say that the (formal) order of A is m . A PsDO A ∈ Ψ m ( R n ) is called uniformly elliptic on R n if there exist positive numbers c and c such that | a ( x, ξ ) | ≥ c | ξ | m whenever x, ξ ∈ R n and | ξ | ≥ c (see [1, Section 3.1 b]).We suppose henceforth in this subsection that(a) A is a PsDO of class Ψ ( R n ) ;(b) A is uniformly elliptic on R n ;(c) the inequality ( A w, w ) R n ≥ k w k R n holds true for every w ∈ C ∞ ( R n ) .Here, as usual, C ∞ ( R n ) denotes the set of all compactly supported functions u ∈ C ∞ ( R n ) .Consider the mapping w
7→ A w, with w ∈ C ∞ ( R n ) , (5.1)as an unbounded linear operator in the Hilbert space H = L ( R n ) . This operator is closablebecause the PsDO A ∈ Ψ ( R n ) acts continuously from L ( R n ) to H − ( R n ) (see [1, Theorem1.1.2]). Here and below, H s ( R n ) denotes the inner product Sobolev space of order s ∈ R over R n .Let A denote the closure of the operator (5.1) in L ( R n ) . It follows from (a) and (b) that Dom A = H ( R n ) (see [1, Sections 2.3 c and 3.1 b]). Owing to (c), the operator A is positivedefinite with the lower bound r = 1 . This implies due to (b) that A is self-adjoint [1, Sections2.3 d and 3.1 b], with Spec A ⊆ [1 , ∞ ) .Thus, A is the operator considered in Section 2. Therefore, the separable Hilbert space H ϕA is defined for every Borel measurable function ϕ : [1 , ∞ ) → (0 , ∞ ) . We denote this space by H ϕA ( R n ) . An important example of A is the operator (1 − ∆) / , with ∆ denoting the Laplaceoperator in R n . In this case, the PsDO A has the symbol a ( x, ξ ) ≡ (1 + | ξ | ) / .If ϕ ( t ) ≡ t s for certain s ∈ R , then it is possible to show that the space H sA ( R n ) := H ϕA ( R n ) coincides with the Sobolev space H s ( R n ) up to equivalence norms. Thus, the A -scale { H sA ( R n ) : s ∈ R } is the Sobolev Hilbert scale. Let us show that the extended A -scale consistsof some generalized Sobolev spaces, namely the spaces H ϕ ( R n ) with ϕ ∈ OR .Let ϕ ∈ OR . By definition, the linear space H ϕ ( R n ) consists of all distributions w ∈ S ′ ( R n ) that their Fourier transform b w is locally Lebesgue integrable over R n and satisfies the condition Z R n ϕ ( h ξ i ) | b w ( ξ ) | dξ < ∞ . Here, as usual, S ′ ( R n ) is the linear topological space of all tempered distributions on R n , and h ξ i := (1 + | ξ | ) / is the smoothed modulus of ξ ∈ R n . The space H ϕ ( R n ) is endowed with the inner product ( w , w ) ϕ, R n := Z R n ϕ ( h ξ i ) c w ( ξ ) c w ( ξ ) dξ and the corresponding norm k w k ϕ, R n := ( w, w ) / ϕ, R n . This space is complete and separablewith respect to this norm and is embedded continuously in S ′ ( R n ) ; the set C ∞ ( R n ) is densein H ϕ ( R n ) [27, Theorem 2.2.1].If ϕ ( t ) ≡ t s for some s ∈ R , then H ϕ ( R n ) becomes the Sobolev space H s ( R n ) . The Hilbertspace H ϕ ( R n ) is an isotropic case of the spaces introduced and investigated by Malgrange[40], H¨ormander [27, Sec. 2.2] (see also [28, Section 10.1]), and Volevich and Paneah [72,Section 2]. These spaces are generalizations of Sobolev spaces to the case where a generalenough function of frequency variables serves as an order of distribution spaces. Theorem 5.1.
Let ϕ ∈ OR . Then the spaces H ϕA ( R n ) and H ϕ ( R n ) coincide as completionsof C ∞ ( R n ) with respect to equivalent norms. It is worth-wile to note that the norm of w ∈ C ∞ ( R n ) in H ϕA ( R n ) is k ϕ ( A ) w k R n because Dom ϕ ( A ) ⊃ Dom A s = H s A ( R n ) ⊃ C ∞ ( R n ); here s is a positive number that satisfies (2.3).According to Theorem 5.1, the space H ϕA ( R n ) , with ϕ ∈ OR , does not depend on A upto equivalence of norms. Theorems 2.1 and 5.1 yield the following explicit description of theextended Hilbert scale generated by the considered operator A : Corollary 5.2.
The extended A -scale consists (up to equivalence of norms) of all the spaces H ϕ ( R n ) with ϕ ∈ OR . Thus, the class { H ϕ ( R n ) : ϕ ∈ OR } consists (up to equivalence if norms) of all Hilbertspaces each of which is an interpolation space between some Sobolev spaces H s ( R n ) and H s ( R n ) with s < s . As we have noted, this class is called the extended Sobolev scaleover R n . It was considered in [50, 52] and [51, Section 2.4.2]. Proof of Theorem . . Choose an integer k ≫ such that − k < σ ( ϕ ) and k > σ ( ϕ ) ,and define the interpolation parameter ψ by formula (2.6) in which s := − k and s := k .According to Theorem 2.5, we have the equality H ϕA ( R n ) = (cid:2) H − kA ( R n ) , H kA ( R n ) (cid:3) ψ . (5.2)Note that each space H ± kA ( R n ) coincides with H ± k ( R n ) up to equivalence of norms. Indeed, A sets an isomorphism between H ( R n ) and L ( R n ) because Dom A = H ( R n ) and / ∈ Spec A . Besides, since the PsDO A of the first order is uniformly elliptic on R n , the operator A has the following lifting property: if u ∈ H ( R n ) and if Au ∈ H s − ( R n ) for some s > ,then u ∈ H s ( R n ) (see [1, Sections 1.8 and 3.1 b]). Hence, A k sets an isomorphism between H k ( R n ) and L ( R n ) . Thus, H kA ( R n ) = H k ( R n ) up to equivalence of norms. Passing here todual spaces with respect to L ( R n ) , we conclude that H − kA ( R n ) = H − k ( R n ) up to equivalenceof norms (see [36, Section 9, Subsection 1]).Thus, it follows from (5.2) that H ϕA ( R n ) = (cid:2) H − k ( R n ) , H k ( R n ) (cid:3) ψ (5.3) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 19 up to equivalence of norms. Indeed, since the identity mapping sets the isomorphisms I : H ∓ kA ( R n ) ↔ H ∓ k ( R n ) and since ψ is an interpolation parameter, the identity mapping realizes the isomorphism I : (cid:2) H − kA ( R n ) , H kA ( R n ) (cid:3) ψ ↔ (cid:2) H − k ( R n ) , H k ( R n ) (cid:3) ψ . This yields (5.3) due to (5.2).Thus, Theorem 5.1 follows directly from (5.3) and H ϕ ( R n ) = (cid:2) H − k ( R n ) , H k ( R n ) (cid:3) ψ . (5.4)The letter equality is proved in [51, Theorem 2.19]. Besides, (5.4) is a special case of (5.3)because H ϕ ( R n ) = H ϕA ( R n ) if A = (1 − ∆) / . Indeed, since the Fourier transform reduces A = (1 − ∆) / to the operator of multiplication by h ξ i , we conclude that Dom ϕ ( A ) ⊆ H ϕ ( R n ) and k ϕ ( A ) u k R n = k u k ϕ, R n for every u ∈ Dom ϕ ( A ) . Hence, H ϕA ( R n ) is a subspace of H ϕ ( R n ) .But C ∞ ( R n ) and then H ϕA ( R n ) are dense in H ϕ ( R n ) . Thus, H ϕA ( R n ) coincides with H ϕ ( R n ) if A = (1 − ∆) / . (cid:3) Ending this subsection, we note the following: in contrast to the spaces H s ( R n ) , the innerproduct Sobolev spaces H s (Ω) := (cid:8) w ↾ Ω : w ∈ H s ( R n ) (cid:9) , with s ∈ R , over a domain Ω ⊂ R n do not form a Hilbert scale even if Ω is a bounded domain withinfinitely smooth boundary and if we restrict ourselves to the spaces of order s ≥ [57,Corollary 2.3]. An explicit description of all Hilbert spaces that are interpolation ones for anarbitrary chosen pair of Sobolev spaces H s (Ω) and H s (Ω) , where −∞ < s < s < ∞ , isgiven in [52, Theorem 2.4] provided that Ω is a bounded domain with Lipschitz boundary.These interpolation spaces are (up to equivalence of norms) the generalized Sobolev spaces H ϕ (Ω) := (cid:8) v := w ↾ Ω : w ∈ H ϕ ( R n ) (cid:9) such that ϕ belongs to OR and satisfies (2.3), the Hilbert norm in H ϕ (Ω) being naturallydefined by the formula k v k ϕ, Ω := inf (cid:8) k w k ϕ, R n : w ∈ H ϕ ( R n ) , v = w ↾ Ω (cid:9) . Γ be an arbitrary closed (i.e. compact and boundaryless) infinitelysmooth manifold of dimension n ≥ . We suppose that a certain positive C ∞ -density dx isgiven on Γ . We put H := L (Γ) , where L (Γ) is the complex Hilbert space of all squareintegrable functions over Γ with respect to the measure induced by this density. Let ( · , · ) Γ and k · k Γ stand respectively for the inner product and norm in L (Γ) .Following [1, Section 2.1], we let Ψ m (Γ) , where m ∈ R , denote the class of all PsDOs on Γ whose representations in every local chart on Γ belong to Ψ m ( R n ) . If A ∈ Ψ m (Γ) , we saythat the (formal) order of A is m . A PsDO A ∈ Ψ m (Γ) is called elliptic on Γ if for everypoint x ∈ Γ there exist positive numbers c and c such that | a x ( x, ξ ) | ≥ c | ξ | m whenever x ∈ U ( x ) and ξ ∈ R n and | ξ | ≥ c , with a x ( x, ξ ) be the local symbol of A corresponding toa certain coordinate neighbourhood U ( x ) of x (see [1, Section 3.1 b]).We suppose in this subsection that (a) A is a PsDO of class Ψ (Γ) ;(b) A is elliptic on Γ ;(c) the inequality ( A f, f ) Γ ≥ k f k holds true for every f ∈ C ∞ (Γ) .Let A denote the closure, in H = L (Γ) , of the linear operator f
7→ A f , with f ∈ C ∞ (Γ) .Note that this operator is closable in L (Γ) because the PsDO A ∈ Ψ (Γ) acts continuouslyfrom L (Γ) to H − (Γ) [1, Theorem 2.1.2]. Here and below, H s (Γ) stands for the inner productSobolev space of order s ∈ R over Γ . It follows from (a)–(c) that the operator A is positivedefinite and self-adjoint in L (Γ) , with Dom A = H (Γ) and Spec A ⊆ [1 , ∞ ) (see [1, Sections2.3 c, d and 3.1 b]).Thus, A is the operator considered in Section 2, and the separable Hilbert space H ϕA isdefined for every Borel measurable function ϕ : [1 , ∞ ) → (0 , ∞ ) . We denote this space by H ϕA (Γ) . An important example of A is the operator (1 − ∆ Γ ) / , where Γ is endowed with aRiemann metric (then the density dx is induced by this metric).If ϕ ( t ) ≡ t s for some s ∈ R , then the space H sA (Γ) := H ϕA (Γ) coincides with the Sobolevspace H s (Γ) up to equivalence norms [1, Corollary 5.3.2]. Thus, the A -scale { H sA (Γ) : s ∈ R } is the Sobolev Hilbert scale over Γ . We will prove that the extended A -scale consists of thegeneralized Sobolev spaces H ϕ (Γ) with ϕ ∈ OR . Let us give their definition with the help oflocal charts on Γ .We arbitrarily choose a finite atlas from the C ∞ -structure on Γ ; let this atlas be formedby κ local charts π j : R n ↔ Γ j , with j = 1 , . . . , κ . Here, the open sets Γ , . . . , Γ κ form acovering of Γ . We also arbitrarily choose functions χ j ∈ C ∞ (Γ) , with j = 1 , . . . , κ , that forma partition of unity on Γ such that supp χ j ⊂ Γ j .Let ϕ ∈ OR . By definition, the linear space H ϕ (Γ) is the completion of the linear manifold C ∞ (Γ) with respect to the inner product ( f , f ) ϕ, Γ := κ X j =1 (( χ j f ) ◦ π j , ( χ j f ) ◦ π j ) ϕ, R n (5.5)of functions f , f ∈ C ∞ (Γ) . Thus, H ϕ (Γ) is a Hilbert space. Let k · k ϕ, Γ denote the norminduced by the inner product (5.5). If ϕ ( t ) ≡ t s for certain s ∈ R , then H ϕ (Γ) becomes theSobolev space H s (Γ) . Theorem 5.3.
Let ϕ ∈ OR . Then the spaces H ϕA (Γ) and H ϕ (Γ) coincide as completions of C ∞ (Γ) with respect to equivalent norms. Note that the norm of f ∈ C ∞ (Γ) in H ϕA (Γ) is k ϕ ( A ) f k Γ because Dom ϕ ( A ) ⊃ Dom A s ⊃ C ∞ (Γ); here s is a positive integer that satisfies (2.3).Theorem 5.3 specifically entails Corollary 5.4.
Let ϕ ∈ OR . Then the space H ϕA (Γ) does not depend on A up to equivalenceof norms. Besides, the space H ϕ (Γ) does not depend (up to equivalence of norms) on ourchoice of the atlas and partition of unity on Γ . Owing to Theorems 2.1 and 5.3, we obtain the following explicit description of the extendedHilbert scale generated by the considered PsDO A on Γ : N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 21
Corollary 5.5.
The extended A -scale consists (up to equivalence if norms) of all the spaces H ϕ (Γ) with ϕ ∈ OR . Thus, the class { H ϕ (Γ) : ϕ ∈ OR } consists (up to equivalence if norms) of all Hilbert spaceseach of which is an interpolation space between some Sobolev inner-product spaces H s (Γ) and H s (Γ) with s < s . This class is called the extended Sobolev scale over Γ . Theorem 5.6.
Suppose that the manifold Γ is endowed with a Riemann metric, and let ϕ ∈ OR . Then the space H ϕ (Γ) admits the following three equivalent definitions in the sensethat they introduce the same Hilbert space up to equivalence of norms: (i) Operational definition.
The Hilbert space H ϕ (Γ) is the completion of C ∞ (Γ) withrespect to the norm k ϕ ((1 − ∆ Γ ) / ) f k Γ of f ∈ C ∞ (Γ) . (ii) Local definition.
The Hilbert space H ϕ (Γ) consists of all distributions f ∈ D ′ (Γ) such that ( χ j f ) ◦ π j ∈ H ϕ ( R n ) for every j ∈ { , . . . , κ } and is endowed with the innerproduct (5.5) of distributions f , f ∈ H ϕ (Γ) . (iii) Interpolational definition.
Let integers s and s satisfy the conditions s < σ ( ϕ ) and s > σ ( ϕ ) , and let ψ be the interpolation parameter defined by (2.6) . Then H ϕ (Γ) := (cid:2) H s (Γ) , H s (Γ) (cid:3) ψ . Here, as usual, D ′ (Γ) is the linear topological space of all distributions on Γ , and ( χ j f ) ◦ π j stands for the representation of the distribution χ j f ∈ D ′ (Γ) in the local chart π j . Wenaturally interpret D ′ (Γ) as the dual of C ∞ (Γ) with respect to the extension of the innerproduct in L (Γ) . This extension is denoted by ( · , · ) Γ as well. Of course, the C ∞ -density dx is now induced by the Riemann metric. Remark . It follows directly from Theorem 5.3 that in the operational definition we maychange (1 − ∆ Γ ) / for the more general PsDO A considered in this subsection.Let us prove Theorems 5.3 and 5.6. Proof of Theorem . . Let the integer k ≫ and the interpolation parameter ψ be the sameas those in the proof of Theorem . . Then H ϕA (Γ) = (cid:2) H − kA (Γ) , H kA (Γ) (cid:3) ψ (5.6)due to Theorem 2.5. Here, H ± kA (Γ) = H ± k (Γ) up to equivalence of norms, which is demon-strated in the same way as that in the proof of Theorem . . Therefore, (5.6) implies that H ϕA (Γ) = (cid:2) H − k (Γ) , H k (Γ) (cid:3) ψ (5.7)up to equivalence of norms. Hence, we have the dense continuous embedding H k (Γ) ֒ → H ϕA (Γ) ,which entails the density of C ∞ (Γ) in H ϕA (Γ) .Owing to (5.7), it remains to show that (cid:2) H − k (Γ) , H k (Γ) (cid:3) ψ = H ϕ (Γ) (5.8)up to equivalence of norms. We will deduce this formula from (5.4) with the help of certainoperators of flattening and sewing of the manifold Γ .Let us define the flattening operator by the formula T : f (( χ f ) ◦ π , . . . , ( χ κ f ) ◦ π κ ) for every f ∈ C ∞ (Γ) . (5.9) The mapping (5.9) extends by continuity to isometric linear operators T : H ϕ (Γ) → ( H ϕ ( R n )) κ (5.10)and T : H ∓ k (Γ) → ( H ∓ k ( R n )) κ . (5.11)Since ψ is an interpolation parameter, it follows from the boundedness of the operators (5.11)that a restriction of the first operator acts continuously T : (cid:2) H − k (Γ) , H k (Γ) (cid:3) ψ → (cid:2) ( H − k ( R n )) κ , ( H k ( R n )) κ (cid:3) ψ . (5.12)Here, the target space equals ( H ϕ ( R n )) κ due to (5.4) and the definition of the interpolationwith the parameter ψ . Thus, the operator (5.12) acts continuously T : (cid:2) H − k (Γ) , H k (Γ) (cid:3) ψ → (cid:0) H ϕ ( R n ) (cid:1) κ . (5.13)We define the sewing operator by the formula K : w κ X j =1 Θ j (cid:0) ( η j w j ) ◦ π − j (cid:1) for every w := ( w , . . . , w κ ) ∈ (cid:0) C ∞ ( R n ) (cid:1) κ . (5.14)Here, for each j ∈ { , . . . , κ } , the function η j ∈ C ∞ ( R n ) is chosen such that η j = 1 in aneighbourhood of π − j (supp χ j ) . Besides, for every function ω : Γ j → C , we put (Θ j ω )( x ) := ω ( x ) whenever x ∈ Γ j and put (Θ j ω )( x ) := 0 whenever x ∈ Γ \ Γ j . Thus, K w ∈ C ∞ (Γ) forevery w ∈ ( C ∞ ( R n )) κ .The mapping K is left inverse to the flattening operator (5.9). Indeed, given f ∈ C ∞ (Γ) ,we have the following equalities: KT f = κ X j =1 Θ j (cid:16)(cid:0) η j (( χ j f ) ◦ π j ) (cid:1) ◦ π − j (cid:17) = κ X j =1 Θ j (cid:0) ( η j ◦ π − j )( χ j f ) (cid:1) = κ X j =1 Θ j ( χ j f ) = κ X j =1 χ j f = f. Thus,
KT f = f for every f ∈ C ∞ (Γ) . (5.15)There exists a number c > such that k K w k ϕ, Γ ≤ c κ X l =1 k w l k ϕ, R n whenever w ∈ (cid:0) C ∞ ( R n ) (cid:1) κ . (5.16) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 23
Indeed, k K w k ϕ, Γ = κ X j =1 k ( χ j K w ) ◦ π j k ϕ, R n = κ X j =1 (cid:13)(cid:13)(cid:13) κ X l =1 (cid:0) χ j Θ l (( η l w l ) ◦ π − l ) (cid:1) ◦ π j (cid:13)(cid:13)(cid:13) ϕ, R n = κ X j =1 (cid:13)(cid:13)(cid:13) κ X l =1 ( η l,j w l ) ◦ β l,j (cid:13)(cid:13)(cid:13) ϕ, R n ≤ c κ X l =1 k w l k ϕ, R n . Here, η l,j := ( χ j ◦ π l ) η l ∈ C ∞ ( R n ) , and β l,j : R n → R n is a C ∞ -diffeomorphism such that β l,j = π − l ◦ π j in a neighbourhood of supp η l,j and that β l,j ( t ) = t whenever | t | is sufficientlylarge. The last inequality is a consequence of the fact that the operator of the multiplicationby a function from C ∞ ( R n ) and the operator v v ◦ β l,j of change of variables are boundedon the space H ϕ ( R n ) . These properties of H ϕ ( R n ) follow by (5.4) from their known analogsfor the Sobolev spaces H ∓ k ( R n ) .According to (5.16), the mapping (5.14) extends by continuity to a bounded linear operator K : ( H ϕ ( R n )) κ → H ϕ (Γ) (5.17)and, specifically, to bounded linear operators K : ( H ∓ k ( R n )) κ → H ∓ k (Γ) . (5.18)Hence, a restriction of the first operator (5.18) acts continuously K : (cid:0) H ϕ ( R n ) (cid:1) κ = (cid:2) ( H − k ( R n )) κ , ( H k ( R n )) κ (cid:3) ψ → (cid:2) H − k (Γ) , H k (Γ) (cid:3) ψ (5.19)in view of (5.4).According to (5.10) and (5.19), we have the bounded operator KT : H ϕ (Γ) → [ H − k (Γ) , H k (Γ)] ψ . Besides, owing to (5.13) and (5.17), we get the bounded operator KT : [ H − k (Γ) , H k (Γ)] ψ → H ϕ (Γ) . These operators are identical mappings in view of (5.15) and the density of C ∞ (Γ) in theirtarget spaces. Thus, the required equality (5.8) holds true up to equivalence of norms. (cid:3) Proof of Theorem . . Let us prove that the initial definition of H ϕ (Γ) as the completion of C ∞ (Γ) with respect to the inner product (5.5) is equivalent to each of definitions (i) – (iii).The initial definition is tantamount to (i) due to Theorem 5.3 in the A = (1 − ∆ Γ ) / case.Hence, the initial definition is equivalent to (iii) in view of Theorem 2.5.To prove the equivalence of this definition and (ii), it suffices to show that C ∞ (Γ) is dense inthe space defined by (ii). We arbitrarily choose a distribution f ∈ D ′ (Γ) such that ( χ j f ) ◦ π j ∈ H ϕ ( R n ) for every j ∈ { , . . . , κ } . Given such j , we take a sequence ( w rj ) ∞ r =1 ⊂ C ∞ ( R n ) suchthat w ( r ) j → ( χ j f ) ◦ π j in H ϕ ( R n ) as r → ∞ . Let T and K be the flattening and sewingmappings used in the proof of Theorem 5.3. These mappings are well defined respectively on D ′ (Γ) and ( S ′ ( R n )) κ , with the formulas KT f = f and (5.16) being valid whenever f ∈ D ′ (Γ) and w ∈ ( H ϕ ( R n )) κ . Therefore, putting w ( r ) := ( w ( r )1 , . . . , w ( r ) κ ) , we conclude that K w ( r ) ∈ C ∞ (Γ) and that k K w ( r ) − f k ϕ, Γ = k K ( w ( r ) − T f ) k ϕ, Γ ≤ c κ X l =1 k w ( r ) l − ( χ l f ) ◦ π l k ϕ, R n → as r → ∞ . Thus, C ∞ (Γ) is dense in the space defined by (ii), and the initial definition is then equivalentto (ii). (cid:3) At the end of this subsection, we give a description of the space H ϕ (Γ) in terms of sequencesinduced by the spectral decomposition of the self-adjoint operator A . Since this operator ispositive definite and since Dom A = H (Γ) , its inverse A − is a compact self-adjoint operatoron L (Γ) (recall that H (Γ) is compactly embedded in L (Γ) ). Hence, the Hilbert space L (Γ) has an orthonormal basis E := ( e j ) ∞ j =1 formed by eigenvectors of A . Let λ j ≥ be thecorresponding eigenvalue of A , i.e. Ae j = λ j e j . We may and will enumerate the eigenvectors e j so that λ j ≤ λ j +1 whenever j ≥ , with λ j → ∞ as j → ∞ . Since A is elliptic on Γ , each e j ∈ C ∞ (Γ) . We suppose that the PsDO A is classical (i.e. polyhomogeneous); see, e.g., [1,Definitions 1.5.1 and 2.1.3]. Then λ j ∼ e c j /n as j → ∞ , (5.20)where e c is a positive number that does not depend on j [1, Section 6.1 b]. Every distribution f ∈ D ′ (Γ) expands into the series f = ∞ X j =1 κ j ( f ) e j in D ′ (Γ); (5.21)here, κ j ( f ) := ( f, e j ) Γ is the value of the distribution f at the test function e j [1, Section6.1 a]. Theorem 5.8.
Let ϕ ∈ OR . Then the space H ϕ (Γ) consists of all distributions f ∈ D ′ (Γ) such that k f k ϕ, Γ , E := ∞ X j =1 ϕ ( j /n ) | κ j ( f ) | < ∞ , (5.22) and the norm in H ϕ (Γ) is equivalent to the (Hilbert) norm k · k ϕ, Γ , E . If f ∈ H ϕ (Γ) , then theseries (5.21) converges in H ϕ (Γ) .Proof. It follows from (5.20) and ϕ ∈ OR that there exists a number c ≥ such that c − ϕ ( λ j ) ≤ ϕ ( j /n ) ≤ c ϕ ( λ j ) whenever ≤ j ∈ Z . (5.23)Since Spec A = { λ j : j ≥ } , we have k ϕ ( A ) f k = ∞ X j =1 ϕ ( λ j ) | κ j ( f ) | < ∞ for every f ∈ Dom ϕ ( A ) . Hence, the norm k · k ϕ, Γ , E is equivalent to the norm in H ϕ (Γ) on Dom ϕ ( A ) ⊃ C ∞ (Γ) (see Theorem 5.3). N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 25 If f ∈ H ϕ (Γ) , we consider a sequence ( f k ) ∞ k =1 ⊂ C ∞ (Γ) such that f k → f in H ϕ (Γ) as k → ∞ . There exist positive numbers c and c such that ∞ X j =1 ϕ ( j /n ) | κ j ( f k ) | = k f k k ϕ, Γ , E ≤ c k f k k ϕ, Γ ≤ c < ∞ for every integer k ≥ . Passing here to the limit as k → ∞ and taking κ j ( f k ) → κ j ( f ) intoaccount, we conclude by Fatou’s lemma that every distribution f ∈ H ϕ (Γ) satisfies (5.22).Assume now that a distribution f ∈ D ′ (Γ) satisfies (5.22), and prove that f ∈ H ϕ (Γ) .Owing to our assumption and (5.23), we have the convergent orthogonal series ∞ X j =1 ϕ ( λ j ) κ j ( f ) e j =: h in L (Γ) . (5.24)Consider its partial sum h k := k X j =1 ϕ ( λ j ) κ j ( f ) e j for each k , and note that ϕ − ( A ) h k = k X j =1 κ j ( f ) e j ∈ C ∞ (Γ) . Since h k → h in L (Γ) as k → ∞ , the sequence ( ϕ − ( A ) h k ) ∞ k =1 is Cauchy in H ϕA (Γ) . Denotingits limit by g , we get g = lim k →∞ ϕ − ( A ) h k = ∞ X j =1 κ j ( f ) e j in H ϕ (Γ) . (5.25)Hence, f = g ∈ H ϕ (Γ) in view of (5.21).Thus, a distribution f ∈ D ′ (Γ) belongs to H ϕ (Γ) if and only if (5.22) is satisfied. Besides,given f ∈ H ϕ (Γ) , we have k f k ϕ, Γ = lim k →∞ k ϕ − ( A ) h k k ϕ, Γ ≍ lim k →∞ k h k k = k h k = ∞ X j =1 ϕ ( λ j ) | κ j ( f ) | ≍ k f k ϕ, Γ , E by (5.23), (5.24), and (5.25) where g = f (as usual, the symbol ≍ means equivalence ofnorms). The last assertion of the theorem is due to (5.25). (cid:3) Remark . Let < m ∈ R . Analogs of Theorems 5.1 and 5.3 hold true for PsDOs of order m .Namely, suppose that a PsDO A belongs to Ψ m ( R n ) or Ψ m (Γ) and satisfies conditions (b)and (c). Let ϕ ∈ OR , and put ϕ m ( t ) := ϕ ( t m ) whenever t ≥ (evidently, ϕ m ∈ OR ). Thenthe equality of spaces H ϕA ( R n or Γ) = H ϕ m ( R n or Γ) (5.26)holds in the sense that these spaces coincide as completions of C ∞ ( R n ) or C ∞ (Γ) with respectto equivalent norms. This implies that Corollaries 5.2, 5.4, and 5.5 remain true in this (moregeneral) case. The proof of (5.26) is very similar to the proofs of Theorems 5.1 and 5.3. We only observe that H kA ( V ) = H km ( V ) for every k ∈ Z whenever V = R n or V = Γ because ord A = m , which gives H ϕA ( V ) = (cid:2) H − kA ( V ) , H kA ( V ) (cid:3) ψ = (cid:2) H − km ( V ) , H km ( V ) (cid:3) ψ = H ϕ m ( V ) (5.27)with equivalence of norms; here the integer k > and the interpolation parameter ψ are thesame as those in the proof of Theorem 5.1. The first equality in (5.27) is due to Theorem 2.5,whereas the third is a direct consequence of this theorem and Theorems 5.1 and 5.3. Note ifthe PsDO A ∈ Ψ m (Γ) is classical, then A /m is the closure (in L (Γ) ) of some classical PsDO A ∈ Ψ (Γ) as was established by Seeley [65]. Hence, H ϕA (Γ) = H ϕ m A /m (Γ) = H ϕ m (Γ) immediately due to Theorem 5.3. Ending this remark, note that Theorem 5.8 remains true ifthe order of the classical PsDO A is m . It follows from the fact every eigenvector of A is alsoan eigenvector of A /m .6. Spectral expansions in spaces with two norms
We will obtain some abstract results on the convergence of spectral expansions in a Hilbertspace endowed with a second norm. In the next section, we will apply these results (togetherwith results of Section 5) to the investigation of the convergence of spectral expansions inducedby normal elliptic operators.6.1. As in Section 2, H is a separable infinite-dimensional complex Hilbert space. Let L bea normal (specifically, self-adjoint) unbounded linear operator in H . Let E be the resolutionof the identity (i.e., the spectral measure) generated by L , we considering E as an operator-valued function E = E ( δ ) of δ ∈ B ( C ) . Here, as usual, B ( C ) denotes the class of all Borelsubsets of the complex plane C . Then f = Z C dEf (6.1)for every f ∈ H . Besides, let N be a normed space. (We use the standard notation k·k N for thenorm in N . As above, k·k and ( · , · ) denote the norm and inner product in H .) Suppose that N and H are embedded algebraically in a certain linear space. We find sufficient conditions forthe convergence of the spectral expansion (6.1) in the space N . Put e B λ := { z ∈ C : | z | ≤ λ } for every number λ > . Definition 6.1.
Let f ∈ H . We say that the spectral expansion (6.1) converges uncondition-ally in the space N at the vector f if E ( δ ) f ∈ N whenever δ ∈ B ( C ) and if for an arbitrarynumber ε > there exists a bounded set γ = γ ( ε ) ∈ B ( C ) such that k f − E ( δ ) f k N < ε whenever γ ⊆ δ ∈ B ( C ) . (6.2)Note that E ( δ ) f = Z δ dEf N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 27 for all f ∈ H and δ ∈ B ( C ) . If the spectrum of L is countable, say Spec L = { z j : 1 ≤ j ∈ Z } where j = k ⇒ z j = z k , then (6.1) becomes f = ∞ X j =1 E ( { z j } ) f. (6.3)If moreover | z j | → ∞ as j → ∞ , Definition 6.1 will mean that the series (6.3) converges to f in N under an arbitrary permutation of its terms.Let I stand for the identity operator on H , and let k · k H → N and k · k H → H denote the normsof bounded linear operators on the pair of spaces H and N and on the space H , respectively. Theorem 6.2.
Let R and S be bounded linear operators on (whole) H such that they arecommutative with L and that R is a bounded operator from H to N . (6.4) Then the spectral expansion (6.1) converges unconditionally in the space N at every vector f ∈ RS ( H ) . Moreover, the degree of this convergence admits the estimate k f − E ( δ ) f k N ≤ k R k H → N · k g k · k S ( I − E ( δ )) k H → H · r g ( δ ) (6.5) for every δ ∈ B ( C ) and with some decreasing function r g ( δ ) ∈ [0 , of δ ∈ B ( C ) such that r g ( e B λ ) → as λ → ∞ . Here, g ∈ H is an arbitrary vector satisfying f = RSg , and thefunction r g ( δ ) does not depend on S and R . Note, if T is a bounded linear operator on H and if M is an unbounded linear operatorin H , the phrase “ T is commutative with M ” means that T M f = M T f for every vector f ∈ (Dom M ) ∩ Dom(
M T ) (see, e.g., [20, Chapter IV, § 3, Section 4]). Proof of Theorem . . Choose a vector f ∈ RS ( H ) ⊆ N ∩ H arbitrarily. If f = 0 , theconclusion of this theorem will be trivial; we thus suppose that f = 0 . Consider a nonzerovector g ∈ H such that f = RSg . Choose a set δ ∈ B ( C ) arbitrarily. Since the operators R and S are bounded on H and commutative with L , they are also commutative with E ( δ ) .Therefore, E ( δ ) f = E ( δ )( RS ) g = ( RS ) E ( δ ) g ∈ N due to (6.4). Hence, k f − E ( δ ) f k N = k RS ( I − E ( δ )) g k N = k RS ( I − E ( δ )) g k N ≤ k R k H → N · k S ( I − E ( δ )) k H → H · k ( I − E ( δ )) g k . (6.6)Put r g ( δ ) := k ( I − E ( δ )) g k · k g k − ; (6.7)then (6.6) yields the required estimate (6.5). It follows plainly from (6.7) that r g ( δ ) viewedas a function of δ ∈ B ( C ) is required. (cid:3) Remark . Let R be a bounded operator on H . If the norms in N and H are compatible,condition (6.4) is equivalent to the inclusion R ( H ) ⊆ N . Indeed, assume that these normsare compatible and that R ( H ) ⊆ N , and show that R satisfies (6.4). According to the closedgraph theorem, the operator R : H → e N is bounded if and only if it is closed; here, e N isthe completion of the normed space N . Therefore, it is enough to prove that this operator isclosable. Assume that a sequence ( f k ) ∞ k =1 ⊂ H satisfies the following two conditions: f k → in H and Rf k → h in e N for certain h ∈ e N , as k → ∞ . Then Rf k → in H because R isbounded on H . Hence, h = 0 as the norms in N and H are compatible. Thus, the operator R : H → e N is closable. Remark . Borel measurable bounded functions of L are important examples of the boundedoperators on H commuting with L . If S = η ( L ) for a bounded Borel measurable function η : Spec L → C , the third factor on the right of (6.5) will admit the estimate k S ( I − E ( δ )) k H → H ≤ sup (cid:8) | η ( z ) | (1 − χ E ( δ ) ( z )) : z ∈ Spec L (cid:9) ≤ sup (cid:8) | η ( z ) | : z ∈ (Spec L ) \ δ (cid:9) . (6.8)(As usual, χ E ( δ ) stands for the characteristic function of the set E ( δ ) .) Hence, if η ( z ) → as | z | → ∞ , then lim λ →∞ k S ( I − E ( e B λ )) k H → H = 0 (as well as the fourth factor r g ( δ ) if δ = e B λ ).6.2. Assume now that the normal operator L has pure point spectrum, i.e. the Hilbert space H has an orthonormal basis ( e j ) ∞ j =1 formed by some eigenvectors e j of L . Unlike the previouspart of this subsection, we suppose that L is either unbounded in H or bounded on H . Thus, f = ∞ X j =1 ( f, e j ) e j (6.9)in H for every f ∈ H . Let λ j denote the eigenvalue of L such that Le j = λ j e j . Note thatthe expansions (6.1) and (6.3) become (6.9) provided that all the proper subspaces of L areone-dimensional. Let P k denote the orthoprojector on the linear span of the eigenvectors e , . . . , e k . Theorem 6.5.
Let ω, η : Spec L → C \ { } be Borel measurable bounded functions, andconsider the bounded linear operators R := ω ( L ) and S := η ( L ) on H . Assume that R satisfies (6.4) . Then the series (6.9) converges unconditionally (i.e. under each permutation of itsterms) in the space N at every vector f ∈ RS ( H ) . Moreover, the degree of this convergenceadmits the estimate (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 ( f, e j ) e j (cid:13)(cid:13)(cid:13)(cid:13) N ≤ k R k H → N · k g k · k S ( I − P k ) k H → H · r g,k (6.10) for every integer k ≥ and with some decreasing sequence ( r g,k ) ∞ k =1 ⊂ [0 , that tends to zeroand does not depend on S and R . Here, g := ( RS ) − f ∈ H .Proof. Since
RSe j = ( ωη )( λ j ) e j for every integer j ≥ and since ( ωη )( t ) = 0 for every t ∈ Spec L , we conclude that each e j ∈ N in view of hypothesis (6.4). Thus, the left-handside of (6.10) makes sense. Besides, the operator RS = ( ωη )( L ) is algebraically reversible;hence, the vector g := ( RS ) − f ∈ H is well defined for every f ∈ RS ( H ) . We suppose that f = 0 because the conclusion of this theorem is trivial in the f = 0 case. Choosing an integer N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 29 k ≥ arbitrarily, we get ( RS ) P k g = RS k X j =1 ( g, e j ) e j = k X j =1 ( g, e j ) RSe j = k X j =1 ( g, e j )( ωη )( λ j ) e j = P k ( RS ) ∞ X j =1 ( g, e j ) e j = P k ( RS ) g. Hence, (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 ( f, e j ) e j (cid:13)(cid:13)(cid:13)(cid:13) N = k f − P k f k N = k RSg − P k ( RS ) g k N = k RS ( I − P k ) g k N = k RS ( I − P k ) g k N ≤ k R k H → N · k S ( I − P k ) k H → H · k ( I − P k ) g k . (6.11)Putting r g,k := k ( I − P k ) g k · k g k − , (6.12)we see that (6.11) yields the required estimate (6.10). It follows plainly from (6.12) that thesequence ( r g,k ) ∞ k =1 is required. Hence, the series (6.9) converges in N . This convergence is un-conditional because the hypotheses of the theorem are invariant with respect to permutationsof terms of this series. (cid:3) Remark . The third factor on the right of (6.10) admits the estimate k S ( I − P k ) k H → H ≤ sup j ≥ k +1 | η ( λ j ) | (6.13)for each integer k ≥ . Indeed, since S ( I − P k ) f = η ( L ) ∞ X j = k +1 ( f, e j ) e j = ∞ X j = k +1 ( f, e j ) η ( λ j ) e j for every f ∈ H (the convergence holds in H ), we have k S ( I − P k ) f k = ∞ X j = k +1 | ( f, e j ) η ( λ j ) | ≤ (cid:0) sup j ≥ k +1 | η ( λ j ) | (cid:1) · k f k , which gives (6.13). Specifically, if η ( t ) → as | t | → ∞ and if | λ j | → ∞ as j → ∞ , then lim k →∞ k S ( I − P k ) k H → H = 0 (as well as the fourth factor r g,k ).It is worthwhile to note that the hypotheses of Theorem 6.5 do not depend on the choiceof a basis of H . They hence imply the unconditional convergence of the series (6.9) in N at every vector f ∈ RS ( H ) for any orthonormal basis of H formed by eigenvectors of L .Remark also that Theorem 6.5 reinforces the conclusion of Theorem 6.2 under the hypothesesof Theorem 6.5. Indeed, owing to Theorem 6.2, the series (6.9) converges in N at every f ∈ RS ( H ) if its terms corresponding to equal eigenvalues are grouped together and if | λ j | → ∞ as j → ∞ . Theorem 6.5 contains M. G. Krein’s theorem [34] according to which the series (6.9) con-verges in N at every f ∈ L ( H ) if L is a self-adjoint compact operator in H obeying (6.4).The latter theorem generalizes (to abstract operators) the Hilbert – Schmidt theorem aboutthe uniform decomposability of sourcewise representable functions with respect to eigenfunc-tions of a symmetric integral operator. If L is a positive definite self-adjoint operator withdiscrete spectrum and if R = L − σ and S = L − τ for certain σ, τ ≥ and if R satisfies (6.4),Krasnosel’ski˘ı and Pustyl’nik [33, Theorem 22.1] proved that the left-hand side of (6.10) is o ( λ − τk ) as k → ∞ . This result follows from (6.10) in view of (6.13).7. Applications to spectral expansions induced by elliptic operators
This section is devoted to applications of results of Sections 5 and 6 to the investigation ofthe convergence (in the uniform metric) of spectral expansions induced by uniformly ellipticoperators on R n and by elliptic operators on a closed manifold Γ ∈ C ∞ . We find explicitcriteria of the convergence of these expansions in the normed space C q , with q ≥ , onthe function class H ϕ , with ϕ ∈ OR , and evaluate the degree of this convergence. Besides,we consider applications of the spaces H ϕ (Γ) to the investigation of the almost everywhereconvergence of the spectral expansions.7.1. Let ≤ n ∈ Z and < m ∈ R . We suppose in this subsection that L is a PsDOof class Ψ m ( R n ) and that L is uniformly elliptic on R n . We may and will consider L as aclosed unbounded operator in the Hilbert space H := L ( R n ) with Dom L = H m ( R n ) (see [1,Sections 2.3 d and 3.1 b]). We also suppose that L is a normal operator in L ( R n ) . Then L generates a resolution of the identity E = E ( δ ) , and the spectral expansion (6.1) holds forevery function f ∈ L ( R n ) . Note that the spectrum of L may be uncountable and may nothave any eigenfunctions. Hence, the expansion (6.1) may not be represented in the form ofthe series (6.3) or (6.9). For example, if L = − ∆ , then the spectrum of L coincides with [0 , ∞ ) and is continuous. Definition 7.1.
Let a normed function space N lie in S ′ ( R n ) . We say that the expansion(6.1) (where H = L ( R n ) ) converges unconditionally in N on a function class Υ if Υ ⊂ L ( R n ) and if this expansion satisfies Definition 6.1 for every f ∈ Υ .We consider the important case where N = C q b ( R n ) for an integer q ≥ and use generalizedSobolev spaces H ϕ ( R n ) as Υ . Here, C q b ( R n ) denotes the Banach space of q times continuouslydifferentiable functions f : R n → C whose partial derivatives ∂ α f are bounded on R n whenever | α | ≤ q . As usual, α = ( α , . . . , α n ) ∈ Z n + and | α | = α + · · · + α n . This space is endowedwith the norm k f k C,q, R n := X | α |≤ q sup (cid:8) | ∂ α f ( x ) | : x ∈ R n (cid:9) . Theorem 7.2.
Let ≤ q ∈ Z and ϕ ∈ OR . The spectral expansion (6.1) converges uncondi-tionally in the normed space C q b ( R n ) on the function class H ϕ ( R n ) if and only if ∞ Z t q + n − ϕ ( t ) dt < ∞ . (7.1) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 31
Remark . If we replace the lover limit in (7.1) with an arbitrary number k > , we willobtain an equivalent condition on the function ϕ ∈ OR . This is due to the fact that everyfunction ϕ ∈ OR is bounded together with /ϕ on each compact interval [1 , k ] where k > .This follows from property (2.3), in which we put t = 1 .The next result allows us to estimate the degree of the convergence stipulated by Theo-rem 7.2. Theorem 7.4.
Let ≤ q ∈ Z and φ , φ ∈ OR . Suppose that φ ( t ) → ∞ as t → ∞ and that ∞ Z t q + n − φ ( t ) dt < ∞ . (7.2) Consider the function ϕ := φ φ , which evidently belongs to OR and satisfies (7.1) . Then thedegree of the convergence of the spectral expansion (6.1) in the normed space C q b ( R n ) on theclass H ϕ ( R n ) admits the estimate k f − E ( e B λ ) f k C,q, R n ≤ c · k f k ϕ, R n · sup (cid:8) ( φ ( t )) − : t ≥ h λ i /m (cid:9) · θ f ( λ ) (7.3) for every function f ∈ H ϕ ( R n ) and each number λ > . Here, c is a certain positive num-ber that does not depend on f and λ , and θ f ( λ ) is a decreasing function of λ such that ≤ θ f ( λ ) ≤ whenever λ > and that θ f ( λ ) → as λ → ∞ . As to (7.3), recall that h λ i := (1 + | λ | ) / . Remark . Suppose that a function ϕ ∈ OR satisfies (7.1); then it may be represented inthe form ϕ = φ φ for some functions φ , φ ∈ OR subject to the hypotheses of Theorem 7.4.Indeed, considering the function η ( t ) := ∞ Z t τ q + n − ϕ ( τ ) dτ < ∞ of t ≥ and choosing a number ε ∈ (0 , / , we put φ ( t ) := η − ε ( t ) and φ ( t ) := ϕ ( t ) η ε ( t ) whenever t ≥ . Then φ ( t ) → ∞ as t → ∞ , and ∞ Z t q + n − φ ( t ) dt = ∞ Z t q + n − ϕ ( t ) η ε ( t ) dt = − ∞ Z dη ( t ) η ε ( t ) = η (1) Z dηη ε < ∞ . To show that φ , φ ∈ OR , it suffices to prove the inclusion η ∈ OR . Since ϕ ∈ OR , thereexist numbers a > and c ≥ such that c − ≤ ϕ ( λζ ) /ϕ ( ζ ) ≤ c for all ζ ≥ and λ ∈ [1 , a ] .Assuming t ≥ and ≤ λ ≤ a , we therefore get η ( λt ) = ∞ Z λt τ q + n − ϕ ( τ ) dτ = λ q + n ∞ Z t ζ q + n − ϕ ( λζ ) dζ ≤ c λ q + n ∞ Z t ζ q + n − ϕ ( ζ ) dζ ≤ c a q + n η ( t ) and η ( λt ) = λ q + n ∞ Z t ζ q + n − ϕ ( λζ ) dζ ≥ c − λ q + n ∞ Z t ζ q + n − ϕ ( ζ ) dζ ≥ c − η ( t ); i.e. η ∈ OR . Before we prove Theorems 7.2 and 7.4, we will illustrate them with three examples. Asabove, ≤ q ∈ Z . As in Theorem 7.4, we let c denote a positive number that does not dependon f and λ . Example . Let us restrict ourselves to the Sobolev spaces H s ( R n ) , with s ∈ R . Owing toTheorem 7.2, the spectral expansion (6.1) converges unconditionally in C q b ( R n ) on the class H s ( R n ) if and only if s > q + n/ . Let s > q + n/ , and put r := s − q − n/ > . If < ε < r/m , then the degree of this convergence admits the following estimate: k f − E ( e B λ ) f k C,q, R n ≤ c k f k s, R n h λ i ε − r/m for all f ∈ H s ( R n ) and λ > . Here, k · k s, R n is the norm in H s ( R n ) . This estimate followsfrom Theorem 7.4, in which we put φ ( t ) := t r − mε and φ ( t ) := t s − r + mε for every t ≥ .Choosing a number ε > arbitrarily and putting φ ( t ) := t r log − ε − / (1 + t ) and φ ( t ) := t s − r log ε +1 / (1 + t ) (7.4)for every t ≥ in this theorem, we obtain the sharper estimate k f − E ( e B λ ) f k C,q, R n ≤ c k f k s, R n h λ i − r/m log ε +1 / (1 + h λ i ) for the same f and λ .Using the generalized Sobolev spaces H ϕ ( R n ) , with ϕ ∈ OR , we may establish the uncon-ditional convergence of (6.1) in C q b ( R n ) at some functions f / ∈ H q + n/ ( R n ) := [ s>q + n/ H s ( R n ) and evaluate its degree. (Note that this union is narrower than H q + n/ ( R n ) .) Example . Choosing a number ̺ > arbitrarily and putting ϕ ( t ) := t q + n/ log ̺ +1 / (1 + t ) for every t ≥ , (7.5)we conclude by Theorem 7.2 that the spectral expansion (6.1) converges unconditionally in C q b ( R n ) on the class H ϕ ( R n ) . This class is evidently broader than H q + n/ ( R n ) . If < ε < ̺ ,then the degree of this convergence admits the estimate k f − E ( e B λ ) f k C,q, R n ≤ c k f k ϕ, R n log ε − ̺ (1 + h λ i ) for all f ∈ H ϕ ( R n ) and λ > . This estimate follows from Theorem 7.4, in which we represent ϕ as the product of the functions φ ( t ) := log ̺ − ε (1 + t ) and φ ( t ) := t q + n/ log ε +1 / (1 + t ) . Using iterated logarithms, we may obtain weaker sufficient conditions for the unconditionalconvergence of (6.1) in C q b ( R n ) . The next example involves the double logarithm. Example . Choose a number ̺ > arbitrarily, and consider the function ϕ ( t ) := t q + n/ (log(1 + t )) / (log log(2 + t )) ̺ +1 / of t ≥ . (7.6)According to Theorem 7.2, the spectral expansion (6.1) converges unconditionally in C q b ( R n ) on the class H ϕ ( R n ) . If < ε < ̺ , then the degree of this convergence admits the estimate k f − E ( e B λ ) f k C,q, R n ≤ c k f k ϕ, R n (cid:0) log log(2 + h λ i ) (cid:1) ε − ̺ N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 33 for all f ∈ H ϕ ( R n ) and λ > . The estimate follows from Theorem 7.4 provided that werepresent ϕ as the product of the functions φ ( t ) := (log log(2 + t )) ̺ − ε and φ ( t ) := t q + n/ (log(1 + t )) / (log log(2 + t )) ε +1 / . Let us turn to the proofs of Theorems 7.2 and 7.4. The proofs are based on the followingversion of H¨ormander’s embedding theorem [27, Theorem 2.2.7]:
Proposition 7.6.
Let ≤ q ∈ Z and ϕ ∈ OR . Then condition (7.1) implies the continuousembedding H ϕ ( R n ) ֒ → C q b ( R n ) . Conversely, if { w ∈ H ϕ ( R n ) : supp w ⊂ G } ⊆ C q ( R n ) (7.7) for an open nonempty set G ⊂ R n , then condition (7.1) is satisfied.Proof. We previously recall the definition of the H¨ormander space B p,k , which the embeddingtheorem deals with. Let ≤ p ≤ ∞ , and let a function k : R n → (0 , ∞ ) satisfy the followingcondition: there exist positive numbers c and ℓ such that k ( ξ + ζ ) ≤ (1 + c | ξ | ) ℓ k ( ζ ) for all ξ, ζ ∈ R n (7.8)(the class of all such functions k is denoted by K ). According to [27, Definition 2.2.1], thecomplex linear space B p,k consists of all distributions w ∈ S ′ ( R n ) that their Fourier transform b w is locally Lebesgue integrable over R n and that the product k b w belongs to the Lebesguespace L p ( R n ) . The space B p,k is endowed with the norm k k b w k L p ( R n ) and is complete withrespect to it.According to [27, Theorem 2.2.7] and its proof, the condition (1 + | ξ | ) q k ( ξ ) ∈ L p ′ ( R n ) (7.9)implies the inclusion B p,k ⊂ C q b ( R n ) ; here, as usual, the conjugate parameter p ′ ∈ [1 , ∞ ] isdefined by /p + 1 /p ′ = 1 . Moreover, if the set { w ∈ B p,k : supp w ⊂ G } lies in C q ( R n ) for an open nonempty set G ⊂ R n , then condition (7.9) is satisfied. Note that the inclusion B p,k ⊂ C q b ( R n ) is continuous because its components are continuously embedded in a Hausdorffspace, e.g. in S ′ ( R n ) .The Hilbert space H ϕ ( R n ) is the H¨ormander space B ,k provided that k ( ξ ) = ϕ ( h ξ i ) forevery ξ ∈ R n and that k satisfies (7.8). Owing to [51, Lemma 2.7], the function k ( ξ ) := ϕ ( h ξ i ) of ξ ∈ R n satisfies a weaker condition than (7.8); namely, there exist positive numbers c and ℓ such that k ( ξ + ζ ) ≤ c (1 + | ξ | ) ℓ k ( ζ ) for all ξ, ζ ∈ R n . However, there exists a function k ∈ K that both functions k/k and k /k are bounded on R n (see [27, the remark at the end of Section 2.1]). Hence, the spaces H ϕ ( R n ) and B ,k areequal with equivalence of norms. Thus, Proposition 7.6 holds true if we change (7.1) for thecondition (1 + | ξ | ) q /k ( ξ ) ∈ L ( R n ) . The latter is equivalent to Z R n h ξ i q dξϕ ( h ξ i ) < ∞ (7.10)It remains to show that (7.1) ⇔ (7.10). Passing to spherical coordinates with r := | ξ | and changing variables t = √ r , weobtain Z R n h ξ i q dξϕ ( h ξ i ) = c ∞ Z (1 + r ) q r n − drϕ ( √ r ) = c ∞ Z t q +1 ( t − n/ − dtϕ ( t )= c + c ∞ Z t q +1 ( t − n/ − dtϕ ( t ) . Here, c := n mes e B , with the second factor being the volume of the unit ball in R n , and c := c Z t q +1 ( t − n/ − dtϕ ( t ) < ∞ because the function /ϕ is bounded on [1 , and because n/ − > − . Hence,(7.10) ⇐⇒ ∞ Z t q +1 ( t − n/ − dtϕ ( t ) < ∞ ⇐⇒ ∞ Z t q + n − dtϕ ( t ) < ∞ ⇐⇒ (7.1) . (cid:3) We systematically use the following auxiliary result:
Lemma 7.7.
Suppose that a function χ ∈ OR is integrable over [1 , ∞ ) . Then χ is boundedon [1 , ∞ ) , and tχ ( t ) → as t → ∞ .Proof. Let us prove by contradiction that tχ ( t ) → as t → ∞ . Assume the contrary; i.e.,there exists a number ε > and a sequence ( t j ) ∞ j =1 ⊂ [1 , ∞ ) such that t j → ∞ as j → ∞ andthat t j χ ( t j ) ≥ ε for each j ≥ . Since χ ∈ OR , there are numbers a > and c ≥ such that c − ≤ χ ( λτ ) /χ ( τ ) ≤ c for all τ ≥ and λ ∈ [1 , a ] . Since χ is integrable over [1 , ∞ ) , we have ∞ X k =0 a k +1 Z a k χ ( t ) dt < ∞ . (7.11)Choosing an integer j ≥ arbitrarily, we find an integer k ( j ) ≥ such that a k ( j ) ≤ t j < a k ( j )+1 and observe that c − ≤ χ ( t ) /χ ( t j ) ≤ c whenever t ∈ [ a k ( j ) , a k ( j )+1 ] . Hence, a k ( j )+1 Z a k ( j ) χ ( t ) dt ≥ a k ( j )+1 Z a k ( j ) c − χ ( t j ) dt ≥ c − ε t − j ( a k ( j )+1 − a k ( j ) ) > c − ε (1 − a − ) for each integer j ≥ , which contradicts (7.11) because c − ε (1 − a − ) > and k ( j ) → ∞ as j → ∞ . Thus, our assumption is wrong; i.e., tχ ( t ) → as t → ∞ . It follows from this thatthe function χ ∈ OR is bounded on [1 , ∞ ) because it is bounded on each compact subintervalof [1 , ∞ ) . (cid:3) N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 35
Proof of Theorem . . Sufficiency. Assume that ϕ satisfies (7.1) and prove that the spectralexpansion (6.1) converges unconditionally in the normed space C q b ( R n ) on the class H ϕ ( R n ) .Note first that H ϕ ( R n ) ⊂ L ( R n ) because the function /ϕ is bounded on [1 , ∞ ) ; the latterproperty follows from (7.1) due to Lemma 7.7. We put A := I + L ∗ L and observe that A isa positive definite self-adjoint unbounded linear operator in the Hilbert space H = L ( R n ) and that Spec A ⊆ [1 , ∞ ) . Here, I is the identity operator in L ( R n ) . It follows from thetheorem on composition of PsDOs [1, Theorem 1.2.4] that A ∈ Ψ m ( R n ) is uniformly ellipticon R n . Consider the functions χ ( t ) := ϕ ( t / (2 m ) ) of t ≥ and ω ( z ) := ( χ (1 + | z | )) − of z ∈ C , and put R := ω ( L ) = (1 /χ )( A ) and S := I in Theorem 6.2. Since the function /χ is bounded on [1 , ∞ ) , the operator R is bounded on L ( R n ) , and Spec χ ( A ) . It followsfrom the latter property that H χA = Dom χ ( A ) ; hence, the operator χ ( A ) sets an isometricisomorphism between H χA and L ( R n ) . Thus, R ( L ( R n )) = H χA = H ϕ ( R n ) ⊂ C q b ( R n ) due to (5.26), Proposition 7.6, and our assumption (7.1). Since the norms in the spaces L ( R n ) and C q b ( R n ) are compatible, the operator R acts continuously from L ( R n ) to N = C q b ( R n ) ,as was shown in Remark 6.3. Thus, the operators R and S satisfy all the hypotheses of The-orem 6.2. According to this theorem, the spectral expansion (6.1) converges unconditionallyin the space C q b ( R n ) at every vector f ∈ ( RS )( L ( R n )) = H ϕ ( R n ) . The sufficiency is proved Necessity.
Assume now that the spectral expansion (6.1) converges unconditionally in C q b ( R n ) on the class H ϕ ( R n ) . Then f ∈ H ϕ ( R n ) implies f = E ( C ) f ∈ C q b ( R n ) by Defini-tion 6.1. Hence, ϕ satisfies (7.1) due to Proposition 7.6. The necessity is also proved. (cid:3) Proof of Theorem . . Consider the function χ j ( t ) := φ j ( t / (2 m ) ) of t ≥ for each j ∈ { , } and the functions η ( z ) := ( χ (1 + | z | )) − and ω ( z ) := ( χ (1 + | z | )) − of z ∈ C . Setting A := I + L ∗ L , we put S := η ( L ) = (1 /χ )( A ) and R := ω ( L ) = (1 /χ )( A ) in Theorem 6.2.The functions η and ω are bounded on C by its hypotheses (note that the boundedness of ω follows from (7.2) in view of Lemma 7.7). Hence, the operators R and S are bounded onthe Hilbert space H = L ( R n ) . It follows from (7.2) that R acts continuously from L ( R n ) to N = C q b ( R n ) , as was shown in the proof of Theorem . (the sufficiency). According toTheorem 6.2 and Remark 6.4, we have the estimate k f − E ( e B λ ) f k C,q, R n ≤ c ′ · k g k R n · sup (cid:8) ( φ ( h z i /m ) − : z ∈ C , | z | ≥ λ (cid:9) · r g ( e B λ ) (7.12)for all f ∈ RS ( L ( R n )) and λ > . Here, c ′ denotes the norm of the bounded operator R : L ( R n ) → C q b ( R n ) , whereas k · k R n stands for the norm in L ( R n ) , and g ∈ L ( R n ) satisfies f = RSg . Note that RS = (1 /χ )( A ) where χ ( t ) := χ ( t ) χ ( t ) = ϕ ( t / (2 m ) ) for every t ≥ . Since Spec χ ( A ) , the operator χ ( A ) sets an isometric isomorphism between H χA and L ( R n ) . The inverse operator RS sets an isomorphism between L ( R n ) and H ϕ ( R n ) because the spaces H χA and H ϕ ( R n ) coincide up to equivalence of norms by (5.26). Hence, c ′ k g k R n ≤ c k f k ϕ, R n for some number c > that does not depend on f and λ . Thus, formula(7.12) yields the required estimate (7.3) if we put θ f ( λ ) := r g ( e B λ ) . (cid:3) Γ be a compact boundaryless C ∞ -manifold of dimension n ≥ endowed with a positive C ∞ -density dx . We suppose here that L is a PsDO of class Ψ m (Γ) for some m > and that L is elliptic on Γ . We may and will consider L as a closed unboundedoperator in the Hilbert space H := L (Γ) with Dom L = H m (Γ) (see [1, Sections 2.3 d and3.1 b]). We also suppose that L is a normal operator in L (Γ) . Then the Hilbert space L (Γ) has an orthonormal basis E := ( e j ) ∞ j =1 formed by some eigenvectors e j ∈ C ∞ (Γ) of L (see,e.g., [67, Section 15.2]). Thus, the spectral expansion f = ∞ X j =1 κ j ( f ) e j , with κ j ( f ) := ( f, e j ) Γ , (7.13)holds in L (Γ) for every f ∈ L (Γ) . (Recall that ( · , · ) Γ and k · k Γ respectively stand for theinner product and norm in L (Γ) .) These eigenvectors are enumerated so that | λ j | ≤ | λ j +1 | whenever j ≥ , with λ j denoting the eigenvalue of L such that Le j = λ j e j . Note that | λ j | → ∞ as j → ∞ . Moreover, if L is a classical PsDO, then | λ j | ∼ e c j m/n as j → ∞ , (7.14)where e c is a certain positive number that does not depend on j .As usual, C q (Γ) denotes the Banach space of all functions u : Γ → C that are q timescontinuously differentiable on Γ . The norm in this space is denoted by k · k C,q, Γ .For the spectral expansion (7.13), the following versions of Theorems 7.2 and 7.4 hold true: Theorem 7.8.
Let ≤ q ∈ Z and ϕ ∈ OR . The series (7.13) converges unconditionally inthe normed space C q (Γ) on the function class H ϕ (Γ) if and only if ϕ satisfies (7.1) . Theorem 7.9.
Let ≤ q ∈ Z , and assume that the PsDO L is classical. Suppose thatcertain functions φ , φ ∈ OR satisfy the hypotheses of Theorem . , and consider the function ϕ := φ φ ∈ OR subject to (7.1) . Then the degree of the convergence of (7.13) in the normedspace C q (Γ) on the class H ϕ (Γ) admits the estimate (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c · k f k ϕ, Γ · sup (cid:8) ( φ ( j /n )) − : k + 1 ≤ j ∈ Z (cid:9) · θ f,k (7.15) for every function f ∈ H ϕ (Γ) and each integer k ≥ . Here, c is a certain positive numberthat does not depend on f and k , and ( θ f,k ) ∞ k =1 is a decreasing sequence that lies in [0 , andtends to zero. We illustrate these theorems with analogous examples to those given in the previous sub-section. Let ≤ q ∈ Z , and let c denote a positive number that does not depend on thefunction f and integer k from Theorem 7.9. Dealing with estimates of the form (7.15), wesuppose that the PsDO L is classical. Example . Owing to Theorem 7.8, the series (7.13) converges unconditionally in C q (Γ) on the Sobolev class H s (Γ) if and only if s > q + n/ . This fact is known (see, e.g., [69,Chapter XII, Exercise 4.5] in the q = 0 case). Let s > q + n/ , and put r := s − q − n/ > .If < ε < r/n , then (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c k f k s, Γ ( k + 1) ε − r/n N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 37 for all f ∈ H s (Γ) and k ≥ , with k · k s, Γ being the norm in H s (Γ) . This estimate followsfrom Theorem 7.9, in which we put φ ( t ) := t r − nε and φ ( t ) := t s − r + nε for every t ≥ . Theestimate admits the following refinement: (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c k f k s, Γ ( k + 1) − r/n log ε +1 / ( k + 1) for the same f and k , we choosing a real number ε > arbitrarily. This estimate follows fromTheorem 7.9 applied to the functions (7.4). Example . We choose a number ̺ > arbitrarily and define a function ϕ by formula(7.5). According to Theorem 7.8, the series (7.13) converges unconditionally in C q (Γ) on theclass H ϕ (Γ) . This fact is known at least in the q = 0 case (see [69, Chapter XII, Exercise 4.8]).If < ε < ̺ , then (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c k f k ϕ, Γ log ε − ̺ ( k + 1) for all f ∈ H ϕ (Γ) and k ≥ . This estimate follows from Theorem 7.9 if we represent ϕ in theform used in Example 7.1.2. Comparing this result with the previous example, we see that H ϕ (Γ) is broader than the union H q + n/ (Γ) := [ s>q + n/ H s (Γ) . Example . We choose a number ̺ > arbitrarily and define a function ϕ by formula(7.6). Owing to Theorem 7.8, the series (7.13) converges unconditionally in C q b ( R n ) on theclass H ϕ ( R n ) . This class is broader than that used in Example 7.2.2. If < ε < ̺ , then (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c · k f k ϕ, Γ · (log log( k + 2)) ε − ̺ for all f ∈ H ϕ (Γ) and k ≥ . This bound follows from Theorem 7.9 if we represent ϕ in theform given in Example 7.1.3.These results are applicable to multiple trigonometric series. Indeed, if Γ = T n and A = ∆ Γ ,then (7.13) becomes the expansion of f into the n -multiple trigonometric series (as usual, T := { e iτ : 0 ≤ τ ≤ π } ). It is known [22, Section 6] that this series is unconditionallyuniformly convergent (on Γ ) on every H¨older class C s (Γ) of order s > n/ . The exponent n/ is critical here; namely, there exists a function f ∈ C n/ (Γ) whose trigonometric seriesdiverges at some point of T n . These results consist a multi-dimensional generalization ofBernstein’s theorem for trigonometric series. Since C s (Γ) ⊂ H s (Γ) , Example 7.2.1 gives aweaker sufficient condition for this convergent. The next Examples 7.2.2 and 7.2.3 treat thecase of the critical exponent with the help of generalized Sobolev spaces.The proofs of Theorems 7.8 and 7.9 are similar to the proofs of Theorems 7.2 and 7.4, weusing Theorem 6.5 (instead of Theorem 6.2) and the following analog of Proposition 7.6: Proposition 7.10.
Let ≤ q ∈ Z and ϕ ∈ OR . Then condition (7.1) is equivalent to theembedding H ϕ (Γ) ⊆ C q (Γ) . Moreover, this embedding is compact under condition (7.1) . Proof.
Suppose first that ϕ satisfies condition (7.1). Then the continuous embedding H ϕ ( R n ) ֒ → C q b ( R n ) holds true by Proposition 7.6. Let κ , χ j , and π j be the same as those inthe definition of H ϕ (Γ) . Choosing f ∈ H ϕ (Γ) arbitrarily, we get the inclusion ( χ j f ) ◦ π j ∈ H ϕ ( R n ) ֒ → C q b ( R n ) for each j ∈ { , . . . , κ } . Hence, each χ j f ∈ C q (Γ) , which implies that f = κ X j =1 χ j f ∈ C q (Γ) . Thus, H ϕ (Γ) ⊆ C q (Γ) ; this embedding is continuous because both the spaces are completeand continuously embedded in D ′ (Γ) . Let us prove that it is compact.We showed in Remark 7.5 that ϕ = φ φ for some functions φ and φ satisfying the hy-potheses of Theorem 7.4. Since φ ( t ) /ϕ ( t ) = 1 /φ ( t ) → as t → ∞ , the compact embedding H ϕ (Γ) ֒ → H φ (Γ) holds true. Indeed, let T and K be the bounded operators (5.10) and (5.17).If a sequence ( f k ) is bounded in H ϕ (Γ) , then the sequence ( T f k ) is bounded in ( H ϕ ( R n )) κ . Itfollows from this by [27, Theorem 2.2.3] that the latter sequence contains a convergent sub-sequence ( T f k ℓ ) in ( H φ ( R n )) κ . Hence, the subsequence of vectors f k ℓ = KT f k ℓ is convergentin H φ (Γ) . Thus, the embedding H ϕ (Γ) ֒ → H φ (Γ) is compact. As we showed in the previousparagraph, the continuous embedding H φ (Γ) ֒ → C q (Γ) holds true because φ satisfies (7.2).Therefore, the embedding H ϕ (Γ) ֒ → C q (Γ) is compact.Assume now that the embedding H ϕ (Γ) ⊆ C q (Γ) holds true, and prove that ϕ satisfies (7.1).We suppose without loss of generality that Γ is not contained in Γ ∪ · · · ∪ Γ κ , choose anopen nonempty set U ⊂ Γ which satisfies U ∩ Γ j = ∅ whenever j = 1 , and put G := π − ( U ) .Consider an arbitrary distribution w ∈ H ϕ ( R n ) subject to supp w ⊂ G . Owing to (5.17) andour assumption, we have the inclusion u := K ( w, , . . . , | {z } κ − ) ∈ H ϕ (Γ) ⊆ C q (Γ) . Hence, w = ( χ u ) ◦ π ∈ C q ( R n ) ; note that the letter equality is true because χ = 1 on U .Thus, (7.7) holds true, which implies (7.1) due to Proposition 7.6. (cid:3) Proof of Theorem . . Sufficiency is proved in the same manner as the proof of the sufficiencyin Theorem 7.2. We only replace R n with Γ and use Theorem 6.5 instead of Theorem 6.2 andProposition 7.10 instead of Proposition 7.6. Necessity.
Assume that the series (7.13) converges in C q (Γ) on the class H ϕ (Γ) . Then H ϕ (Γ) ⊆ C q (Γ) , which implies (7.1) by Proposition 7.10. (cid:3) Proof of Theorem . . It is very similar to the proof of Theorem . . Replacing R n with Γ inthis proof and using Theorem 6.5 and Remark 6.6 instead of Theorem 6.2 and Remark 6.4,we obtain the following analog of the estimate (6.10): (cid:13)(cid:13)(cid:13)(cid:13) f − k X j =1 κ j ( f ) e j (cid:13)(cid:13)(cid:13)(cid:13) C,q, Γ ≤ c ′ · k g k Γ · sup j ≥ k +1 (cid:8) ( φ ( h λ j i /m )) − (cid:9) · r g,k (7.16)for every function f ∈ H ϕ (Γ) and each integer k ≥ . Here, c ′ denotes the norm of thebounded operator R : L (Γ) → C q (Γ) , and g := ( RS ) − f ∈ L (Γ) . Reasoning in the sameway as that given after formula (7.12), we arrive at the inequality c ′ k g k Γ ≤ c ′′ k f k ϕ, Γ where N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 39 the number c ′′ > does not depend on f and k . Besides, owing to the inclusion φ ∈ OR andasymptotic formula (7.14), the exist two positive numbers c and c such that c φ ( j /n ) ≤ φ ( h λ j i /m ) ≤ c φ ( j /n ) for every integer j ≥ . Thus, formula (7.16) yields the required estimate (7.15) if we put θ f,k := r g,k and c := c ′′ /c . (cid:3) Γ with respect to the measureinduced by the C ∞ -density dx . These conditions are formulated in terms of belonging of f to some generalized Sobolev spaces on Γ . Put S ∗ ( f, x ) := sup ≤ k< ∞ (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 κ j ( f ) e j ( x ) (cid:12)(cid:12)(cid:12)(cid:12) for all f ∈ L (Γ) and x ∈ Γ ; thus, S ∗ ( f, x ) is the majorant of partial sums of (7.13). Considerthe function log ∗ t := max { , log t } of t ≥ ; it pertains to OR . We suppose that the PsDO L is classical. Theorem 7.11.
The series (7.13) converges a.e. on Γ on the function class H log ∗ (Γ) . Be-sides, there exists a number c > such that k S ∗ ( f, · ) k Γ ≤ c k f k log ∗ , Γ for every f ∈ H log ∗ (Γ) . If f ∈ H log ∗ (Γ) , then the convergence of the series (7.13) may be violated under a permu-tation of its terms. To ensure that the convergence does not depend on their order, we shouldsubject f to a stronger condition. Theorem 7.12.
Assume that a function ϕ ∈ OR (nonstrictly) increases and satisfies ∞ Z dtt (log t ) ϕ ( t ) < ∞ . (7.17) Then the series (7.13) converges unconditionally a.e. on Γ on the function class H ϕ log ∗ (Γ) . These theorems are proved in [51, Section 2.3.2], the second being demonstrated in the casewhere ϕ varies slowly at infinity in the sense of Karamata. The proofs rely on Theorem 5.8 andgeneral forms of the classical Menshov–Rademacher [46, 60] and Orlicz [58] theorems abouta.e. convergence of orthogonal series. We give these brief proofs for the sake of completeness. Proof of Theorem . . Note that the orthonormal basis E of L (Γ) consists of eigenvectors ofthe operator A := ( I + L ∗ L ) /m , to which Theorem 5.8 is applicable. Owing to this theorem,we have ∞ X j =1 (log ( j + 1)) | κ j ( f ) | ≍ ∞ X j =1 (log ∗ ( j /n )) | κ j ( f ) | ≍ k f k ∗ , Γ < ∞ whenever f ∈ H log ∗ (Γ) , with ≍ meaning equivalence of norms. Now Theorem 7.11 followsfrom the Menshov–Rademacher theorem, which remains true for general complex orthogonalseries formed by square integrable functions (see, e.g., [45, 48, 56]). (cid:3) Proof of Theorem . . Let f ∈ H ϕ log ∗ (Γ) , and put ω j := ϕ ( j /n ) for every integer j ≥ .Owing to Theorem 5.8 applied to A := ( I + L ∗ L ) /m , we have ∞ X j =2 (log j ) ω j | κ j ( f ) | ≍ k f k ϕ log ∗ , Γ < ∞ . (7.18)Besides, condition (7.17) implies that ∞ X j =3 j (log j ) ω j ≤ ∞ Z dττ (log τ ) ϕ ( τ /n ) = ∞ Z /n n t n − dtt n n (log t ) ϕ ( t ) < ∞ . (7.19)The conclusion of Theorem 7.12 follows from (7.18) and (7.19) due to the Orlicz theorem (inUl’janov’s equivalent statement [71, Section 9, Subsection 1]), which remains true for generalcomplex orthogonal series [49, Theorem 2] (see also [48, Theorem 3]). (cid:3) As to Theorems 7.11 and 7.12, note the following: if we restrict ourselves to the Sobolevspaces, we will assert only that the series (7.13) converges unconditionally a.e. on Γ on thefunction class H (Γ) := S s> H s (Γ) (cf. [44]). This class is significantly narrower than thespaces used in these theorems. Using the extended Sobolev scale, we express in adequateforms the hypotheses of the Menshov–Rademacher and Orlicz theorems. References [1] M. S. Agranovich,
Elliptic operators on closed manifolds , Encyclopaedia Math. Sci., Springer, Berlin,vol. 63, 1994, pp. 1–130.[2] M. S. Agranovich,
Elliptic boundary problems , Encyclopaedia Math. Sci., Springer, Berlin, vol. 79, 1997,pp. 1–144.[3] M. S. Agranovich, M. I. Vishik,
Elliptic problems with parameter and parabolic problems of general form ,Uspehi Mat. Nauk (1964), 53–161 (Russian). [English translation in Russian Math. Surveys (1964),53–157.][4] Y. Ameur, A new proof of Donoghue’s interpolation theorem , J. Funct. Spaces Appl. (2004), 253–265.[5] Y. Ameur, Interpolation between Hilbert spaces , in: A. Aleman etc. (eds.) Analysis of Operators on Func-tion Spaces, The Serguei Shimorin Memorial Volume, Trends Math., pp. 63–115, Birkh¨auser/Springer,Cham, 2019.[6] A. Anop, R. Denk, A. Murach,
Elliptic problems with rough boundary data in generalized Sobolev spaces ,Commun. Pure Appl. Anal. (2021), no. 2, 697–735.[7] A. V. Anop, T. M. Kasirenko, Elliptic boundary-value problems in H¨ormander spaces , Methods Funct.Anal. Topology (2016), no. 4, 295–310.[8] A. V. Anop, A. A. Murach, Parameter-elliptic problems and interpolation with a function parameter ,Methods Funct. Anal. Topology (2014), no. 2, 103–116.[9] A. V. Anop, A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale ,Ukrainian Math. J. (2014), no. 7, 969–985.[10] V. G. Avakumovi´c, O jednom O-inverznom stavu , Rad Jugoslovenske Akad. Znatn. Umjetnosti (1936), 167–186.[11] J. Bergh, J. L¨ofstr¨om,
Interpolation Spaces , Springer, Berlin, 1976.[12] Yu. M. Berezansky,
Expansions in Eigenfunctions of Selfadjoint Operators , American Mathematical So-ciety, Providence, RI, 1968.[13] N. H. Bingham, C. M. Goldie, J. L. Teugels,
Regular Variation , Cambridge University Press, Cambridge,1989.[14] M. Braverman, O. Milatovic, M. Shubin,
Essential selfadjointness of Schr¨odinger-type operators on man-ifolds , Russian. Math. Surveys (2002), no. 4, 641–692. N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 41 [15] V. V. Buldygin, K.-H. Indlekofer, O. I. Klesov, J. G. Steinebach,
Pseudo-Regularly Varying Functionsand Generalized Renewal Processes , Springer, Cham, 2018.[16] W. F. Donoghue,
The interpolation of quadratic norms , Acta Math. (1967), no. 3–4, 251–270.[17] S. D. Eidel’man,
Parabolic equations , Encyclopaedia Math. Sci., Springer, Berlin, vol. 63, 1994, pp.205–316.[18] M. Fan,
Qudratic interpolation and some operator inequalities , J. Math. Inequal. (2011), no. 3, 413–427.[19] C. Foia¸s, J.-L. Lions, Sur certains th´eor`emes d’interpolation , Acta Sci. Math. (Szeged) (1961), no.3–4, 269–282.[20] Functional Analysis (ed. S.G. Krein), Nauka, Moscow, 1972 (Russian).[21] M. Gaffney,
A special Stokes’s theorem for complete Riemannian manifolds , Ann. of Math. (2) (1954),no. 1, 140–145.[22] B. I. Golubov, Multiple Fourier series and integrals , J. Math. Sci (N. Y.) (1984), no. 6, 639–673.[23] M. Hegland, Variable Hilbert scales and their interpolation inequalities with applications to Tikhonovregularization , Appl. Anal. (1995), no. 1–4, 207–223.[24] M. Hegland, Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities ,J. Integral Equations Appl. (2010), no. 2, 285–312.[25] M. Hegland, R. S. Anderssen, Dilational interpolatory inequalities , Math. Comp. (2011), no. 274,1019–1036.[26] M. Hegland, B. Hofmann, Errors of regularisation under range inclusions using variable Hilbert scales ,Inverse Probl. Imaging (2011), no. 3, 619–643.[27] L. H¨ormander, Linear Partial Differential Operators , Springer, Berlin, 1963.[28] L. H¨ormander,
The Analysis of Linear Partial Differential Operators. II , Springer, Berlin, 1983.[29] L. H¨ormander,
The Analysis of Linear Partial Differential Operators. III , Springer, Berlin, 1985.[30] Q. Jin, U. Tautenhahn,
Implicit iteration methods in Hilbert scales under general smoothness conditions ,Inverse Problems (2011), no. 4, article no. 045012, 27 pp.[31] Kasirenko T. M., Murach O. O., Elliptic problems with boundary conditions of higher orders in H¨ormanderspaces , Ukrainian Math. J. (2018), no. 11, 1727–1748.[32] V. Koshmanenko, M. Dudkin, The Method of Rigged Spaces in Singular Perturbation Theory of Self-adjoint Operators , Birkh¨auser/Springer, Cham, 2016.[33] M. A. Krasnosel’ski˘ı, P. P. Zabre˘ıko, E. I. Pustyl’nik, P. E. Sobolevski˘ı,
Integral Operators in Spaces ofSummable Functions , Noordhoff International Publishing, Leiden, 1976.[34] M. G. Krein,
On completely continuous operators in function spaces with two norms , Works of Instituteof Mathematics of Academy of Sciences of Ukraine (1947), no. 9, 104–129 (Russian).[35] S. G. Krein,
On one interpolation theorem in the theory of operators , Dokl. Akad. Nauk. SSSR (1960), no. 3, 491–494 (Russian).[36] S. G. Krein, Yu. I. Petunin,
Scales of Banach spaces , Russian Math. Surveys (1966), no. 2, 85–159.[37] P. D. Lax, Hyperbolic Partial Differential Equations , American Mathematical Society, Providence, RI,2006.[38] J.-L. Lions,
Espaces interm´ediaires entre espaces hilbertiens et applications , Bull. Math. Soc. Sci. Math.Phys. R. P. Roumanie (1958), no. 4, 419–432.[39] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I , Springer,Berlin, 1972.[40] B. Malgrange,
Sur une classe d’op´eratuers diff´erentiels hypoelliptiques , Bull. Soc. Math. France (1957),283–306.[41] P. Mathe and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales , InverseProblems (2003), no. 3, 789–803.[42] P. Math´e, U. Tautenhahn, Interpolation in variable Hilbert scales with application to innverse problems ,Inverse Problems (2006), no. 6, 2271–2297.[43] W. Matuszewska, On a generalization of regularly increasing functions , Studia Math. (1964), 271–279.[44] C. Meaney, On almost-everywhere convergent eigenfunction expansions of the Laplace–Beltrami operator ,Math. Proc. Camb. Phil. Soc. (1982), no. 1, 129–131. [45] C. Meaney, Remarks on the Rademacher–Menshov theorem , Proc. Centre Math. Appl. Austral. Nat.Univ. (2007), 100–110.[46] D. Menschoff, Sur les series de fonctions orthogonales I , Fund. Math. (1923), 82–105.[47] V. A. Mikhailets, A. A. Murach, Interpolation with a function parameter and refined scale of spaces ,Methods Funct. Anal. Topology (2008), no. 1, 81–100.[48] V. A. Mikhailets, A. A. Murach, General forms of the Menshov–Rademacher, Orlicz, and Tandori theo-rems on orthogonal series , Methods Funct. Anal. Topology (2011), no. 4, 330–340.[49] V. A. Mikhailets, A. A. Murach, On the unconditional almost-everywhere convergence of general orthog-onal series , Ukrainian. Math. J. (2012), no. 10, 1543–1550.[50] V. A. Mikhailets, A. A. Murach, Extended Sobolev scale and elliptic operators , Ukrainian Math. J. (2013), no. 3, 435–447.[51] V. A. Mikhailets, A. A. Murach, H¨ormander Spaces, Interpolation, and Elliptic Problems , De Gruyter,Berlin, 2014.[52] V. A. Mikhailets, A. A. Murach,
Interpolation Hilbert spaces between Sobolev spaces , Results Math. (2015), no. 1, 135–152.[53] O. Milatovic, F. Truc, Self-adjoint extensions of differential operators on Riemannian manifolds , Ann.Global Anal. Geom. (2016), no. 1, 87–103.[54] A. A. Murach, On elliptic systems in H¨ormander spaces , Ukrainian Math. J. (2009), no. 3, 467–477.[55] A. A. Murach, T. N. Zinchenko, Parameter-elliptic operators on the extended Sobolev scale , MethodsFunct. Anal. Topology (2013), no. 1, 29–39.[56] F. M´oricz, K. Tandori, An improved Menshov–Rademacher theorem , Proc. Amer. Math. Soc. (1996),no. 3, 877–885.[57] A. Neubauer,
When do Sobolev spaces form a Hilbert scale? , Proc. Amer. Math. Soc. (1988), no. 2,557-562.[58] W. Orlicz,
Zur Theorie der Orthogonalreihen , Bull. Intern. Acad. Sci. Polon. Cracovie (1927), 81–115.[59] V. I. Ovchinnikov,
The methods of orbits in interpolation theory , Mathematical Reports, Vol. 1, Part 2,pp. 349–515, Harwood Academic Publishers, 1984.[60] H. Rademacher,
Einige S¨atze ¨uber Reihen von allgemeinen Orthogonalfunctionen , Math. Annalen (1922), 111–138.[61] J. Peetre, On interpolation functions , Acta Sci. Math. (Szeged) (1966), 167–171.[62] J. Peetre, On interpolation functons, II , Acta Sci. Math. (Szeged) (1968), 91–92.[63] E. I. Pustyl‘nik, On permutation-interpolation Hilbert spaces , Russian Math. (Iz. VUZ) (1982), no. 5,52–57.[64] M. Reed, B. Simon, Methods of Modern Mathematical Physics. Vol. 1: Functional Analysis , AcademicPress, New York – London, 1972.[65] R. Seeley,
Complex powers of an elliptic operator , Proc. of Symposia in Pure Math., Vol. 10, pp. 288–307,Amer. Math. Soc., 1967.[66] E. Seneta,
Regularly Varying Functions , Springer, Berlin, 1976.[67] M. A. Shubin,
Pseudodifferential Operators and Spectral Theory [2-nd edn], Springer, Berlin, 2001.[68] B. Simon,
Loewner’s Theorem on Monotone Matrix Functions , Springer, Cham, 2019.[69] M. E. Taylor,
Pseudodifferential Operators , Princeton University Press, Princeton, 1981.[70] U. Tautenhahn, U. H¨amarik, B. Hofmann, Y. Shao,
Conditional stability estimates for ill-posed PDEproblems by using interpolation , Numer. Funct. Anal. Optim. (2013), no. 12, 1370–1417.[71] P. L. Ulj’anov, Solved and unsolved problems in the theory of trigonometric and orthogonal series (Rus-sian), Uspehi Mat. Nauk (1964), no. 1, 3–69 (English translation in: Russian Math. Surveys (1964),no. 1, 1–62).[72] L. R. Volevich, B. P. Paneah, Certain spaces of generalized functions and embedding theorems (Russian),Uspehi Mat. Nauk (1965), no. 1, 3–74 (English translation in: Russian Math. Surveys (1965),no. 1, 1–73).[73] T. N. Zinchenko, A. A. Murach, Douglis–Nirenberg elliptic systems in H¨ormander spaces , UkrainianMath. J. (2012), no. 11, 1672–1687. N EXTENDED HILBERT SCALE AND ITS APPLICATIONS 43 [74] T. N. Zinchenko, A. A. Murach,
Petrovskii elliptic systems in the extended Sobolev scale , J. Math. Sci.(N. Y.) (2014), no. 5, 721–732.
Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchen-kivs’ka, Kyiv, 01024, Ukraine
Email address : [email protected] Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchen-kivs’ka, Kyiv, 01024, Ukraine
Email address : [email protected]
5d Mittelstr., Oranienburg, 16515, Germany
Email address ::