Interpolation Hilbert spaces between Sobolev spaces
aa r X i v : . [ m a t h . F A ] O c t INTERPOLATION HILBERT SPACESBETWEEN SOBOLEV SPACES
VLADIMIR A. MIKHAILETS, ALEKSANDR A. MURACH
Abstract.
We explicitly describe all Hilbert function spaces that are interpolation spaceswith respect to a given couple of Sobolev inner product spaces considered over R n or a half-space in R n or a bounded Euclidean domain with Lipschitz boundary. We prove that theseinterpolation spaces form a subclass of isotropic H¨ormander spaces. They are parametrizedwith a radial function parameter which is OR-varying at + ∞ and satisfies some additionalconditions. We give explicit examples of intermediate but not interpolation spaces. Introduction
The fundamental importance of Sobolev spaces for analysis and the theory of partial differ-ential equations is well known. This importance comes partly from interpolation propertiesof the Sobolev scale [10, 23]. Owing to them, it is possible to extend significant properties ofinteger order Sobolev spaces to fractional order spaces. The invariance of spaces with respectto admissible change of variables is one of these properties and enables the Sobolev spaces (ofany real order) to be well-defined over smooth manifolds (see, e.g., [7, Sec. 2.6]).Nevertheless, the Sobolev scale is not sufficiently fine for a number of mathematical prob-lems, which explains a natural need to replace the Sobolev spaces with more general isotropicH¨ormander spaces H ϕ [7, 8]. They are initially defined over R n with the help of the Fouriertransform and a radial weight function ϕ of the scalar argument h ξ i := (1 + | ξ | ) / ≥ .The use of the power function ϕ ( t ) = t s leads to the Sobolev space H ( s ) , with s ∈ R . Theapplication of interpolation with a function parameter allows the classical theory of ellip-tic boundary-value problems to be completely transferred from the Sobolev scale to a moreextensive class of H¨ormander inner product spaces (see [13, 16, 15] and references therein).The latter—refined Sobolev scale—is parametrized with a function ϕ that varies regularly inJ. Karamata’s sense at + ∞ with an arbitrary index s ∈ R .Interpolation with a function parameter between spaces of differentiable functions has in-vestigated less completely [4, 12, 20] than the interpolation between spaces of integrablefunctions. The purpose of this paper is to describe constructively all Hilbert function spaces Mathematics Subject Classification. that are interpolation spaces with respect to a given couple of Sobolev inner product spaces(1.1) (cid:2) H ( s ) , H ( s ) (cid:3) , −∞ < s < s < + ∞ , considered over R n or a half-space in R n or a bounded Euclidean domain. We show that theseinterpolation spaces form a subclass of isotropic H¨ormander spaces. This subclass is charac-terized by the fact that the function ϕ is OR-varying at + ∞ in the sense of V. G. Avakumovi´c[2, 22] and satisfies two additional conditions, which involve the numbers s and s respec-tively. The chosen class of function spaces is broad enough to be effectively used in variousproblems of the modern analysis [16, Sec. 2.4]. In particular, we may define the largest classof H¨ormander inner product spaces over a closed compact manifold in such a way that theydo not depend on a choice of local charts but possess the interpolation property. These resultsand their applications will be given in a separate paper.This paper is organized in the following way. The main result of the paper is formulated asTheorems 2.4 and 2.7 in the next section 2. Here we also define necessary classes of functionparameters and Hilbert function spaces. Section 3 contains auxiliary results. Here we discussthe interpolation with a function parameter of abstract Hilbert spaces, which is the main toolin our proofs. The main result is proved in Section 4. In Section 5 we investigate the classof all Hilbert spaces that are interpolation spaces between Sobolev inner product spaces. Weshow that this class is the complete extension of the scale { H ( s ) | s ∈ R } by the interpolationwithin the category of Hilbert spaces. In the final section 6, we construct explicit and mutuallycomplementary examples of Hilbert spaces that are intermediate but not interpolation spacesfor couples of Sobolev spaces. We were not able to find any examples of this type in theliterature. 2. H ¨ormander spaces and the main results
Suppose that µ : R n → (0 , ∞ ) is a Borel measurable weight function in the following sense:there exist numbers c ≥ and ℓ ≥ such that(2.1) µ ( ξ ) µ ( η ) ≤ c (1 + | ξ − η | ) ℓ for arbitrary ξ, η ∈ R n . By definition (see [7, Sec. 2.2] or [25, Sec. 2]), the H¨ormander space H µ ( R n ) consists ofall L. Schwartz’s distributions u ∈ S ′ ( R n ) that their Fourier transform b u is locally Lebesgueintegrable over R n and µ b u ∈ L ( R n ) . The norm in the complex linear space H µ ( R n ) is definedby the formula k u k H µ ( R n ) := k µ b u k L ( R n ) . The space H µ ( R n ) is Hilbert with respect to this norm and is continuously embedded in S ′ ( R n ) . This space is separable and C ∞ ( R n ) is dense in it. Note that(2.2) H µ ( R n ) ⊆ H ν ( R n ) ⇐⇒ ( ν/µ is bounded on R n ); here µ and ν are weight functions, and the embedding is continuous. NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 3
Among the H¨ormander spaces H µ ( R n ) we only need isotropic spaces. They correspondto radial weight functions µ ( ξ ) = ϕ ( h ξ i ) , where ϕ belongs to the following class of functionparameters.Let OR be the class of all Borel measurable functions ϕ : [1 , ∞ ) → (0 , ∞ ) for which thereexist numbers a > and c ≥ such that(2.3) c − ≤ ϕ ( λt ) ϕ ( t ) ≤ c for arbitrary t ≥ and λ ∈ [1 , a ] , with a and c depending on ϕ . Such functions are said to be OR-varying at + ∞ . This functionclass was introduced by V. G. Avakumovi´c in 1936 and has been comprehensively investigated[2, 22]. We recall some of its known properties (see, e.g., [22, Sec. A.1]). Proposition 2.1. (i) If ϕ ∈ OR , then both the functions ϕ and /ϕ are bounded on everycompact interval [1 , b ] with < b < ∞ . (ii) The following description of the class OR holds: ϕ ∈ OR ⇐⇒ ϕ ( t ) = exp β ( t ) + t Z ε ( τ ) τ dτ , t ≥ , where the real-valued functions β and ε are Borel measurable and bounded on [1 , ∞ ) . (iii) For an arbitrary function ϕ : [1 , ∞ ) → (0 , ∞ ) , condition (2.3) is equivalent to thefollowing: there exist numbers s , s ∈ R , s ≤ s , and c ≥ such that (2.4) ϕ ( t ) t s ≤ c ϕ ( τ ) τ s and ϕ ( τ ) τ s ≤ c ϕ ( t ) t s for all t ≥ and τ ≥ t. Condition (2.4) means that the function ϕ ( t ) /t s is equivalent to an increasing function,whereas the function ϕ ( t ) /t s is equivalent to a decreasing one on [1 , ∞ ) . Here and below wesay that positive functions ψ and ψ are equivalent on a given set if both ψ /ψ and ψ /ψ are bounded on it; this property will be denoted by ψ ≍ ψ .Setting λ := τ /t in condition (2.4) we rewrite it in the equivalent form(2.5) c − λ s ≤ ϕ ( λt ) ϕ ( t ) ≤ cλ s for all t ≥ and λ ≥ . We associate the following notation with ϕ ∈ OR : σ ( ϕ ) := sup { s ∈ R | the left-hand inequality in (2.5) holds } , (2.6) σ ( ϕ ) := inf { s ∈ R | the right-hand inequality in (2.5) holds } . (2.7)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 [11] and [2, Theorem 2.2.2]). Inparticular, if ϕ is regularly varying with an index s ∈ R , then ϕ ∈ OR and σ ( ϕ ) = σ ( ϕ ) = s (see [2, Sec. 1.4.2] or [22, Sec. 1.1]). V. A. MIKHAILETS, A. A. MURACH
Remark . The Matuszewska indices are closely connected with the classical Boyd indices[3]. Let us clarify this connection. Given ϕ ∈ OR , we define ϕ ( t ) := ϕ (1) /ϕ ( t − ) for < t < and obtain a positive function ϕ , which is given on (0 , ∞ ) and satisfies the condition c − λ s ≤ ϕ ( λt ) ϕ ( t ) ≤ c λ s for all t > and λ ≥ by virtue of (2.5). Consider the function m ϕ ( λ ) := sup t> ϕ ( λt ) ϕ ( t ) of λ > . Then [9, Chap. II, § 1, Subsec. 2] σ ( ϕ ) = lim λ → log m ϕ ( λ )log λ , σ ( ϕ ) = lim λ →∞ log m ϕ ( λ )log λ . By definition, the right-hand sides of these equalities are the Boyd indices of the function m ϕ ( λ ) .Let ϕ ∈ OR . By definition, H ϕ ( R n ) is the Hilbert space H µ ( R n ) with µ ( ξ ) := ϕ ( h ξ i ) forall ξ ∈ R n . The inner product in H ϕ ( R n ) is ( u , u ) H ϕ ( R n ) := Z R n ϕ ( h ξ i ) b u ( ξ ) b u ( ξ ) dξ. It induces the norm introduced above. Note that the space H ϕ ( R n ) is well-defined, since thefunction µ ( ξ ) = ϕ ( h ξ i ) of ξ ∈ R n satisfies (2.1). This will be demonstrated in Proposition 3.5,Section 3.We also introduce necessary function spaces over Euclidean domains. Let Ω be a domainin R n . By definition, the linear space H ϕ (Ω) consists of the restrictions v = u ↾ Ω to Ω of alldistributions u ∈ H ϕ ( R n ) . The norm in H ϕ (Ω) is k v k H ϕ (Ω) := inf (cid:8) k u k H ϕ ( R n ) | u ∈ H ϕ ( R n ) , u = v in Ω (cid:9) . The space H ϕ (Ω) is a separable and Hilbert space with respect to the above norm because itis a factor space of the separable Hilbert space H ϕ ( R n ) by { w ∈ H ϕ ( R n ) | supp w ⊆ R n \ Ω } . If ϕ ( t ) = t s with t ≥ for some s ∈ R , then H ϕ (Ω) coincides with the Sobolev space H ( s ) (Ω) of order s (we refer to the definition given in [23, Sec. 4.2.1]). Remark . Let ϕ, χ ∈ RO . It follows from (2.2) that H ϕ (Ω) = H χ (Ω) if and only if ϕ ≍ χ on [1 , ∞ ] . Specifically, the latter property yields σ j ( ϕ ) = σ j ( χ ) for every j ∈ { , } .Let X := [ X , X ] be a couple of complex Hilbert spaces X and X that the continuousembedding X ֒ → X holds. (This assumption is caused by the fact that we are only interestedin the case where X is a couple of inner product Sobolev spaces.) A Hilbert space H is called NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 5 an interpolation space with respect to (or for) the couple X if the following two conditionsare satisfied:(i) H is an intermediate space for this couple; i.e., X ⊆ H ⊆ X , and the embeddingsare continuous;(ii) for every linear operator T given on X , the following implication holds: if the restric-tion of T to X j is a bounded operator on X j for each j ∈ { , } , then the restrictionof T to H is a bounded operator on H .Property (ii) implies the following inequality for norms of the operators: k T k H → H ≤ c max { k T k X → X , k T k X → X } , where c is a positive number independent of T (see [1, Theorem 2.4.2]). If this inequalityholds with c = 1 for every T , then the interpolation space H is called exact.Note that the above properties (i) and (ii) are invariant with respect to the choice of anequivalent norm on H . Therefore, we will describe the interpolation spaces up to equivalenceof norms.The main result of the paper consists of the following two theorems. Here and below wesuppose that Ω is either the whole space R n or an open half-space in R n or a bounded domainin R n with Lipschitz boundary. Theorem 2.4.
Let −∞ < s < s < ∞ . A Hilbert space H is an interpolation space withrespect to the couple of Sobolev spaces [ H ( s ) (Ω) , H ( s ) (Ω)] if and only if H = H ϕ (Ω) up toequivalence of norms for some function parameter ϕ ∈ OR that satisfies condition (2.5) .Remark . Naturally, we mean in this theorem that the number c ≥ in condition (2.5) isindependent of t and λ . This condition is equivalent to the following pair of conditions: (i) s ≤ σ ( ϕ ) and, moreover, s < σ ( ϕ ) if the supremum in (2.6) is not attained; (ii) σ ( ϕ ) ≤ s and, moreover, σ ( ϕ ) < s if the infimum in (2.7) is not attained. Remark . It is useful to note the following. Suppose that a Hilbert space H is an inter-polation space with respect to a given couple of Sobolev spaces [ H ( s ) (Ω) , H ( s ) (Ω)] . Then H is an interpolation space for each wider couple [ H ( s − ε ) (Ω) , H ( s + δ ) (Ω)] , with ε, δ > . Thisresult follows immediately from Theorem 2.4.Let a scale of Hilbert spaces { X s | s ∈ R } be such that the continuous embedding X s ֒ → X s holds whenever s < s . A Hilbert space H is called an interpolation space with respectto this scale if H is an interpolation space with respect to a certain couple [ X s , X s ] with s < s . Theorem 2.7.
A Hilbert space H is an interpolation space with respect to the Sobolev scale { H ( s ) (Ω) | s ∈ R } if and only if H = H ϕ (Ω) up to equivalence of norms for some ϕ ∈ OR . V. A. MIKHAILETS, A. A. MURACH
Remark . There are functions ϕ ∈ OR for which H ϕ ( R n ) is an intermediate but not aninterpolation space for the couple [ H ( s ) ( R n ) , H ( s ) ( R n )] . See Example 6.3.Theorem 2.7 is a consequence of Theorem 2.4 but is of independent interest. Both of themwill be proved in Section 4. 3. Auxiliary abstract results
The exact Hilbert interpolation spaces with respect to a couple of Hilbert spaces werecharacterized (isometrically) by W. F. Donoghue [5] in 1967. V. I. Ovchinnikov [17] laterused this result to describe all Hilbert interpolation spaces, up to equivalence of norms. Thetheorems of Donoghue and Ovchinnikov are fairly deep, but it seems that they have notattract much attention to users of Hilbert spaces. Here we formulate some results of theinterpolation theory in Hilbert spaces, including the Ovchinnikov theorem. It is sufficient torestrict ourselves to separable complex Hilbert spaces.We say that an ordered couple [ X , X ] of Hilbert spaces X and X is admissible if thesespaces are separable and the dense continuous embedding X ֒ → X holds.Let us recall the definition of the interpolation of Hilbert spaces with a function param-eter. It is a natural generalization of the classical interpolation method of J.-L. Lions andS. G. Krein (see, e.g., [10, Chapter 1, Sec. 2, 5] and [9, Chapter 3, Sec. 10]) to the casewhere a general enough function is used, instead of the number θ ∈ (0 , , as an interpolationparameter. The generalization appeared in C. Foia¸s and J.-L. Lions’ paper [6, Section 3.4]and was then studied by several authors.Following [14, Sec. 2.1], we denote by B 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, moreover, /ψ is bounded on every set [ r, ∞ ) with r > .Let a function ψ ∈ B and an admissible couple of Hilbert spaces X = [ X , X ] be given. For X there exists an isometric isomorphism J : X ↔ X such that J is a self-adjoint positiveoperator on X with the domain X . The operator J is called a generating operator for thecouple X . This operator is uniquely determined by X . (Indeed, assume that J is anothergenerating operator for X . Then J = J because J and J are positive and metrically equal.)Using the spectral theorem, we can define an (unbounded) operator ψ ( J ) in X as the Borelfunction ψ of the self-adjoint operator J . Let us denote by [ X , X ] ψ or simply by X ψ the do-main of the operator ψ ( J ) endowed with the inner product ( u , u ) X ψ := ( ψ ( J ) u , ψ ( J ) u ) X and the corresponding norm k u k X ψ = k ψ ( J ) u k X . The space X ψ is Hilbert and separable.A function ψ ∈ B is called an interpolation parameter if the following condition is fulfilledfor all admissible couples X = [ X , X ] and Y = [ Y , Y ] of Hilbert spaces and for an arbitrarylinear mapping T given on X : if the restriction of T to X j is a bounded operator T : X j → Y j for each j ∈ { , } , then the restriction of T to X ψ is also a bounded operator T : X ψ → Y ψ . NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 7 If ψ is an interpolation parameter, then we say that the Hilbert space X ψ is obtained bythe interpolation of X with the function parameter ψ . In this case, we have the continuousand dense embeddings X ֒ → X ψ ֒ → X .The classical result by J.-L. Lions and S. G. Krein consists in that the power function ψ ( t ) := t θ is an interpolation parameter whenever < θ < , the exponent θ being regardedas a number parameter of the interpolation.Let us describe the class of all interpolation parameters (in the sense of the above definition).Let a function ψ : (0 , ∞ ) → (0 , ∞ ) and a number r ≥ be given, then ψ is calledpseudoconcave on the semiaxis ( r, ∞ ) if there exists a concave function ψ : ( r, ∞ ) → (0 , ∞ ) such that ψ ( t ) ≍ ψ ( t ) for t > r . The function ψ is called pseudoconcave in a neighbourhoodof + ∞ if it is pseudoconcave on ( r, ∞ ) , where r is a sufficiently large number. Proposition 3.1.
A function ψ ∈ B is an interpolation parameter if and only if ψ is pseu-doconcave in a neighbourhood of + ∞ . This fact follows from J. Peetre’s [18] description of all interpolation functions for theweighted L p ( R n ) -type spaces (see the monograph [1, Theorem 5.4.4] as well). A proof ofProposition 3.1 is given in, e.g., [14, Section 2.7]. It is useful to note that ψ ∈ B is pseudo-concave in a neighbourhood of + ∞ if and only if ψ is pseudoconcave on every set ( r, ∞ ) with r > . This is evident in view of the definition of the class B .Let X = [ X , X ] be an admissible couple of Hilbert spaces. V. I. Ovchinnikov [17, Theorem11.4.1] has described (up to equivalence of norms) all the Hilbert spaces that are interpolationspaces with respect to X . In connection with our considerations, his result can be restatedas follows. Proposition 3.2.
A Hilbert space H is an interpolation space with respect to X if and onlyif H = X ψ up to equivalence of norms for some function ψ ∈ B that is pseudoconcave in aneighbourhood of + ∞ . This proposition will play a key role in the proof of Theorem 2.4, the main result of thepaper. In this connection the following property of pseudoconcave functions will be of use.
Proposition 3.3.
Let a function ψ ∈ B and a number r ≥ be given. The function ψ ispseudoconcave on ( r, ∞ ) if and only if there exists a number c > such that (3.1) ψ ( t ) ψ ( τ ) ≤ c max (cid:26) , tτ (cid:27) for all t, τ > r. In the r = 0 case, this proposition was proved by J. Peetre [18] (see [1, Theorem 5.4.4] aswell), the condition ψ ∈ B being superfluous. In the r > case, the sufficiency is arguedanalogously; the necessity is proved in [14, Lemma 2.2] by reduction to the r = 0 case.We also need a reiteration theorem for interpolation with a function parameter [14, Theo-rems 2.1 and 2.3]. V. A. MIKHAILETS, A. A. MURACH
Proposition 3.4.
Suppose that f, g, ψ ∈ B and that f /g is bounded in a neighbourhood of + ∞ . Let X be an admissible couple of Hilbert spaces. Then the couple [ X f , X g ] is admissible,and [ X f , X g ] ψ = X ω with equality of norms. Here the function ω ∈ B is given by the formula ω ( t ) := f ( t ) ψ ( g ( t ) /f ( t )) with t > . Moreover, if f, g , and ψ are interpolation parameters,then ω is an interpolation parameter as well. At the end of this section we will prove a result, which shows that H ϕ ( R n ) is well-defined. Proposition 3.5.
Let ϕ ∈ OR ; then the function µ ( ξ ) := ϕ ( h ξ i ) of ξ ∈ R n satisfies (2.1) ,i.e. µ is a weight function.Proof. Let ξ, η ∈ R n . By taking squares it is easy to check the inequality |h ξ i − h η i| ≤| | ξ | − | η | | . Therefore, in the h ξ i ≥ h η i case, we have h ξ ih η i = 1 + h ξ i − h η ih η i ≤ | ξ | − | η | ≤ | ξ − η | . Then, by Proposition 2.1 (iii), we may write ϕ ( h ξ i ) ϕ ( h η i ) ≤ c (cid:18) h ξ ih η i (cid:19) s ≤ c (1 + | ξ − η | ) max { ,s } . Besides, if h η i ≥ h ξ i , then ϕ ( h ξ i ) ϕ ( h η i ) ≤ c (cid:18) h ξ ih η i (cid:19) s = c (cid:18) h η ih ξ i (cid:19) − s ≤ c (1 + | ξ − η | ) max { , − s } . Thus ϕ ( h ξ i ) ϕ ( h η i ) ≤ c (1 + | ξ − η | ) ℓ for all ξ, η ∈ R n , with ℓ := max { , − s , s } . This yields (2.1) for a certain constant c ≥ . (cid:3) (cid:3) Remark . There are Borel measurable functions ϕ : [1 , ∞ ) → (0 , ∞ ) such that ϕ / ∈ OR but µ ( ξ ) := ϕ ( h ξ i ) is a weight function of ξ ∈ R n (see Example 6.2).4. Proof of the main result
Beforehand, we prove two theorems. In the first one we describe the space which is a resultof the interpolation between Sobolev spaces provided the function interpolation parameter isused.
Theorem 4.1.
Let numbers s , s ∈ R be such that s < s , and let a function ψ ∈ B be aninterpolation parameter. Set (4.1) ϕ ( t ) := t s ψ ( t s − s ) for t ≥ . Then ϕ ∈ OR and (4.2) [ H ( s ) (Ω) , H ( s ) (Ω)] ψ = H ϕ (Ω) up to equivalence of norms. If Ω = R n , then (4.2) holds with equality of norms. NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 9
Proof.
First we prove that ϕ ∈ OR . By definition, the function ϕ is Borel measurable on [1 , ∞ ) . Let us prove that ϕ satisfies (2.3). Since the function ψ ∈ B is an interpolationparameter, it is pseudoconcave on ( r, ∞ ) , where we may put r = 1 (see Proposition 3.1 andthe comment to it). Hence, according to Proposition 3.3, we may write ϕ ( λt ) ϕ ( t ) = λ s ψ (( λt ) s − s ) ψ ( t s − s ) ≤ λ s c max { , λ s − s } = c λ s , (4.3) ϕ ( t ) ϕ ( λt ) = λ − s ψ ( t s − s ) ψ (( λt ) s − s ) ≤ λ − s c max { , λ s − s } = c λ − s (4.4)for arbitrary t ≥ , λ ≥ , and a certain number c > that is independent of t and λ . Thisshows (2.5); therefore, ϕ satisfies (2.3) with a = 2 and, hence, belongs to OR .Let us prove that(4.5) [ H ( s ) ( R n ) , H ( s ) ( R n )] ψ = H ϕ ( R n ) with equality of norms. The couple of Sobolev spaces [ H ( s ) ( R n ) , H ( s ) ( R n )] is admissible.Let J denote the pseudodifferential operator whose symbol is the function h ξ i s − s of ξ ∈ R n .Then J is a generating operator for this couple. Using the Fourier transform F : H ( s ) ( R n ) ↔ L ( R n , h ξ i s dξ ) , we reduce J to an operator of multiplication by the function h ξ i s − s . Hence, ψ ( J ) is reduced to an operator of multiplication by the function ψ ( h ξ i s − s ) = h ξ i − s ϕ ( h ξ i ) .Therefore, we may write the following: k u k H ( s ( R n ) ,H ( s ( R n )] ψ = k ψ ( J ) u k H ( s ( R n ) = Z R n | ( \ ψ ( J ) u )( ξ ) | h ξ i s dξ = Z R n | ψ ( h ξ i s − s ) b u ( ξ ) | h ξ i s dξ = Z R n ϕ ( h ξ i ) | b u ( ξ ) | dξ = k u k H ϕ ( R n ) for every u ∈ C ∞ ( R n ) . This implies the equality of spaces (4.5) as C ∞ ( R n ) is dense in both ofthem. (Note that C ∞ ( R n ) is dense in the interpolation space [ H ( s ) ( R n ) , H ( s ) ( R n )] ψ because C ∞ ( R n ) is dense in the Sobolev space H ( s ) ( R n ) , which is embedded continuously and denselyin the interpolation space.)Now formula (4.2) will be deduced from (4.5) in the remaining cases considered for Ω ,namely, where Ω is either an open half-space or a bounded domain in R n with Lipschitsboundary (cf. [24, Sec. 1.1.18], where classical interpolation methods are used). Note thatthe couple of Sobolev spaces in (4.2) is admissible. Let R Ω stand for the operator that restricts distributions u ∈ S ′ ( R n ) to Ω . We have the surjective bounded operators R Ω : H ( s ) ( R n ) → H ( s ) (Ω) , s ∈ R , (4.6) R Ω : H ϕ ( R n ) → H ϕ (Ω) . (4.7)Applying the interpolation with the parameter ψ , we infer, by (4.5), that the boundedness ofthe operators (4.6), with s ∈ { s , s } , implies boundedness of the operator R Ω : H ϕ ( R n ) = [ H ( s ) ( R n ) , H ( s ) ( R n )] ψ → [ H ( s ) (Ω) , H ( s ) (Ω)] ψ . Hence, since the operator (4.7) is surjective, we have the continuous embedding(4.8) H ϕ (Ω) ֒ → [ H ( s ) (Ω) , H ( s ) (Ω)] ψ . Let us prove the inverse inclusion and its continuity. We need to use a linear mapping, say T , that extends every distribution u ∈ [ s ∈ R H ( s ) (Ω) to R n and defines a bounded operator(4.9) T : H ( s ) (Ω) → H ( s ) ( R n ) for each s ∈ R . This mapping is constructed by R. Seeley [21] in the case where Ω is a half-space, andby V. S. Rychkov [19] in the case where Ω is a bounded domain with Lipschitz boundary.Consider the operators (4.9) for s = s and s = s . Since ψ is an interpolation parameter,their boundedness and formula (4.5) yield boundedness of the operator(4.10) T : [ H ( s ) (Ω) , H ( s ) (Ω)] ψ → [ H ( s ) ( R n ) , H ( s ) ( R n )] ψ = H ϕ ( R n ) . The product of the bounded operators (4.7) and (4.10) gives us the bounded identity operator I = R Ω T : [ H ( s ) (Ω) , H ( s ) (Ω)] ψ → H ϕ (Ω) . Thus, together with the continuous embedding (4.8), we have its continuous inverse; i.e., (4.2)holds up to equivalence of norms. (cid:3) (cid:3)
Theorem 4.2.
Let s , s ∈ R , with s < s , and let ψ ∈ B . Suppose that ϕ is defined by (4.1) . Then ψ is an interpolation parameter if and only if ϕ satisfies (2.5) with some number c ≥ that is independent of t and λ .Proof. If ψ is an interpolation parameter, then, as we have proved above, the function ϕ satisfies (4.3) and (4.4) for arbitrary t ≥ and λ ≥ , i.e. (2.5) is fulfilled. NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 11
Conversely, suppose that ϕ satisfies (2.5). Let us prove the inequality (3.1) for ψ . Consid-ering arbitrary t ≥ τ ≥ and applying the right-hand inequality in (2.5), we may write ψ ( t ) ψ ( τ ) = t − s / ( s − s ) ϕ ( t / ( s − s ) ) τ − s / ( s − s ) ϕ ( τ / ( s − s ) ) ≤ λ − s / ( s − s ) c λ s / ( s − s ) = c λ = c max (cid:26) , tτ (cid:27) ; here λ := t/τ ≥ , whereas the number c > does not depend on t and τ . Analogously,considering any τ ≥ t ≥ and applying the left-hand inequality in (2.5), we may write ψ ( t ) ψ ( τ ) ≤ λ s / ( s − s ) c λ − s / ( s − s ) = c = c max (cid:26) , tτ (cid:27) , with λ := τ /t ≥ . Thus, the inequality (3.1) holds for r = 1 . Hence, we conclude, byPropositions 3.1 and 3.3, that ψ is an interpolation parameter. (cid:3) (cid:3) Proof of Theorem . . Necessity. Let a Hilbert space H be an interpolation space withrespect to the couple of Sobolev spaces [ H ( s ) (Ω) , H ( s ) (Ω)] . Then, by Propositions 3.2, 3.1,and Theorem 4.1, we conclude that H = [ H ( s ) (Ω) , H ( s ) (Ω)] ψ = H ϕ (Ω) up to equivalence of norms. Here ψ ∈ B is a certain interpolation function parameter, whereas ϕ is defined by (4.1). The function ϕ satisfies (2.5) in view of Theorem 4.2 and, hence, belongsto OR . The necessity is proved. Sufficiency.
Let a function parameter ϕ ∈ OR satisfy condition (2.5). Suppose that aHilbert space H coincides with H ϕ (Ω) up to equivalence of norms. Starting with ϕ , weconstruct a Borel measurable function ψ such that (4.1) holds. Namely, we set(4.11) ψ ( τ ) := τ − s / ( s − s ) ϕ ( τ / ( s − s ) ) for τ ≥ ,ϕ (1) for < τ < . Note that ψ ∈ B in view of Proposition 2.1 (i) and condition (2.5). (This condition writtenfor t = 1 yields ψ ( τ ) /ϕ (1) ≥ c − for every τ ≥ .) By Theorem 4.2, the function ψ is aninterpolation parameter. Therefore, applying Theorem 4.1, we conclude that(4.12) H = H ϕ (Ω) = [ H ( s ) (Ω) , H ( s ) (Ω)] ψ up to equivalence of norms. Hence H is an interpolation space with respect to the couple [ H ( s ) (Ω) , H ( s ) (Ω)] . The sufficiency is proved. (cid:3) The following proof is simple, but we give it for the sake of completeness.
Proof of Theorem . . Necessity. Let a Hilbert space H be an interpolation space withrespect to the Sobolev scale { H ( s ) (Ω) : s ∈ R } . Then H is an interpolation space with respectto a certain couple [ H ( s ) (Ω) , H ( s ) (Ω)] , with −∞ < s < s < ∞ . Hence, by Theorem 2.4, we conclude that H = H ϕ (Ω) up to equivalence of norms for some ϕ ∈ OR . The necessity isproved. Sufficiency.
Let ϕ ∈ OR , and let a Hilbert space H coincide with H ϕ (Ω) up to equivalence ofnorms. Choose numbers s , s ∈ R such that s < σ ( ϕ ) and σ ( ϕ ) < s . Then condition (2.5)is satisfied. According to Theorem 2.4, H is an interpolation space with respect to the couple [ H ( s ) (Ω) , H ( s ) (Ω)] and, consequently, with respect to the Sobolev scale { H ( s ) (Ω) | s ∈ R } .The sufficiency is proved. (cid:3) The extended Sobolev scale
Here we expound two important interpolation properties of the extended Sobolev scale(5.1) { H ϕ (Ω) | ϕ ∈ OR } (see Theorems 5.1 and 5.2 below). Specifically, they explain why we have chosen such a namefor the class of H¨ormander spaces (5.1).Owing to the first property, every space in (5.1) can be obtained by the interpolation of anappropriate couple of Sobolev spaces provided that a certain interpolation function parameteris used. Just this result can be a basis for further applications of the scale (5.1) in variousproblems. Theorem 5.1.
Let ϕ ∈ OR . Choose numbers s , s ∈ R such that s < σ ( ϕ ) < σ ( ϕ ) < s and define a function ψ by formula (4.11) . Then ψ ∈ B is an interpolation parameter, and (5.2) [ H ( s ) (Ω) , H ( s ) (Ω)] ψ = H ϕ (Ω) up to equivalence of norms. If Ω = R n , then (5.2) holds with equality of norms.Proof. Note that ψ belongs to B and is an interpolation parameter; this has been demonstratedin the proof of Theorem 2.4 (sufficiency). Now, since ϕ satisfies (4.1), Theorem 5.1 is a directcorollary of Theorem 4.1. (cid:3) (cid:3) The second property reveals that the class of spaces (5.1) is closed with respect to interpo-lation with a function parameter.
Theorem 5.2.
Let functions ϕ , ϕ ∈ OR and ψ ∈ B be given. Suppose that ϕ /ϕ is boundedin a neighbourhood of + ∞ and that ψ is an interpolation parameter. Set ϕ ( t ) := ϕ ( t ) ψ (cid:18) ϕ ( t ) ϕ ( t ) (cid:19) for t ≥ . Then ϕ ∈ OR , and (5.3) [ H ϕ (Ω) , H ϕ (Ω)] ψ = H ϕ (Ω) up to equivalence of norms. If Ω = R n , then (5.3) holds with equality of norms. NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 13
Proof.
Let us deduce (5.3) from (5.2) with the help of the reiterated interpolation with afunction parameter. Choose numbers s , s ∈ R such that s < σ ( ϕ j ) and s > σ ( ϕ j ) foreach j ∈ { , } . By Theorem 5.1, we have [ H ( s ) (Ω) , H ( s ) (Ω)] ψ j = H ϕ j (Ω) for each j ∈ { , } . Here the function ψ j ∈ B is the interpolation parameter defined by (4.11) with ϕ = ϕ j .Note that ψ /ψ is bounded in a neighbourhood of + ∞ . According to Proposition 3.4 andTheorem 4.1, we may write [ H ϕ (Ω) , H ϕ (Ω)] ψ = (cid:2) [ H ( s ) (Ω) , H ( s ) (Ω)] ψ , [ H ( s ) (Ω) , H ( s ) (Ω)] ψ (cid:3) ψ = [ H ( s ) (Ω) , H ( s ) (Ω)] ω = H ϕ (Ω) . Here the interpolation parameter ω ∈ B satisfies the equality ω ( τ ) : = ψ ( τ ) ψ (cid:16) ψ ( τ ) ψ ( τ ) (cid:17) = τ − s / ( s − s ) ϕ ( τ / ( s − s ) ) ψ (cid:16) ϕ ( τ / ( s − s ) ) ϕ ( τ / ( s − s ) ) (cid:17) for τ ≥ . Hence, the function ϕ ( t ) := ϕ ( t ) ψ (cid:16) ϕ ( t ) ϕ ( t ) (cid:17) = t s ω ( t s − s ) , t ≥ , belongs to the class OR according to Theorem 4.1. The equality of spaces is written up toequivalence of norms, with the equivalence becoming equality in the case of Ω = R n . (cid:3) (cid:3) Thus the extended Sobolev scale (5.1) is the final extension of the scale { H ( s ) (Ω) | s ∈ R } by the interpolation within the category of Hilbert spaces.6. Examples
Theorem 2.4 allows us to construct three explicit examples of Hilbert spaces that areintermediate for given couples of Sobolev spaces but differ considerably on their interpolationproperties.Namely, for the couples(6.1) [ H ( s ) (Ω) , H ( s ) (Ω)] , with s ≤ < s , we construct examples of intermediate spaces H ϕ (Ω) that each of the following possibilitiesis realized. For every couple (6.1), the space H ϕ (Ω) (i) is an interpolation space;(ii) is not an interpolation space;(iii) is not an interpolation space provided that s = 0 , but is an interpolation spaceprovided that s < . Example 6.1.
First we give an example of a Hilbert space H ϕ (Ω) which is an interpolation(and intermediate) space for every couple (6.1). We obtain the required space H ϕ (Ω) if wetake ϕ ( t ) := 1 + log t for t ≥ . This follows from Theorem 2.4 because σ ( ϕ ) = 0 = σ ( ϕ ) and the supremum in (2.6) is evidently attained. Example 6.2.
Now we give an example of a Hilbert space H ϕ (Ω) such that, for every couple(6.1), H ϕ (Ω) is an intermediate space but is not an interpolation space. Let us put ϕ ( t ) := 1 + (1 + sin log t ) log t for t ≥ and show that the space H ϕ (Ω) is as required.The corresponding radial function µ ( ξ ) := ϕ ( h ξ i ) of ξ ∈ R n is a weight function, i.e. (2.1)is fulfilled. Indeed, given t, τ ≥ , we may write(6.2) ϕ ( t ) ϕ ( τ ) ≤ | ϕ ( t ) − ϕ ( τ ) | ϕ ( τ ) ≤ | ϕ ( t ) − ϕ ( τ ) | ≤ | t − τ | . Here we use the estimate | ϕ ′ ( θ ) | ≤ for every θ ≥ , which is directly verified. It followsfrom (6.2) that the function µ ( ξ ) := ϕ ( h ξ i ) satisfies (2.1) with c = 3 and ℓ = 1 . Thus, theH¨ormander space H ϕ (Ω) is well-defined.Choose numbers s ≤ and s > arbitrarily. Since ≤ ϕ ( t ) ≤ t for t ≥ , wehave the continuous embeddings H ( s ) (Ω) ֒ → H ϕ (Ω) ֒ → H ( s ) (Ω) by virtue of (2.2). Note alsothat ϕ / ∈ OR . Indeed, putting λ := exp( π/ and t k := exp(2 πk − π/ with k = 1 , , , . . . ,we have ϕ ( λt k ) /ϕ ( t k ) = 1 + 2 πk → ∞ as k → ∞ , contrary to the property (2.5) of the classOR. Therefore, by Theorem 2.4, H ϕ (Ω) is not an interpolation space for the couple (6.1).Let us argue the latter conclusion in more detail. If H ϕ (Ω) were an interpolation space forthis couple, then H ϕ (Ω) = H ϕ (Ω) up to equivalence of norms for some ϕ ∈ OR by Theorem2.4. Hence, ϕ ≍ ϕ on [1 , ∞ ) (see Remark 2.3). This contradicts ϕ / ∈ OR . Example 6.3.
Finally, we give an example of a Hilbert space H ϕ (Ω) such that, for each couple [ H (0) (Ω) , H ( s ) (Ω)] with s > , H ϕ (Ω) is an intermediate space but is not an interpolationspace and, moreover, H ϕ (Ω) is an interpolation space for every couple [ H ( s ) (Ω) , H ( s ) (Ω)] with s < < s .Set h ( t ) := (log t ) − / sin(log / t ) and define the function ϕ ( t ) := t h ( t ) + log t if t ≥ , if < t < . Let us show that the space H ϕ (Ω) is as required.Evidently, ϕ ∈ B . A simple calculation shows that tϕ ′ ( t ) /ϕ ( t ) → as t → ∞ . Hence [22,Section 1.2], the function ϕ is slowly varying at infinity in the sense of J. Karamata; i.e.,(6.3) lim t →∞ ϕ ( λ t ) ϕ ( t ) = 1 for each λ > . NTERPOLATION HILBERT SPACES BETWEEN SOBOLEV SPACES 15
Owing to Uniform Convergence Theorem [22, Section 1.2, Theorem 1.1], the convergencein (6.3) is uniform on every compact λ -set in (0 , ∞ ) . Therefore ϕ ∈ OR and, moreover, σ ( ϕ ) = σ ( ϕ ) = 0 (see [2, Section 2.1]). Thus the space H ϕ (Ω) is well-defined.Since both the functions /ϕ ( t ) and ϕ ( t ) /t s are bounded on [1 , ∞ ) for every s > , thecontinuous embeddings H ( s ) (Ω) ֒ → H ϕ (Ω) ֒ → H (0) (Ω) hold.Next, we show that ϕ is not pseudoconcave on ( r, ∞ ) whenever r > . Consider thesequences of numbers t k := exp((2 πk + π/ ) and s k := exp((2 πk + π ) ) , with k = 1 , , , . . . .Straightforward calculations yield h ( t k ) = (2 πk + π/ − and h ( s k ) = 0 ; hence, log ϕ ( t k ) ≥ h ( t k ) log t k = (cid:16) πk + π (cid:17) and ϕ ( s k ) = 1 + (2 πk + π ) . Therefore, ϕ ( t k ) ϕ ( s k ) ≥ exp((2 πk + π/ )(1 + (2 πk + π ) ) → ∞ as k → ∞ . But t k < s k , then, by Proposition 3.3, the function ϕ is not pseudoconcave on ( r, ∞ ) whenever r > .Applying this fact, we will prove that H ϕ (Ω) is not an interpolation space for the couple [ H (0) (Ω) , H (1) (Ω)] . Suppose the contrary; then, by Proposition 3.2, we may write H ϕ (Ω) =[ H (0) (Ω) , H (1) (Ω)] ψ (up to equivalence of norms) for some function parameter ψ ∈ B which ispseudoconcave in a neighbourhood of + ∞ . Hence, H ϕ (Ω) = H ψ (Ω) by Theorem 5.2 appliedto the functions ϕ ( t ) ≡ t and ϕ ( t ) ≡ t . As we have noted in Remark 2.3, the equality H ϕ (Ω) = H ψ (Ω) is equivalent to the property ϕ ≍ ψ on [1 , ∞ ) . Hence, ϕ is pseudoconcavein a neighbourhood of + ∞ , a contradiction.Thus, H ϕ (Ω) is not an interpolation space for the couple (cid:2) H (0) (Ω) , H (1) (Ω) (cid:3) . It follows from this and Theorem 2.4 (see Remark 2.5) that the supremum in (2.6) is notattained (recall that σ ( ϕ ) = 0 = σ ( ϕ ) ). 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