aa r X i v : . [ m a t h . F A ] D ec Function Spaces on Singular Manifolds
H. Amann ∗ Math. Institut, Universit¨at Z¨urich, Winterthurerstr. 190, CH–8057 Z¨urich, Switzerland
Key words
Weighted Sobolev spaces, Bessel potential spaces, Besov spaces, singularities, non-complete Rie-mannian manifolds with boundary
MSC (2000) R n , continue to be valid on a wide class of Riemannian manifoldswith singularities and boundary, provided suitable weights, which reflect the nature of the singularities, areintroduced. These results are of importance for the study of partial differential equations on piece-wise smoothdomains. It is our principal concern in this paper to develop a satisfactory theory of spaces of functions and tensor fields onRiemannian manifolds which may have a boundary and may be non-compact and non-complete. Such a theoryhas to extend the basic results known for function spaces on subdomains of R n with smooth boundary to this moregeneral setting, that is to say, embedding and interpolation properties, point-wise multiplier and trace theorems,duality characterizations and, last but not least, intrinsic local descriptions.Our research is motivated by — and provides the basis for — the study of elliptic and parabolic boundary valueproblems on piece-wise smooth manifolds, on domains in R n with piece-wise smooth boundary, in particular.Such domains occur in a wide variety of problems modeling physical, chemical, biological, and engineeringprocesses by means of differential and pseudodifferential equations. In this connection Sobolev spaces play apredominant role, as is well-known from the theory of partial differential equations on smooth domains. In thepresence of singularities, say edges on the boundary, solutions of differential equations lose their smoothnessnear these singularities. Since the seminal work of V.A. Kondrat ′ ev [22] on elliptic boundary value problems indomains with conical points it is known that an appropriate setting for the study of such problems is provided bySobolev spaces with weights reflecting the nature of the singularity. This has since been exploited by numerousauthors and there is a large number of papers and monographs devoted to elliptic problems on non-smoothdomains. Besides of the early papers by V.G. Maz ′ ya and B.A. Plamenevski˘ı [26]–[28], the first successfulapproaches to this kind of problems, we cite only the following few books and refer the reader to the referencestherein for further information: P. Grisvard [19], M. Dauge [15], S.A. Nazarov and B.A. Plamenevski˘ı [30],V.A. Kozlov, V.G. Maz ′ ya, and J. Rossmann [23], V.G. Maz ′ ya, and J. Rossmann [29] (and many more papersand books by V.G. Maz ′ ya and coauthors), and the numerous contributions of B.-W. Schulze and co-workers onthe L -theory of elliptic pseudo-differential boundary problems on singular manifolds for which [34] may standrepresentatively.Weighted Sobolev spaces of a different type occur as solution spaces for degenerate elliptic equations. This facthas triggered a large amount of research on weighted Sobolev and related function spaces, e.g., A. Kufner [24],H. Triebel [37], H.-J. Schmeisser and H. Triebel [33], and the references therein. Since that work is not directlyrelated to the subject of our paper we do not give more details or cite more recent references. ∗ e-mail: [email protected] H. Amann: Function Spaces on Singular Manifolds
In Section 2 we give a precise definition of our concept of a singular manifold M . It will be seen that, to a largeextent, M is determined by a ‘singularity function’ ρ ∈ C ∞ (cid:0) M, (0 , ∞ ) (cid:1) . The behavior of ρ at the ‘singular ends’of M , that is, near that parts of M at which ρ gets either arbitrarily small or arbitrarily large, reflects the singularstructure of M . It turns out that the basic building blocks for a useful theory of function spaces on singularmanifolds are weighted Sobolev spaces based on the singularity function ρ . More precisely, we denote by K either R or C . Then, given k ∈ N , λ ∈ R , and p ∈ (1 , ∞ ) , the weighted Sobolev space W k,λp ( M ) = W k,λp ( M, K ) isthe completion of D ( M ) , the space of smooth functions with compact support in M , in L , loc ( M ) with respectto the norm u (cid:16) k X i =0 (cid:13)(cid:13) ρ λ + i |∇ i u | g (cid:13)(cid:13) pp (cid:17) /p . (1 . Here ∇ denotes the Levi-Civita covariant derivative and |∇ i u | g is the ‘length’ of the covariant tensor field ∇ i u naturally derived from the Riemannian metric g of M . Of course, integration is carried out with respect to thevolume measure of M . It turns out that W k,λp ( M ) is well-defined, independently — in the sense of equivalentnorms — of the representation of the singularity structure of M by means of the particular singularity function.A very special and simple example of a singular manifold is provided by a bounded smooth domain whoseboundary contains a conical point. More precisely, suppose Ω is a bounded domain in R m whose topolog-ical boundary, bdry(Ω) , contains the origin, and Γ := bdry(Ω) \{ } is a smooth ( m − -dimensional sub-manifold of R m lying locally on one side of Ω . Also suppose that Ω ∪ Γ is near diffeomorphic to a cone { ry ; 0 < r < , y ∈ B } , where B is a smooth compact submanifold of the unit sphere in R m . Then, en-dowing M := Ω ∪ Γ with the Euclidean metric, we get a singular manifold with a single conical singularity, asconsidered in [30] and [23], for example. In this case the weighted norm (1.1) is equivalent to u (cid:16) X | α |≤ k k r λ + | α | ∂ α u k pL p (Ω) (cid:17) /p , where r ( x ) is the Euclidean distance from x ∈ M to the origin. Moreover, W k,λp ( M ) coincides with the space V kp,λ + k (Ω) employed by S.A. Nazarov and B.A. Plamenevski˘ı (cf. p. 319 of [30]) and, in the case p = 2 , by V.A.Kozlov, V.G. Maz ′ ya, and J. Rossmann (see Section 6.2 of [23], for example).As mentioned above, the theory of function spaces on singular manifolds is built on the weighted Sobolevspaces W k,λp ( M ) . We define weighted Sobolev spaces of negative order by duality, and Bessel potential spaces, H s,λp ( M ) , and Besov spaces, B s,λp,p ( M ) , by complex and real interpolation, respectively. A basic result, whichrenders the theory useful, is the fact that these spaces can be characterized locally by their ‘classical’ non-weighted counterparts on R m and on half-spaces. This implies, in particular, H k,λp ( M ) = W k,λp ( M ) for k ∈ N .A linear differential operator on a Riemannian manifold is of the form P ki =0 a i · ∇ i u , where a i is a con-travariant tensor field of order i and · denotes complete contraction. In order to study continuity properties ofsuch operators in the weighted function spaces under consideration we have to have at our disposal point-wisemultiplier theorems for tensor fields. Thus it is mandatory to study spaces of tensor fields on singular manifolds.In the particular case where we can choose the constant map as singularity function, our spaces reduce to non-weighted Sobolev spaces W kp ( M ) , Bessel potential spaces H sp ( M ) , and Besov spaces B sp,p ( M ) , respectively.This is, for example, the case if M is a complete Riemannian manifold without boundary and with boundedgeometry (that is, M has a positive injectivity radius and all covariant derivatives of the curvature tensor arebounded). To the best of our knowledge, this is the only class of Riemannian manifolds for which a generaltheory of function spaces has been developed so far. More precisely:Integer order Sobolev spaces, with particular emphasis on the validity of Sobolev’s embedding theorem, havebeen treated by Th. Aubin [12]–[14] in the case of compact manifolds with boundary, and for complete Rieman-nian manifolds without boundary, making essential use of curvature estimates and the positivity of the injectivityradius. Also see E. Hebey [20] and [21] for the case where M has no boundary.Bessel potential spaces H sp ( M ) , < p < ∞ , s ∈ R , on complete Riemannian manifolds without boundaryhave been introduced and investigated by R.S. Strichartz [36] as domains of the fractional powers of − ∆ M ,where ∆ M is the Laplace-Beltrami operator. H. Triebel [38], [39] (see also [40]) established a general theoryof Triebel-Lizorkin and Besov spaces on complete Riemannian manifolds without boundary and with bounded geometry. His work makes use of a distinguished coordinate system based on the exponential map and of mappingproperties of the Laplace-Beltrami operator.None of the above techniques is available in our situation, where M may be not complete or may not havebounded geometry. In particular, relevant properties of the Laplace-Beltrami operator are not at our disposal,even in the case where M has no boundary. Anyhow, they would not be helpful in the presence of a boundary.B. Ammann, R. Lauter, and V. Nistor [8] introduce a class of complete non-compact Riemannian manifoldswithout boundary and with bounded geometry, called Lie manifolds. This class encompasses, in particular,manifolds with cylindrical ends and manifolds being Euclidean at infinity. In B. Ammann, A.D. Ionescu, andV. Nistor [7] Bessel potential spaces on suitable open subsets of Lie manifolds — called Sobolev spaces thereinand denoted by W s,p — are being investigated to some extent. Lie manifolds are useful for the study of regularityproperties of elliptic differential operators on polyhedral domains in which case the authors are led to introduceweighted Bessel potential spaces, the weight being equivalent to the distance to the non-smooth boundary points(also see [9], [10], and the references therein for related research). The results of the present paper apply to Liemanifolds and polyhedral domains as well and greatly extend and sharpen the investigations of these authors; inparticular, as far as the trace theorem is concerned.There seem to be only very few general results on spaces of tensor fields. J. Eichhorn [17] studies integer orderSobolev spaces of differential forms on complete Riemannian manifolds without boundary and with bounded ge-ometry; also see [18]. Some results on Sobolev spaces of differential forms on compact manifolds with boundarycan be found in G. Schwarz [35]. Of course, there are many ‘ad hoc’ results in the literature, predominantly on L -Sobolev spaces, for Riemannian manifolds (without boundary) possessing specific geometries.Section 3 is of technical nature. There we review some concepts from differential geometry, mainly to fixnotation. Then we prove basic estimates related to the singularity structure of the manifold. They are fundamentalfor the construction of universal retractions by which we can transplant the well-established theory of functionspaces on R m to the singular manifold. For this we first have to establish a localization procedure for tensor-field-valued distributions on M . This is done in Sections 4 and 5. In Section 6 we show that this localizationprocedure induces a corresponding retraction-coretraction system on Sobolev spaces. Then, by interpolation, weextend the retraction-coretraction theorem to Bessel potential and Besov spaces of positive order.After having introduced weighted H¨older spaces in Section 8, we prove in Section 9 point-wise multipliertheorems. Section 10 is devoted to the trace theorem, and in the following section we characterize spaces withvanishing traces. This puts us in position to define, in Section 12, spaces of negative order by duality. Allspaces under consideration possess the retraction-coretraction property induced from the localization procedurefor tensor-field-valued sections constructed in Section 5. By means of this property we can then, in Sections 13and 14, respectively, easily prove interpolation and embedding theorems for weighted spaces of tensor fields onsingular manifolds.Section 15 is concerned with spaces of differential forms. In particular, we establish mapping properties of theexterior differential and codifferential operators, and, as an application, of the gradient and divergence operators.These results are of importance in the study of differential operators on singular manifolds. Such investigations,which will be carried out elsewhere, rely fundamentally on the retraction-coretraction theorems established inthis paper.For simplicity, and being oriented towards differential equations, we restrict our considerations essentially toweighted Sobolev-Slobodeckii spaces. However, we include some brief remarks concerning possible extensionsto spaces of Triebel-Lizorkin type. By a manifold we always mean a smooth, that is, C ∞ manifold with (possibly empty) boundary such that itsunderlying topological space is separable and metrizable. Thus, in the context of manifolds, we work in thesmooth category. A manifold need not be connected, but all connected components are of the same dimension.We denote by H m the closed right half-space R + × R m − in R m , where R = { } . We set Q := ( − , ⊂ R .If κ is a local chart for an m -dimensional manifold M , then we write U κ for the corresponding coordinatepatch dom( κ ) . A local chart κ is normalized if κ ( U κ ) = Q m whenever U κ ⊂ ˚ M , the interior of M , whereas κ ( U κ ) = Q m ∩ H m if U κ ∩ ∂M = ∅ . We put Q mκ := κ ( U κ ) if κ is normalized. H. Amann: Function Spaces on Singular Manifolds
An atlas K for M has finite multiplicity if there exists k ∈ N such that any intersection of more than k coor-dinate patches is empty. It is uniformly shrinkable if it consists of normalized charts and there exists r ∈ (0 , such that (cid:8) κ − ( rQ mκ ) ; κ ∈ K (cid:9) is a cover of M .Given an open subset X of R m or H m and a Banach space E over K , we write k·k k, ∞ for the usual normof BC k ( X, E ) , the Banach space of all u ∈ C k ( X, E ) such that | ∂ α u | E is uniformly bounded for α ∈ N m with | α | ≤ k .By c we denote constants ≥ whose numerical value may vary from occurrence to occurrence; but c is alwaysindependent of the free variables in a given formula, unless an explicit dependence is indicated.Let S be a nonempty set. On R S , the space of all real-valued functions on S , we introduce an equivalencerelation ∼ by setting f ∼ g iff there exists c ≥ such that f /c ≤ g ≤ cf . By we denote the constant function s , whose domain will always be clear from the context.The Euclidean metric on R m , ( dx ) + · · · + ( dx m ) , is denoted by g m . The same symbol is used for itsrestriction to an open subset U of R m or H m , that is, for ι ∗ g m , where ι : U ֒ → R m is the natural embedding.Here and below, we employ the standard notation for pull-back and push-forward operations.Let M = ( M, g ) be an m -dimensional Riemannian manifold. Suppose ρ ∈ C ∞ (cid:0) M, (0 , ∞ ) (cid:1) and K is an atlasfor M . Then ( ρ, K ) is a singularity datum for M if (i) K is uniformly shrinkable, has finite multiplicity, and is orientation preserving if M is oriented. (ii) k e κ ◦ κ − k k, ∞ ≤ c ( k ) , κ, e κ ∈ K , k ∈ N . (iii) κ ∗ ( ρ − g ) ∼ g m , κ ∈ K . (iv) k κ ∗ ( ρ − g ) k k, ∞ ≤ c ( k ) , κ ∈ K , k ∈ N . (v) k κ ∗ ρ k k, ∞ ≤ c ( k ) ρ κ , κ ∈ K , k ∈ N , where ρ κ := κ ∗ ρ (0) = ρ (cid:0) κ − (0) (cid:1) . (vi) 1 /c ≤ ρ ( p ) /ρ κ ≤ c, p ∈ U κ , κ ∈ K . (2 . In (ii) and in similar situations it is understood that only κ, e κ ∈ K with U κ ∩ U e κ = ∅ are being considered. Con-dition (iii) reads more explicitly: κ ∗ ρ ( x ) | ξ | /c ≤ κ ∗ g ( x )( ξ, ξ ) ≤ cκ ∗ ρ ( x ) | ξ | , x ∈ Q mκ , ξ ∈ R m , κ ∈ K . Note that the finite multiplicity of K and the separability of M imply that K is countable.Let ( ρ, K ) and ( e ρ, e K ) be singularity data for M . Set N ( κ ) := { e κ ∈ e K ; U e κ ∩ U κ = ∅ } , κ ∈ K . Then ( ρ, K ) and ( e ρ, e K ) are equivalent if (i) ρ ∼ e ρ ;(ii) card (cid:0) N ( κ ) (cid:1) ≤ c, κ ∈ K ;(iii) k e κ ◦ κ − k k, ∞ ≤ c ( k ) , κ ∈ K , e κ ∈ e K , k ∈ N . (2 . A singularity structure , S ( M ) , for M is a maximal family of equivalent singularity data. A singularity function for M is a function ρ ∈ C ∞ (cid:0) M, (0 , ∞ ) (cid:1) such that there exists an atlas K with ( ρ, K ) ∈ S ( M ) . The set of allsingularity functions is the singularity type , T ( M ) , of M . By a singular manifold we mean a Riemannianmanifold M endowed with a singularity structure S ( M ) . Then M is said to be singular of type T ( M ) . If ρ ∈ T ( M ) , then it is convenient to set [[ ρ ]] := T ( M ) . A singular manifold of type [[ ]] is called uniformly regular .Let ( M, g ) be singular of type [[ ρ ]] . It follows from (2.1)(i)–(iv) that then ( M, ρ − g ) is uniformly regular.Suppose ρ / ∼ . Then either inf ρ = 0 or sup ρ = ∞ , or both. Hence M is not compact but has singular ends. Itfollows from (2.1)(iii) that the diameter of the coordinate patches converges either to zero or to infinity near thesingular ends in a manner controlled by the singularity type T ( M ) . Examples 2.1 (a)
Every compact Riemannian manifold is uniformly regular. (b)
Let M be an m -dimensional Riemannian submanifold of R m possessing a compact boundary. Then M is uniformly regular. (c) R m = ( R m , g m ) and H m = ( H m , g m ) are uniformly regular. P r o o f . For X ∈ { R m , H m } and z ∈ Z m ∩ X we set Q mz := Q m if either X = R m or z ∈ ˚ H m ; otherwise welet Q mz := Q m ∩ H m . We put U z := z + Q mz and κ z ( x ) := x − z for z ∈ Z m ∩ X and x ∈ U z . Then ( , K ) ,where K := { κ z ; Z m ∩ X } , is a singularity datum for X . (d) Let ( M, g ) be singular of type [[ ρ ]] and ϕ : M → N a diffeomorphism. Then ( N, ϕ ∗ g ) is singular oftype [[ ϕ ∗ ρ ]] . Assume ( ρ, K ) is a singularity datum for M and set ϕ ∗ K := { ϕ ∗ κ ; κ ∈ K } . Then ( ϕ ∗ ρ, ϕ ∗ K ) is asingularity datum for N . (e) Let M be singular of type [[ ρ ]] . Suppose ∂M = ∅ . Denote by • ι : ∂M ֒ → M the natural injection andendow ∂M with the induced Riemannian metric g ∂M := • ι ∗ g . Suppose κ : U κ → R m is a local chart for M with U • κ := ∂U κ = U κ ∩ ∂M = ∅ . Put • κ := ι ◦ ( • ι ∗ κ ) : U • κ → R m − , where ι : { } × R m − → R m − , (0 , x ′ ) x ′ . Let K be a normalized atlas for M . Then a normalized atlasfor ∂M is given by • K := { • κ ; κ ∈ K , ∂U κ = ∅ } , the one induced by K . Assume ( ρ, K ) is a singularity datumfor M . Set • ρ := • ι ∗ ρ = ρ | ∂M . Then ( • ρ, • K ) is a singularity datum for ∂M . Thus ∂M is singular of type [[ • ρ ]] . (f) If M is a complete Riemannian manifold without boundary and with bounded geometry, then M isuniformly regular.P r o o f . This follows from Lemma 2.2.6 in [13], for example.In order to describe nontrivial classes of singular manifolds we need some preparation. Let N be a completeRiemannian manifold without boundary and of dimension n . Suppose M is an m -dimensional submanifoldof N . Denote by M the closure of M in N . Then S ( M ) := M \ M is the singularity set of M (in N ). Thus M = ˚ M ∪ ∂M ∪ S ( M ) and S ( M ) is closed in N . In particular, M is not complete if S ( M ) = ∅ .We assume now that M can be described, locally in the neighborhood of S ( M ) , by model cusps and wedgesover such cusps. More precisely: suppose d ∈ N × := N \{ } and B is a submanifold of S d − , the unit spherein R d . Then K d ( B ) := { ry ∈ R d ; 0 < r < , y ∈ B } , where y ∈ B is identified with its image in R d under the natural embedding S d − ֒ → R d , is called model coneover B in R d .Next, let < α < ∞ and assume now that B is a submanifold of Q d − , where d ≥ . Then K dα ( B ) := (cid:8) ( r, r α y ) ∈ R d ; 0 < r < , y ∈ B (cid:9) is the model α -cusp in R d . To allow for a unified treatment we call K d , in abuse of language, model -cusp.Then, given α ∈ [1 , ∞ ) and ℓ ∈ N , K dα ( B, ℓ ) := K dα ( B ) × Q ℓ is the model ( α, ℓ ) -wedge over B in R d + ℓ . Here and below, all references to Q ℓ have to be neglected if ℓ = 0 .Thus K dα ( B,
0) = K dα ( B ) , and a model cusp is a specific instance of a model wedge.If b := dim( B ) , then K dα ( B, ℓ ) is a submanifold of R d + ℓ of dimension b + 1 + ℓ and boundary K dα ( ∂B, ℓ ) .Thus ∂K dα ( B, ℓ ) = ∅ if ∂B = ∅ , which is the case, in particular, if α = 1 and B = S d − , or if b = 0 .Now we suppose ≤ ℓ ≤ m − and S is an ℓ -dimensional submanifold of N without boundary, containedin S ( M ) . We also suppose α ∈ [1 , ∞ ) and B is an ( m − ℓ − -dimensional submanifold of S m − ℓ − if α = 1 ,or of Q m − ℓ − if α > . Then S is called ( α, ℓ ) -wedge of M over B if for each p ∈ S there exists a normalizedlocal chart ϕ for N at p such that S ( M ) ∩ U ϕ = S ∩ U ϕ , ϕ ( M ∩ U ϕ ) = (cid:0) K m − ℓα ( B, ℓ ) × { } (cid:1) ∩ Q n , and ϕ ( S ∩ U ϕ ) = (cid:0) { } × Q ℓ (cid:1) × { } . Thus an ( α, ℓ ) -wedge of M over B looks locally like the model wedge K m − ℓα ( B, ℓ ) in R m .Finally, M is called relatively compact ( sub- ) manifold ( of N ) with smooth cuspidal singularities if M iscompact, S ( M ) = ∅ , and for each connected component Γ of S ( M ) there exist α ∈ [1 , ∞ ) , ℓ ∈ { , . . . , m − } ,and a compact manifold B such that Γ is an ( α, ℓ ) -wedge of M over B . H. Amann: Function Spaces on Singular Manifolds
In the adjacent figure we have depicted a three-dimensional rel-atively compact submanifold M of R with smooth cuspidal singu-larities. More precisely, S ( M ) consists of connected components,namely of one . -cusp, one (2 , -wedge (the upper rim), and three (1 , -wedges (one at the bottom of the figure and two on the innerplateau).Let M be a relatively compact submanifold of N with smoothcuspidal singularities. Denote by Γ the set of connected componentsof S ( M ) . Since S ( M ) is closed in M , it is compact. Hence Γ isa finite set and each Γ ∈ Γ is a compact submanifold of N withoutboundary.Given a nonempty subset S of S ( M ) , we denote by d N ( p, S ) the Riemannian distance in N from p ∈ N to S .For each Γ ∈ Γ we can find a relatively compact open neighborhood U Γ in N such that d N (cid:0) p, S ( M ) (cid:1) = d N ( p, Γ) for p ∈ U Γ and d N ( · , Γ) is smooth on U Γ . Moreover, there exists a unique α Γ ∈ [1 , ∞ ) such that Γ is an (cid:0) α Γ , dim(Γ) (cid:1) -wedge of M over some compact manifold B Γ of dimension m − dim(Γ) − . Theorem 2.2
Let M be a relatively compact manifold with smooth cuspidal singularities.Choose ρ ∈ C ∞ (cid:0) M, (0 , (cid:1) satisfying ρ ( p ) ∼ (cid:0) d N ( p, Γ) (cid:1) α Γ for p near Γ ∈ Γ . Then M is a singular manifoldof type [[ ρ ]] . P r o o f. H. Amann [4].In the case of the manifold M depicted above, ρ behaves near S ( M ) like the power α of the Euclideandistance in R to S ( M ) , where α = 2 . near the vertex of the cusp, α = 2 near the upper rim, and α = 1 nearthe remaining three wedges.For manifolds with non-smooth cuspidal singularities we refer to [4]. There it is no longer assumed that B Γ isa compact manifold, but B Γ itself can have (non-) smooth cuspidal singularities. This covers the case of cornersand intersecting wedges. In addition, in [4] we consider singular manifolds which are not relatively compact; forexample: subdomains of R m with ‘outlets to infinity’. It is the purpose of this section to provide technical estimates on which much of what follows is based. First weprepare some results on tensor bundles and covariant derivatives. For general background information we referto J. Dieudonn´e [16], for instance.Let M = ( M, g ) be an m -dimensional Riemannian manifold. We denote by T M and T ∗ M the (complexified,if K = C ) tangent and cotangent bundle, respectively. Then, given σ, τ ∈ N , T στ M := T M ⊗ σ ⊗ T ∗ M ⊗ τ is the ( σ, τ ) -tensor bundle of M , that is, the vector bundle of all tensors on M being contravariant of order σ and covariant of order τ . We use obvious conventions if σ = 0 or τ = 0 . In particular, T M = M × K , a trivialvector bundle. We write T στ M for the C ∞ ( M ) -module of all smooth sections of T στ M , the smooth ( σ, τ ) -tensorfields on M . For abbreviation, T M := T M and T ∗ M := T M .For ν ∈ N × we set J ν := { , . . . , m } ν . Then, given local coordinates κ = ( x , . . . , x m ) and setting ∂∂x ( i ) := ∂∂x i ⊗ · · · ⊗ ∂∂x i σ , dx ( j ) := dx j ⊗ · · · ⊗ dx j τ for ( i ) = ( i , . . . , i σ ) ∈ J σ , ( j ) ∈ J τ , the local representation of a ∈ T στ M with respect to these coordinates isgiven by a = a ( i )( j ) ∂∂x ( i ) ⊗ dx ( j ) (3 . with a ( i )( j ) ∈ C ∞ ( U κ ) . Here and below, we use the summation conventions whereby expressions are summed overall possible values of repeated indices. We write g ♭ : T M → T ∗ M for the conjugate linear (fiber-wise defined) Riesz isomorphism. Thus h g ♭ X, Y i = g ( Y, X ) , X, Y ∈ T M, (3 . where h· , ·i : T ∗ M × T M → C ∞ ( M ) (3 . is the (fiber-wise defined) duality pairing. The inverse of g ♭ , denoted by g ♯ , satisfies h α, Y i = g ( Y, g ♯ α ) , α ∈ T ∗ M, X ∈ T M. Denoting by g ∗ the adjoint Riemannian metric on T ∗ M it follows from (3.2) that h α, g ♯ β i = h g ♭ g ♯ α, g ♯ β i = g ( g ♯ β, g ♯ α ) = g ∗ ( α, β ) , α, β ∈ T ∗ M. (3 . From this we obtain, in local coordinates, g ♭ X = g ij X j dx i , g ♯ α = g ij α j ∂∂x i for X = X i ∂∂x i , α = α j dx j , (3 . where g = g ij dx i ⊗ dx j and [ g ij ] is the inverse of the matrix [ g ij ] .We let h· , ·i : T στ M × T τσ M → C ∞ ( M ) (3 . be the natural extension of (3.3). Thus, given p ∈ M , we write ( T τσ M ) p for the fiber of T τσ M over p . Then, fordecomposable tensors u ⊗ α ∈ ( T στ M ) p and v ⊗ β ∈ ( T τσ M ) p , h u ⊗ α, v ⊗ β i p := σ Y i =1 h β i , u i i p τ Y j =1 h α j , v j i p , where u = u ⊗ · · · ⊗ u σ ∈ ( T σ M ) p and α = α ⊗ · · · ⊗ α τ ∈ ( T τ M ) p , etc. Hence ( T στ M ) ′ = T τσ M with respect to the ‘tensor product duality pairing’ (3.6). This is consistent with ( T M ) ′ = T ∗ M .Suppose σ + τ ≥ . We put ( G τσ a )( α , . . . , α τ , X , . . . , X σ ) := a ( g ♭ X , . . . , g ♭ X σ , g ♯ α , . . . , g ♯ α τ ) (3 . for a ∈ T στ M , α , . . . , α τ ∈ T ∗ M , and X , . . . , X σ ∈ T M . This induces a conjugate linear bijection G τσ : T στ M → T τσ M, ( G τσ ) − = G στ . Consequently, ( ·|· ) g : T στ M × T στ M → C ∞ ( M ) , ( a, b )
7→ h a, G τσ b i (3 . is an inner product (a vector bundle metric) on T στ M , the inner product induced by g . It follows from (3.5) that,in local coordinates, ( a | b ) g = g ( i )( j ) g ( k )( ℓ ) a ( i )( k ) b ( j )( ℓ ) , a, b ∈ T στ M, (3 . where g ( i )( j ) := g i j · · · g i σ j σ , g ( k )( ℓ ) := g k ℓ · · · g k τ ℓ τ (3 . for ( i ) , ( j ) ∈ J σ and ( k ) , ( ℓ ) ∈ J τ . Of course, ( a | b ) g = ab for a, b ∈ T M = C ∞ ( M ) . Clearly, |·| g : T στ M → C ( M ) , a q ( a | a ) g H. Amann: Function Spaces on Singular Manifolds is called (vector bundle) norm induced by g . (We do not notationally indicate the dependence on ( σ, τ ) . This willbe clear from the context.) Note that | a | g = g ∗ ( a, a ) for a ∈ T M . For this reason we also write | a | g ∗ for | a | g if a ∈ T τ M .Let ϕ : M → N be a diffeomorphism onto some manifold N . Then one verifies ϕ ∗ (cid:0) ( a | b ) g (cid:1) = ( ϕ ∗ a | ϕ ∗ b ) ϕ ∗ g . We denote by ∇ = ∇ g the (complexified, if K = C ) Levi-Civita connection on T M . It has a unique extensionover T στ satisfying, for X ∈ T M , (i) ∇ X f = h df, X i , f ∈ C ∞ ( M );(ii) ∇ X ( a ⊗ b ) = ∇ X a ⊗ b + a ⊗ ∇ X b, a ∈ T σ τ M, b ∈ T σ τ M ;(iii) ∇ X h a, b i = h∇ X a, b i + h a, ∇ X b i , a ∈ T στ M, b ∈ T τσ M. (3 . Then the covariant (Levi-Civita) derivative is the linear map ∇ = ∇ g : T στ M → T στ +1 M, a
7→ ∇ a, defined by h∇ a, b ⊗ X i := h∇ X a, b i , b ∈ T τσ M, X ∈ T M. Since it satisfies ∇ g = 0 , it commutes with g ♭ and g ♯ . From this we infer ∇ X ( a | b ) g = ( ∇ X a | b ) g + ( a |∇ X b ) g , a, b ∈ T στ M, X ∈ T M. (3 . Thus ∇ is a metric connection on T στ M = (cid:0) T στ M, ( ·|· ) g (cid:1) .Let ϕ : M → N be a diffeomorphism. The uniqueness of the Levi-Civita connection implies ϕ ∗ ( ∇ g a ) = ∇ ϕ ∗ g ( ϕ ∗ a ) , a ∈ T στ M. For k ∈ N we define ∇ k : T στ M → T στ + k M, a
7→ ∇ k a by ∇ a := a and ∇ k +1 := ∇ ◦ ∇ k .Now we are ready for the proof of the needed estimates. In the following, dV g denotes the Lebesgue volumemeasure for M . Furthermore, given a ∈ T στ M and a local chart κ , we write [ κ ∗ a ] for the ( m σ × m τ ) -matrixwhose general entry equals ( κ ∗ a ) ( i )( j ) = ( a ◦ κ − ) ( i )( j ) , with ( i ) ∈ J σ and ( j ) ∈ J τ . Lemma 3.1
Let ( ρ, K ) be a singularity datum for ( M, g ) . Then the following estimates hold uniformly withrespect to κ ∈ K : (i) κ ∗ g ∼ ρ κ g m , κ ∗ g ∗ ∼ ρ − κ g m . (ii) ρ − κ k κ ∗ g k k, ∞ + ρ κ k κ ∗ g ∗ k k, ∞ ≤ c ( k ) , k ∈ N . (iii) κ ∗ ( dV g ) ∼ ρ mκ dV g m . (iv) If r, σ, τ ∈ N , then P ri =0 |∇ iκ ∗ g ( κ ∗ a ) | g m ∼ P | α |≤ r | ∂ α [ κ ∗ a ] | g m for a ∈ T στ M . (v) Given σ, τ ∈ N , κ ∗ ( | a | g ) ∼ ρ σ − τκ | κ ∗ a | g m , a ∈ T στ M, and | κ ∗ b | g ∼ ρ σ − τκ κ ∗ ( | b | g m ) , b ∈ T στ Q mκ . P r o o f. (1) The first part of claim (i) is immediate from (2.1)(iii) and (vi).(2) By (i) and the symmetry of g the spectrum of the matrix [ κ ∗ g ] is contained in an interval of the form ρ κ [1 /c, c ] for κ ∈ K . Hence [ κ ∗ g ] − has its spectrum in ρ − κ [1 /c, c ] for κ ∈ K . This implies the second part ofstatement (i) and k κ ∗ g ∗ k ∞ ≤ cρ − κ , κ ∈ K . (3 . Furthermore, ρ − κ κ ∗ g = (cid:16) κ ∗ ρρ κ (cid:17) κ ∗ ( ρ − g ) . (3 . Thus assertion (ii) follows from (2.1)(iv)–(vi), (3.13), (3.14), Leibniz’ rule, and the formulas for derivatives ofinverses (cf. Lemma 1.4.2 in H. Amann [3]).(3) Writing, as usual, √ g := p det[ g ] , statement (iii) follows from (i) and κ ∗ ( dV g ) = √ κ ∗ g dV g m .(4) Recall that, setting ∇ i := ∇ ∂ i with ∂ i = ∂/∂x i , ∇ i X = ( ∂ i X k + Γ kij X j ) ∂∂x k , X = X k ∂∂x k , (3 . where the Christoffel symbols Γ kij are given by kij = g kℓ ( ∂ i g ℓj + ∂ j g ℓi − ∂ ℓ g ij ) . (3 . Suppose a ∈ T στ M has the local representation (3.1). Correspondingly, ∇ a = ∇ k a ( i )( j ) ∂∂x ( i ) ⊗ dx ( j ) ⊗ dx k . Then it follows from (3.11) and (3.15) that ∇ k a ( i )( j ) = ∂ k a ( i )( j ) + σ X s =1 Γ i s kℓ a ( i ,...,ℓ,...,i σ )( j ) − τ X t =1 Γ ℓkj t a ( i )( j ,...,ℓ,...,i τ ) , (3 . where ℓ is at position s in the first sum and at position t in the second sum (and the terms are added up from ℓ = 1 to ℓ = m ). We set ∇ ( k ) := ∇ k r ◦ · · · ◦ ∇ k and ∂ ( k ) := ∂ k r ◦ · · · ◦ ∂ k for ( k ) ∈ J r and r ∈ N × . Then, writing ∇ r a = (cid:0) ∇ ( k ) a ( i )( j ) (cid:1) ∂∂x ( i ) ⊗ dx ( j ) ⊗ dx ( k ) , we obtain from (3.17) ∇ ( k ) a ( i )( j ) = ∂ ( k ) a ( i )( j ) + b ( i )( j )( k ) , (3 . where b ( i )( j )( k ) is a linear combination of the elements of (cid:8) ∂ α a ( e ı )( e ) ; | α | ≤ r − , ( e ı ) ∈ J σ , ( e ) ∈ J τ (cid:9) , the coefficients being polynomials in the derivatives of the Christoffel symbols of order at most r − − | α | .We deduce from (ii) and (3.16) k Γ kij ◦ κ − k ℓ, ∞ ≤ c ( ℓ ) , ≤ i, j, k ≤ m, κ ∈ K , ℓ ∈ N . (3 . Hence (3.18) implies r X i =0 |∇ iκ ∗ g ( κ ∗ a ) | g m ≤ c X | α |≤ r | ∂ α [ κ ∗ a ] | g m , a ∈ T στ M, κ ∈ K . By solving system (3.18) for ∂ α a ( i )( j ) we obtain an analogous expression for ∂ ( k ) a ( i )( j ) in terms of ∇ ( ℓ ) ( κ ∗ a ) , ℓ ∈ J σ , ≤ σ ≤ r − . Thus, invoking (3.19) once more, we get the second half of assertion (iv).(5) The first part of (v) follows from (3.9), (3.10), and (ii). The second part is then deduced by applying thisresult to a := κ ∗ b . From (2.1)(v) and (vi) and Lemma 3.1(ii) we find by the arguments of step (2) (cid:13)(cid:13) κ ∗ (cid:0) ( ρ − g ) ∗ (cid:1)(cid:13)(cid:13) k, ∞ ≤ c ( k ) , κ ∈ K , k ∈ N . (3 . This, in combination with (2.1)(iii) and (iv), is close to the statement that all covariant derivatives of the curvaturetensor of ( M, ρ − g ) are bounded. Note however that, taking (2.2) into consideration, (2.1)(iv) and (3.20) areonly true for atlases in S ( M ) .Let M be a manifold and K an atlas for it consisting of normalized charts. A family (cid:8) ( π κ , χ κ ) ; κ ∈ K (cid:9) is a(uniform) localization system subordinate to K if (i) π κ ∈ D (cid:0) U κ , [0 , (cid:1) and { π κ ; κ ∈ K } is a partition of unity subordinate to { U κ ; κ ∈ K } ;(ii) χ κ = κ ∗ χ with χ ∈ D (cid:0) Q m , [0 , (cid:1) and χ | supp( κ ∗ π κ ) = ;(iii) k κ ∗ π κ k k, ∞ + k κ ∗ χ κ k k, ∞ ≤ c ( k ) , κ ∈ K , k ∈ N . (3 . The crucial assumption, besides (i), is the uniform estimate (iii). Assumption (ii) will simplify some formulas.In principle, it would suffice to require that χ κ be a cut-off function for supp( π κ ) .It should also be noted that, for the purpose of this paper, we could replace π κ in (3.21)(i) by π κ . In fact, thensome of the computations below would even become simpler. However, in applications to differential equationsit will be important that we can use a partition of unity whose square root is smooth. For this reason we employcondition (3.21)(i). Lemma 3.2
Let ( ρ, K ) be a singularity datum for M . Then there exists a localization system subordinate to K . P r o o f. Fix r ∈ (0 , such that r U := (cid:8) κ − ( rQ mκ ) ; κ ∈ K (cid:9) is a cover of M . Choose e π ∈ D (cid:0) Q m , [0 , (cid:1) with e π | rQ m = . Set e π κ := κ ∗ e π . Since r U covers M and has finite multiplicity, ≤ X κ e π κ ( p ) ≤ c, p ∈ M. Put π κ := e π κ .qP e κ e π e κ . Then π κ ∈ D (cid:0) U κ , [0 , (cid:1) and P κ π κ = , where κ ∗ ( π κ ) has its support in supp( e π ) .Fix χ ∈ D (cid:0) Q m , [0 , (cid:1) with χ | supp( e π ) = . Set χ κ := κ ∗ χ . Then conditions (3.21)(i) and (ii) are satisfied.The validity of (3.21)(iii) is a consequence of (2.1)(ii). Given locally convex spaces X and Y , we denote by L ( X , Y ) the space of continuous linear maps from X into Y ,and L ( X ) := L ( X , X ) . By L is( X , Y ) we mean the set of all topological isomorphisms in L ( X , Y ) . If X and Y are Banach spaces, then L ( X , Y ) is endowed with the uniform operator norm. We write h· , ·i X for the dualitypairing between X ′ and X , that is, h x ′ , x i X is the value of x ′ ∈ X ′ at x ∈ X .Let M = ( M, g ) be a Riemannian manifold. Suppose V = ( V, π, M ) is a K -vector bundle over M . For asubset S of M we denote by V S the restriction of V to S , that is, V S = π − ( S ) . If k ∈ N ∪ {∞} and S is openin M , then C k ( S, V ) is the C k ( S ) -module of C k -sections over S .We denote by V ′ = V ∗ the dual vector bundle and by h· , ·i the fiber-wise defined duality pairing between V ′ and V . We also assume that V is equipped with an inner product and write |·| V for the corresponding vectorbundle norm.Given an open subset S of M and q ∈ [1 , ∞ ] , the Lebesgue space L q ( S, V ) = (cid:0) L q ( S, V ) , k·k q (cid:1) is the Banachspace of all (equivalence classes of measurable) sections v of V over S such that k v k q = k v k L q ( S,V ) := (cid:13)(cid:13) | v | V (cid:13)(cid:13) L q ( S ) < ∞ , where L q ( S ) = L q ( S, K ; dV g ) .In the following, we write U ⊂⊂ V to mean that U and V are open, U is relatively compact, and U ⊂ V .Since M is locally compact, separable, and metrizable it is σ -compact. Thus there exists a sequence ( M j ) such that M j ⊂⊂ M j +1 and S j M j = M . Hence L , loc ( M, V ) , the vector space of sections v of V such that v | S ∈ L ( S, V ) for every S ⊂⊂ M , is a Fr´echet space.We denote by D ( ˚ M , V ) and D ( M, V ) the spaces of smooth sections of V being compactly supported in ˚ M and M , respectively. For S ⊂⊂ ˚ M , or S ⊂⊂ M , we write D S ( ˚ M , V ) , respectively D S ( M, V ) , for the linearsubspace of all v ∈ D ( ˚ M , V ) , respectively v ∈ D ( M, V ) , with supp( v ) ⊂ S . Then D S ( ˚ M , V ) and D S ( M, V ) are Fr´echet spaces (e.g., Section VII.2 of J. Dieudonn´e [16]). If S ⊂⊂ S , then D S ( ˚ M , V ) ⊂ D S ( ˚ M , V ) and D S ( ˚ M , V ) induces on D S ( ˚ M , V ) its original topology. Hence we can endow D ( ˚ M , V ) with the LF topology(the strict inductive limit topology) with respect to all such subspaces of D ( ˚ M , V ) . Similarly, D ( M, V ) is giventhe LF topology with respect to the subspaces D S ( M, V ) . Then D ′ ( ˚ M , V ) := D ( ˚ M , V ′ ) ′ w ∗ (4 . is the space of distribution sections on ˚ M , endowed with the weak ∗ topology.Given v ∈ L , loc ( ˚ M , V ) , (cid:16) u
7→ h v, u i D := Z M h v, u i dV g (cid:17) ∈ D ′ ( ˚ M , V ) , (4 . and the map L , loc ( ˚ M , V ) → D ′ ( ˚ M , V ) , v
7→ h v, ·i D is linear, continuous, and injective. We identify v ∈ L , loc ( ˚ M , V ) with the distribution section (4.2) and consider L , loc ( ˚ M , V ) as a linear subspace of D ′ ( ˚ M , V ) . Then D ( ˚ M , V ) ֒ → D ( M, V ) d ֒ → L , loc ( M, V ) d ֒ → L , loc ( ˚ M , V ) ֒ → D ′ ( ˚ M , V ) , (4 . where ֒ → means ‘continuous’ and d ֒ → ‘continuous and dense’ embedding. Given f ∈ C ∞ ( M ) , the point-wisemultiplication u f u belongs to L (cid:0) D ( ˚ M , V ′ ) (cid:1) . Hence, setting ( f T )( u ) := T ( f u ) , T ∈ D ′ ( ˚ M , V ) , u ∈ D ( ˚ M , V ′ ) , it follows ( T f T ) ∈ L (cid:0) D ′ ( ˚ M , V ) (cid:1) . We often identify f with this ‘point-wise multiplication’ operator.Suppose k, ℓ ∈ N satisfy k + ℓ ≥ and E = (cid:0) K k × ℓ , ( · , · ) HS (cid:1) , where ( · , · ) HS : E × E → K , ( a, b ) trace( b ∗ a ) is the Hilbert-Schmidt inner product, b ∗ ∈ K ℓ × k being the conjugate matrix of b . Then E × E → K , ( a, b ) ( a | b ) HS (4 . is a separating bilinear form, the duality pairing of E , by which we identify E ′ with E .Consider the trivial bundle M × E . As usual, we write D ( M, E ) for D ( M, M × E ) etc. By juxtaposition ofthe rows of a matrix a ∈ K k × ℓ we fix an isomorphism from K k × ℓ onto K n , where n = kℓ . By means of it weidentify D ( M, E ) with D ( M ) n , etc. Then T ( u ) = n X i =1 T i ( u i ) , ( T, u ) ∈ D ′ ( ˚ M , E ) × D ( ˚ M , E ) , (4 . where u = ( u , . . . , u n ) ∈ D ( ˚ M ) n , etc.Assume X = (cid:0) X , ( ·|· ) g m (cid:1) with X ∈ { R m , H m } . Let S ( X , E ) be the Schwartz space of rapidly decreasingsmooth E -valued functions on X . Then S (˚ X , E ) is the closure of D (˚ X , E ) in S ( X , E ) , and S ′ (˚ X , E ) := S (˚ X , E ) ′ w ∗ is the space of E -valued tempered distributions on ˚ X . Since ˚ X = R m if X = R m , our notation is consistent withthe well-known fact D ( R m , E ) d ֒ → S ( R m , E ) .Set V := (cid:0) X × E, ( ·|· ) HS (cid:1) and note that h v, ·i D , defined by (4.2) and (4.4), is for each v ∈ D ( M, V ) contin-uous with respect to the topology induced by S ( X , E ) on D (˚ X , E ) . From this it follows D (˚ X , E ) d ֒ → S (˚ X , E ) ֒ → S ( X , E ) ֒ → S ′ (˚ X , E ) ֒ → D ′ (˚ X , E ) . (4 . By mollifying we further obtain D (˚ X , E ) d ֒ → D ′ (˚ X , E ) . (4 . For u ∈ S ′ ( R m , E ) we let r + be the restriction of u to ˚ H m in the sense of distributions, that is, h r + u, ϕ i S (˚ H m ,E ) = h u, ϕ i S ( R m ,E ) , ϕ ∈ S (˚ H m , E ) . Then r + ∈ L (cid:0) S ′ ( R m , E ) , S ′ (˚ H m , E ) (cid:1) .If no confusion seems likely we use the same symbol for a linear map and its restriction to a linear subspaceof its domain. Furthermore, in a diagram arrows always represent continuous linear maps.Recall that a retraction X → Y , where X and Y are locally convex spaces, is a continuous linear map pos-sessing a continuous right inverse, a coretraction. Thus the following lemma guarantees that r + is a retraction. Lemma 4.1
There exists an extension operator e + such that the diagram e + e + r + r + S ( H m , E ) S ′ (˚ H m , E ) S ( R m , E ) S ′ ( R m , E ) S ( H m , E ) S ′ (˚ H m , E ) d d d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting and r + e + = id . P r o o f. As in (4.5) we identify S ( X , E ) with S ( X ) n and S ′ (˚ X , E ) with S ′ (˚ X ) n . Then the assertion followsfrom Theorems 4.2.2 and 4.2.4 in [3] (with F := K ).It is a consequence of this lemma, (4.3), (4.6), and (4.7) that D ( X , E ) ֒ → S ( X , E ) d ֒ → S ′ (˚ X , E ) d ֒ → D ′ (˚ X , E ) and D ( X , E ) d ֒ → D ′ (˚ X , E ) , (4 . due to D (˚ X , E ) ⊂ D ( X , E ) . Let A be a countable index set. Suppose X α is for each α ∈ A a locally convex space. We endow Q α X α withthe product topology, that is, the coarsest locally convex topology for which all projections pr β : Q α X α → X β , x = ( x α ) x β are continuous. By L α X α we mean the locally convex direct sum. Thus L α X α is the vectorsubspace of Q α X α consisting of all finitely supported x ∈ Q α X α , equipped with the inductive topology, thatis, the finest locally convex topology for which all injections X β → L α X α are continuous. Let h· , ·i α be the X α -duality pairing. Then hh · , · ii : Y α X ′ α × M α X α → K , ( x ′ , x ) X α h x ′ α , x α i α is a separating bilinear form, and (cf. Corollary 1 in Section IV.4.3 of H.H. Schaefer [32]) (cid:16)M α X α (cid:17) ′ w ∗ = Y α ( X α ) ′ w ∗ (5 . with respect to hh · , · ii , (that is, hh · , · ii is the L α X α -duality pairing).Throughout the rest of this paper we assume • M = ( M, g ) is an m -dimensional singular manifold . • ρ ∈ T ( M ) . • σ, τ ∈ N and V = V στ := (cid:0) T στ M, ( ·|· ) g (cid:1) . It follows that we can choose • a singularity datum ( ρ, K ) , • a localization system (cid:8) ( π κ , χ κ ) ; κ ∈ K (cid:9) subordinate to K . (5 . For K ⊂ M we put K K := { κ ∈ K ; U κ ∩ K = ∅ } . Then, given κ ∈ K , X κ := ( R m if κ ∈ K \ K ∂M , H m otherwise , endowed with the Euclidean metric g m .We set E = E στ := (cid:0) K m σ × m τ , ( ·|· ) HS (cid:1) and consider the trivial bundles V κ := (cid:0) X κ × E, ( ·|· ) g m (cid:1) for κ ∈ K . For abbreviation, D (˚ X , E ) := M κ D (˚ X κ , E ) , D ( X , E ) := M κ D ( X κ , E ) , and D ′ (˚ X , E ) := Y κ D ′ (˚ X κ , E ) . It follows from (5.1) that D ′ (˚ X , E ) = D (˚ X , E ′ ) ′ w ∗ , where E ′ = E τσ .We introduce linear maps ϕ κ : D ( M, V ) → D ( X κ , E ) , u κ ∗ ( π κ u ) and ψ κ : D ( X κ , E ) → D ( M, V ) , v κ π κ κ ∗ v κ for κ ∈ K . Here and in similar situations it is understood that a partially defined and compactly supported sectionof a vector bundle is extended over the whole base manifold by identifying it with the zero section outside itsoriginal domain. Moreover, ϕ : D ( M, V ) → D ( X , E ) , u ( ϕ κ u ) and ψ : D ( X , E ) → D ( M, V ) , v P κ ψ κ v κ . The following retraction theorem shows, in particular, that these maps are well-defined and possess unique con-tinuous linear extensions to distribution sections.
Theorem 5.1
The diagram ϕϕ ψψ D ( M, V ) D ′ ( ˚ M, V ) D ( X , E ) D ′ (˚ X , E ) D ( M, V ) D ′ ( ˚ M, V ) d d d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting and ψ ◦ ϕ = id . P r o o f. (1) We set ˚ ϕ κ u := √ κ ∗ g κ ∗ ( π κ u ) , u ∈ D ( ˚ M , V ′ ) , κ ∈ K . (5 . Suppose K ⊂⊂ ˚ M . Then L κ := κ (cid:0) K ∩ dom( χ κ ) (cid:1) ⊂⊂ ˚ X κ . Assume u ∈ D K ( ˚ M , V ′ ) . Then κ ∗ ( π κ u ) belongsto D L κ (˚ X κ , V ′ κ ) . Since √ κ ∗ g ∈ C ∞ ( Q mκ ) , it follows ˚ ϕ κ ∈ L (cid:0) D K ( ˚ M , V ′ ) , D (˚ X κ , V ′ κ ) (cid:1) , κ ∈ K , due to D L κ (˚ X κ , V ′ κ ) ֒ → D (˚ X κ , V ′ ) . This being true for each K ⊂⊂ ˚ M , we obtain ˚ ϕ κ ∈ L (cid:0) D ( ˚ M , V ′ ) , D (˚ X κ , V ′ κ ) (cid:1) , κ ∈ K . (2) We put ˚ ψ κ v := π κ κ ∗ (cid:16)(cid:0) √ κ ∗ g (cid:1) − χv (cid:17) , v ∈ D (˚ X κ , V ′ κ ) , κ ∈ K . (5 . Suppose L κ ⊂⊂ ˚ X κ and set K κ := κ − (cid:0) L κ ∩ dom( χ ) (cid:1) . Then K κ ⊂⊂ ˚ M . Similarly as above, we find that ˚ ψ κ maps D L κ (˚ X κ , V ′ κ ) continuously into D ( ˚ M , V ′ ) . Consequently, ˚ ψ κ ∈ L (cid:0) D (˚ X κ , V ′ κ ) , D ( ˚ M , V ′ ) (cid:1) . (3) Set ˚ ϕu := (˚ ϕ κ u ) , u ∈ D ( ˚ M , V ′ ) . Assume K ⊂⊂ ˚ M . Since K is uniformly shrinkable there exist r ∈ (0 , and a finite subset L K of K such that (cid:8) κ − ( rQ mκ ) ; κ ∈ L K (cid:9) is a cover of K . Put M K := { κ ∈ K ; there exists e κ ∈ L K with U e κ ∩ U κ = ∅ } . Then M K is a finite set, due to the finite multiplicity of K . Since ˚ ϕ κ u = 0 for u ∈ D K ( ˚ M , V ′ ) and κ ∈ K \ M K it follows from step (1) that ˚ ϕ maps D K ( ˚ M , V ′ ) continuously into the closed linear subspace (cid:8) v ∈ D (˚ X , E ′ ) ; v κ = 0 for κ ∈ K \ M K (cid:9) of D (˚ X , E ′ ) , hence into D (˚ X , E ′ ) . Since this is true for all K ⊂⊂ ˚ M , ˚ ϕ ∈ L (cid:0) D ( ˚ M , V ′ ) , D (˚ X , E ′ ) (cid:1) . (5 . (4) Put ˚ ψ v := X κ ˚ ψ κ v κ , v = ( v κ ) ∈ D (˚ X , E ′ ) . Let L be a finite subset of K and put X L := (cid:8) v ∈ D (˚ X , E ′ ) ; v κ = 0 if κ ∈ K \ L (cid:9) . Step (2) implies that ˚ ψ maps X L continuously into D ( ˚ M , V ′ ) . Thus, since this holds for all finite subset L of K , ˚ ψ ∈ L (cid:0) D (˚ X , E ′ ) , D ( ˚ M , V ′ ) (cid:1) . (5 . (5) For u ∈ D ( ˚ M , V ′ ) and κ ∈ K it follows from π κ χ κ = π κ and χ κ = κ ∗ χ that (˚ ψ κ ◦ ˚ ϕ κ ) u = π κ u . Hence P κ π κ = implies (˚ ψ ◦ ˚ ϕ ) u = X κ ψ κ ( ϕ κ u ) = X κ π κ u = u, u ∈ D ( ˚ M , V ′ ) . Thus ˚ ψ is a retraction from D (˚ X , E ′ ) onto D ( ˚ M , V ′ ) , and ˚ ϕ is a coretraction.(6) Steps (3) and (4) and relations (4.1) and (5.1) imply Ψ := (˚ ϕ ) ′ ∈ L (cid:0) D ′ (˚ X , E ) , D ′ ( ˚ M , V ) (cid:1) and Φ := (˚ ψ ) ′ ∈ L (cid:0) D ′ ( ˚ M , V ) , D ′ (˚ X , E ) (cid:1) . By step (5), Ψ ◦ Φ = (˚ ψ ◦ ˚ ϕ ) ′ = (id D ( ˚ M,V ′ ) ) ′ = id D ′ ( ˚ M,V ) . (7) Suppose v ∈ D ( M, V ) and u ∈ D (˚ X , E ′ ) . Then, see (4.2), hh Φ v, u ii = h v, ˚ ψ u i D = X κ h v, ˚ ψ κ u κ i D = X κ Z M π κ (cid:10) v, ( √ κ ∗ g ) − κ ∗ ( χu κ ) (cid:11) dV g = X κ Z U κ κ ∗ (cid:0) h κ ∗ ( π κ v ) , u κ i dV g m (cid:1) = X κ Z X κ h ϕ κ v, u κ i dV g m = hh ϕv, u ii . This proves ϕ = Φ |D ( M, V ) . By the arguments of steps (1) and (3), with ˚ M replaced by M and ˚ X κ by X κ , respectively, we find ϕ ∈ L (cid:0) D ( M, V ) , D ( X , E ) (cid:1) . (8) Let v ∈ D ( X , E ) and u ∈ D ( ˚ M , V ′ ) . Then h Ψ v , u i D = hh v , ˚ ϕu ii = X κ Z X κ (cid:10) v κ , κ ∗ ( π κ u ) (cid:11) √ κ ∗ g dV g m = X κ Z Q mκ κ ∗ (cid:0) h π κ κ ∗ v κ , u i dV g (cid:1) = X κ Z M h ψ κ v κ , u i dV g = Z M h ψ v , u i dV g = h ψ v , u i D . Consequently, ψ = Ψ | D ( X , E ) . Modifying the arguments of steps (2) and (4) in the obvious way gives Ψ ∈ L (cid:0) D ( X , E ) , D ( M, V ) (cid:1) .(9) By collecting what has been proved so far we see that the diagram ϕ Φ ψ Ψ D ( M, V ) D ′ ( ˚ M, V ) D ( X , E ) D ′ (˚ X , E ) D ( M, V ) D ′ ( ˚ M, V ) d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting, where the embeddings symbolized by the vertical arrows follow from (4.3) and (4.8). Furthermore, Ψ is a retraction and Φ is a coretraction. Thus we read off this diagram that Ψ (cid:0) D ( X , E ) (cid:1) is dense in D ′ ( ˚ M , V ) (cf. Lemma 4.1.6 in [3]).Let U be a neighborhood of in D ′ ( ˚ M , V ) . Then there exists u ∈ D ( X , E ) such that Ψ u ∈ U . Hence Ψ u = ψ u ∈ D ( M, V ) shows that U ∩ D ( M, V ) = ∅ . This implies that D ( M, V ) is dense in D ′ ( ˚ M , V ) . Since Φ and Ψ are continuous linear extensions of ϕ and ψ , respectively, they are uniquely determined by the densityof the ‘vertical’ embeddings in the above diagram. Thus we can denote Φ and Ψ also by ϕ and ψ , respectively,without fearing confusion. This establishes the theorem. Henceforth, we always assume • < p < ∞ , λ ∈ R . Suppose k ∈ N . The weighted Sobolev space W k,λp ( V ; ρ ) of ( σ, τ ) -tensor fields is the completion of D ( M, V ) in L , loc ( M, V ) with respect to the norm u (cid:16) k X i =0 (cid:13)(cid:13) ρ λ + τ − σ + i |∇ i u | g (cid:13)(cid:13) pp (cid:17) /p . (6 . If ρ ′ ∈ T ( M ) , then ρ ′ ∼ ρ and we obtain an equivalent norm by replacing ρ in (6.1) by ρ ′ . Thus the topology of W k,λp ( V ; ρ ) depends on the singularity type T ( M ) only. Henceforth, we simply write W k,λp ( V ) for W k,λp ( V ; ρ ) and denote the norm (6.1) by k·k k,p ; λ . Moreover, L λp ( V ) := W ,λp ( V ) and k·k p ; λ := k·k ,p ; λ . If T ( M ) = [[ ]] ,then all these spaces are independent of λ and we obtain the ‘standard’ Sobolev spaces W kp ( V ) . The readershould be careful not to confuse W k, p ( V ) with W kp ( V ) .We also define weighted spaces of bounded smooth ( σ, τ ) -tensor fields by BC k,λ ( V ) := (cid:0)(cid:8) u ∈ C k ( M, V ) ; k u k k, ∞ ; λ < ∞ (cid:9) , k·k k, ∞ ; λ (cid:1) , where k u k k, ∞ ; λ := max ≤ i ≤ k (cid:13)(cid:13) ρ λ + τ − σ + i |∇ i u | g (cid:13)(cid:13) ∞ . The topology of BC k,λ ( V ) is independent of the particular choice of ρ ∈ T ( M ) .The following basic retraction theorems show that these spaces can be characterized by means of local coordi-nates, similarly as in the case of function spaces on compact manifolds. Below we make free use, usually withoutfurther mention, of the theory of function spaces on R m and H m . Everything for which we do not give specificreferences can be found in H. Triebel [37], for example.Let E α be a Banach space for each α in a countable index set. Then E := Q α E α . For ≤ q ≤ ∞ we denoteby ℓ q ( E ) the linear subspace of E consisting of all x = ( x α ) such that k x k ℓ q ( E ) := ( (cid:0)P α k x α k qE α (cid:1) /q , ≤ q < ∞ , sup α k x α k E α , q = ∞ , is finite. Then ℓ q ( E ) is a Banach space with norm k·k ℓ q ( E ) , and ℓ p ( E ) ֒ → ℓ q ( E ) , ≤ p < q ≤ ∞ . (6 . We also set c c ( E ) := L α E α . Then c c ( E ) ֒ → ℓ q ( E ) , ≤ q ≤ ∞ , c c ( E ) d ֒ → ℓ q ( E ) , q < ∞ . (6 . Furthermore, c ( E ) is the closure of c c ( E ) in ℓ ∞ ( E ) . If each E α is reflexive, then ℓ p ( E ) is reflexive as well, and ℓ p ( E ) ′ = ℓ p ′ ( E ′ ) with respect to the dualitypairing hh · , · ii := P α h· , ·i α . Of course, p ′ := p/ ( p − , E ′ := Q α E ′ α , and h· , ·i α is the E α -duality pairing.Let (5.2) be chosen. For ≤ q ≤ ∞ we set ϕ λq,κ := ρ λ + m/qκ ϕ κ , ψ λq,κ := ρ − λ − m/qκ ψ κ , κ ∈ K , and ϕ λq u := ( ϕ λq,κ u ) , ψ λq v := X κ ψ λq,κ v κ for u ∈ D ′ ( ˚ M , V ) and v ∈ D ′ (˚ X , E ) . If the dependence on ( σ, τ ) is important, then we write ϕ λq, ( σ,τ ) , etc. Note ( ϕ λp,κ , ψ λp,κ ) = ( ϕ p,κ , ψ p,κ ) if ρ = .Suppose F is a symbol for one of the standard function spaces, say, Sobolev, Slobodeckii, Besov spaces, etc.,on R m . Then we put F := Q κ F κ and F κ := F ( X κ , E ) . For example, W kp = Q κ W kp,κ = Q κ W kp ( X κ , E ) . Theorem 6.1
Suppose k ∈ N . The diagram ϕ λp ϕ λp ψ λp ψ λp D ( M, V ) W k,λp ( V ) D ( X , E ) ℓ p ( W kp ) D ( M, V ) W k,λp ( V ) d d d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) ϕ λp ψ λp D ′ ( ˚ M, V ) D ′ (˚ X , E ) D ′ ( ˚ M, V ) d d d ✲ ✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting and ψ λp ◦ ϕ λp = id . P r o o f. (1) It is an obvious consequence of Theorem 5.1 that ψ λp is a retraction from D ( X , E ) onto D ( M, V ) ,and from D ′ (˚ X , E ) onto D ′ ( ˚ M , V ) , and that ϕ λp is a coretraction in each case.(2) Estimate (3.21)(iii), Leibniz’ rule, and κ ∗ ( π κ u ) = ( κ ∗ π κ ) κ ∗ u imply, due to χ κ | supp( π κ ) = , k κ ∗ ( π κ u ) k W kp,κ ≤ c k κ ∗ ( χ κ u ) k W kp ( Q mκ ,E ) , κ ∈ K . (6 . From Lemma 3.1(iv) we deduce k κ ∗ ( χ κ u ) k pW kp ( Q mκ ,E ) = Z Q mκ χ X | α |≤ k | ∂ α ( κ ∗ u ) | pg m dV g m ≤ k X i =0 Z Q mκ χ |∇ iκ ∗ g ( κ ∗ u ) | pg m dV g m . (6 . By part (v) of Lemma 3.1 we get, due to ∇ i u ∈ D ( M, T στ + i M ) for u ∈ D ( M, V ) , |∇ iκ ∗ g ( κ ∗ u ) | g m ∼ κ ∗ ( ρ τ − σ + iκ |∇ i u | g ) , κ ∈ K . Thus, observing Lemma 3.1(iii) and (2.1)(vi), Z Q mκ χ |∇ iκ ∗ g ( κ ∗ u ) | pg m dV g m ∼ Z κ ( U κ ) κ ∗ (cid:16)(cid:0) χ κ ρ τ − σ + i − m/pκ |∇ i u | g (cid:1) p dV g (cid:17) ∼ ρ − mκ Z M χ κ (cid:0) ρ τ − σ + i |∇ i u | g (cid:1) p dV g for κ ∈ K . Thus we get from (6.4) and (6.5) k ϕ λp,κ u k pW kp,κ ≤ c k X i =0 Z M χ κ (cid:0) ρ λ + τ − σ + i |∇ i u | g (cid:1) p dV g . The finite multiplicity of K implies ≤ P κ χ κ ≤ c M . Consequently, k ϕ λp u k ℓ p ( W kp ) ≤ c k u k k,p ; λ , u ∈ D ( M, V ) . (6 . Since D ( M, V ) is dense in W k,λp ( V ) it follows ϕ λp ∈ L (cid:0) W k,λp ( V ) , ℓ p ( W kp ) (cid:1) .(3) Similarly as in the preceding step we find k ψ λp,κ v κ k k,p ; λ ≤ c k v κ k W kp,κ , κ ∈ K . Since χ κ | im( ψ λp,κ ) = it follows from the finite multiplicity of K and H¨older’s inequality that |∇ i ( ψ λp v ) | pg = (cid:12)(cid:12)(cid:12) X κ χ κ ∇ i ( ψ λp,κ v κ ) (cid:12)(cid:12)(cid:12) pg ≤ c X κ |∇ i ( ψ λp,κ v κ ) | pg . Consequently, k ψ λp v k k,p ; λ ≤ c k v k ℓ p ( W kp ) , v ∈ ℓ p ( W kp ) . Since ψ λp ϕ λp u = u for u ∈ W k,λp ( V ) we have shown that ψ λp is a retraction from ℓ p ( W kp ) onto W k,λp ( V ) .(4) For each κ ∈ K it holds S ( X κ , E ) d ֒ → W kp ( X κ , E ) . This is well-known if X κ = R m (e.g., [37]) and followsfrom (4.4.3) in [3] if X κ = H m . Furthermore, D ( X κ , E ) d ֒ → S ( X κ , E ) . In fact, this is standard knowledge if X κ = R m ; otherwise it follows from Section 4.2 in [3]. Hence D ( X κ , E ) d ֒ → W kp ( X κ , E ) , κ ∈ K . (6 . Thus, since c c ( W kp ) is dense in ℓ p ( W kp ) , we obtain D ( X , E ) d ֒ → ℓ p ( W kp ) . (6 . (5) Analogously we find W kp ( X κ , E ) ֒ → D ′ (˚ X κ , E ) for κ ∈ K . From this and the definition of the producttopology it follows ℓ p ( W kp ) ֒ → Y κ W kp ( X κ , E ) ֒ → D (˚ X , E ) . Since D ( X , E ) d ֒ → D ′ (˚ X , E ) we thus obtain from (6.8) that ℓ p ( W kp ) d ֒ → D (˚ X , E ) . The theorem is proved. Corollary 6.2
Suppose M = R m or M = H m , and V = M × K . Then the above definition yields the usualSobolev spaces. P r o o f. This follows from (6.7) and Example 2.1(c).
Theorem 6.3
Suppose k ∈ N . The diagram ϕ λ ∞ ϕ λ ∞ ψ λ ∞ ψ λ ∞ D ( M, V ) BC k,λ ( V ) D ( X , E ) ℓ ∞ ( BC k ) D ( M, V ) BC k,λ ( V ) ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) ϕ λ ∞ ψ λ ∞ D ′ ( ˚ M , V ) D ′ (˚ X , E ) D ′ ( ˚ M, V ) ✲ ✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting and ψ λ ∞ ◦ ϕ λ ∞ = id . P r o o f. This is verified by modifying the preceding proof in the obvious way.
Remark 6.4
Define e ϕ λq and e ψ λq by replacing π κ in the definition of ( ϕ λq , ψ λq ) by χ κ . Then e ϕ λq and e ψ λq possessthe same mapping properties as ϕ λq and ψ λq . (Of course, e ψ λq is not a retraction.)P r o o f. This is clear from the preceding proofs. We denote by [ · , · ] θ the complex and by ( · , · ) θ,q , ≤ q ≤ ∞ , the real interpolation functor for < θ < .Definitions and proofs of the results from interpolation theory which we use below without further mention canbe found in [37]. (Also see Section I.2 of [1] for a summary.) We write X . = Y if X and Y are Banach spaceswhich are equal, except for equivalent norms.For s ≥ we define weighted Bessel potential spaces of ( σ, τ ) -tensor fields by H s,λp = H s,λp ( V ) := [ W k,λp , W k +1 ,λp ] s − k , k < s < k + 1 , k ∈ N , [ W k − ,λp , W k +1 ,λp ] / , s = k ∈ N × ,L λp , s = 0 , where W k,λp = W k,λp ( V ) . Similarly, weighted Besov spaces are defined for s > by B s,λp = B s,λp ( V ) := ( ( W k,λp , W k +1 ,λp ) s − k,p , k < s < k + 1 , k ∈ N , ( W k − ,λp , W k +1 ,λp ) / ,p , s = k ∈ N × . In the remainder of this paper • F ∈ { H, B } . This allows us to develop the theory of Bessel potential and Besov spaces to a large extent in one and the samesetting.
Theorem 7.1
Let (5.2) be chosen and s > . Then ψ λp is a retraction from ℓ p ( F sp ) onto F s,λp , and ϕ λp is acoretraction. P r o o f. Suppose k, ℓ ∈ N satisfy k < ℓ . Theorem 6.1 implies that the diagram ϕ λp ϕ λp ψ λp ψ λp W ℓ,λp W k,λp ℓ ( W ℓp ) ℓ ( W kp ) W ℓ,λp W k,λp d d d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting, and ψ λp ◦ ϕ λp = id . From this it follows that ψ λp is a retraction from (cid:2) ℓ p ( W kp ) , ℓ p ( W ℓp ) (cid:3) θ onto [ W k,λp , W ℓ,λp ] θ and from (cid:0) ℓ p ( W kp ) , ℓ p ( W ℓp ) (cid:1) θ,p onto ( W k,λp , W ℓ,λp ) θ,p for < θ < .By Theorem 1.18.1 in [37] we obtain, using obvious notation, (cid:2) ℓ p ( W kp ) , ℓ p ( W ℓp ) (cid:3) θ = ℓ p (cid:0) [ W kp , W ℓp ] θ (cid:1) , (cid:0) ℓ p ( W kp ) , ℓ p ( W ℓp ) (cid:1) θ,p . = ℓ p (cid:0) ( W kp , W ℓp ) θ,p (cid:1) . Since [ W kp,κ , W ℓp,κ ] θ . = H (1 − θ ) k + θℓp,κ , the assertion follows.For ξ , ξ ∈ R and < θ < we set ξ θ := (1 − θ ) ξ + θξ . Corollary 7.2 (i) H k,λp ( V ) . = W k,λp ( V ) , k ∈ N . (ii) Suppose ≤ s < s < ∞ and θ ∈ (0 , . Then [ H s ,λp , H s ,λp ] θ . = H s θ ,λp , ( B s ,λp , B s ,λp ) θ,p = B s θ ,λp , provided s > in the latter case. P r o o f. (i) follows from H kp,κ . = W kp,κ for k ∈ N .(ii) is a consequence of the reiteration theorems for the complex and real interpolation functors.The following theorem shows that weighted Bessel potential and Besov spaces can be characterized locally byintrinsic norms, since this is the case for the spaces F sp,κ . In particular, B sp,κ . = W sp,κ for s / ∈ N . For this reasonwe call W s,λp = W s,λp ( V ) := B s,λp , s ∈ R + \ N , weighted Slobodeckii space . Theorem 7.3
Let (5.2) be selected. Suppose s ≥ with s > if F = B . Then u ∈ L , loc ( M, V ) belongsto F s,λp ( V ) iff κ ∗ ( π κ u ) ∈ F sp,κ and ||| u ||| F s,λp := (cid:16)X κ (cid:0) ρ λ + m/pκ k κ ∗ ( π κ u ) k F sp,κ (cid:1) p (cid:17) /p < ∞ . Moreover, |||·||| F s,λp is a norm for F s,λp . P r o o f. Let X and Y be Banach spaces, r ∈ L ( X, Y ) a retraction, and e ∈ L ( Y, X ) a coretraction. Then k ey k ≤ k e k k y k = k e k k rey k ≤ k e k k r k k ey k , y ∈ Y, implies k·k Y ∼ k e · k X . Thus the assertion follows from Theorem 7.1, setting e := ϕ λp .Of course, |||·||| F s,λp depends on the particular singularity datum ( ρ, K ) and on the chosen localization systemsubordinate to K . Since F s,λp has been invariantly defined it follows that another choice of these data results in anequivalent norm. Theorem 7.4 F s,λp ( V ) is a reflexive Banach space. P r o o f. Since F sp,κ is reflexive (cf. Theorem 4.4.4 of [3] if X κ = H m ), ℓ p ( F sp ) is reflexive. Theorem 7.1implies that F s,λp ( V ) is isomorphic to a closed linear subspace of ℓ p ( F sp ) (e.g., Lemma I.2.3.1 in [1]). Hence F s,λp ( V ) is reflexive as well.The following theorem shows that the weighted Bessel potential and Besov spaces are natural with respectto ∇ . Theorem 7.5
Suppose s ≥ with s > if F = B , and k ∈ N × . Then ∇ k ∈ L (cid:0) F s + k,λp ( V στ ) , F s,λp ( V στ + k ) (cid:1) . P r o o f. Since ∇ k u is a ( σ, τ + k ) -tensor field if u is a ( σ, τ ) -tensor field, it is obvious that ∇ k ∈ L (cid:0) W s + k,λp ( V στ ) , W s,λp ( V στ + k ) (cid:1) for s ∈ N . Now we obtain the assertion by interpolation, due to Corollary 7.2. Remarks 7.6 (a)
We consider the simplest case: M = ( R m , g m ) and V = M × K with T ( M ) = [[ ]] . Bythe arguments of the proof of Lemma 3.2 we construct π ∈ D (cid:0) Q m , [0 , (cid:1) such that (cid:8) π ( · + z ) ; z ∈ Z m (cid:9) is a partition of unity subordinate to the open covering { z + Q m ; z ∈ Z m } of R m . Consequently, fixing χ ∈ D (cid:0) Q m , [0 , (cid:1) with χ | supp( π ) = , it follows that (cid:8) π ( · + z ) , χ ( · + z ) ; z ∈ Z (cid:9) is a localization system subordinate to the ‘translation atlas’ constructed in the proof of Example 2.1(c). Hence Theorem 7.3 guaranteesthat u (cid:16) X z ∈ Z m k πu ( · + z ) k p F sp ( R m ) (cid:17) /p = (cid:16) X z ∈ Z m k π ( · − z ) u k p F sp ( R m ) (cid:17) /p (7 . is an equivalent norm for F sp ( R m ) , where s > if F = B . This assertion is equivalent to the ‘localization princi-ple’ of Theorem 2.4.7 of [40] for the Bessel potential spaces H sp ( R m ) with s ≥ and the Besov spaces B sp ( R m ) with s > . (b) Of course, it is natural to define B s,λp,q ( V ) with ≤ q ≤ ∞ by replacing ( · , · ) θ,p in the definition of B s,λp ( V ) by ( · , · ) θ,q . However, in this case the proof of Theorem 7.1 does not apply. In fact, it follows fromTheorem 2.4.7 in [40] that there is no characterization of B sp,q ( R m ) analogous to (7.1) if p = q . For this reasonthe spaces B s,λp,q ( V ) with q = p are less useful and we refrain from considering them here. (cid:3) In the case where M = R m , a retraction-coretraction pair ( ψ p , ϕ p ) based on a localization system equivalent tothe one of Remark 7.6(a) has been introduced in H. Amann, M. Hieber, and G. Simonett [6]. In that paper, besidesestablishing the analogue of (7.1), it is shown that ( ψ p , ϕ p ) is useful to localize partial differential equations forderiving maximal regularity results. This localization technique has since been applied by several authors for thestudy of parabolic equations on R m (eg., [25] and the references therein). An abstract formulation has been givenby S. Angenent [11]. As mentioned in the introduction, the retraction-coretraction pair ( ψ λp , ϕ λp ) is part of thefundament on which we build (elsewhere) a theory of parabolic equations on singular manifolds. Let (5.2) be chosen. For k < s < k + 1 with k ∈ N we denote by BC sκ := BC s ( X κ , E ) the Banach space of all u ∈ BC k ( X κ , E ) such that ∂ α u is uniformly ( s − k ) -H¨older continuous for | α | = k , endowed with one of itsstandard norms.From BC k +1 κ ֒ → BC sκ ֒ → BC kκ and Theorem 6.3 it follows ℓ ∞ ( BC k +1 ) BC k +1 ,λ ℓ ∞ ( BC s ) ℓ ∞ ( BC k ) BC k,λ ψ λ ∞ ψ λ ∞ ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ❄ ❄ Now we define BC s,λ := BC s,λ ( V ) , the weighted space of ( s - ) H¨older continuous ( σ, τ ) -tensor fields , to be theimage space of ψ λ ∞ | ℓ ∞ ( BC s ) , so that the diagram ℓ ∞ ( BC k +1 ) BC k +1 ,λ ℓ ∞ ( BC s ) BC s,λ ℓ ∞ ( BC k ) BC k,λ ψ λ ∞ ψ λ ∞ ψ λ ∞ ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ❄ ❄ ❄ is commuting. Of course, this definition depends on the choice of the singularity datum ( ρ, K ) and the localizationsystem subordinate to K . The following theorem shows, however, that the topology of BC s,λ is determined bythe singularity type T ( M ) only. Theorem 8.1
Suppose k < s < k + 1 with k ∈ N . (i) ψ λ ∞ is a retraction onto BC s,λ and ϕ λ ∞ is a coretraction. (ii) BC s,λ is a Banach space and u
7→ ||| u ||| s, ∞ ; λ := sup κ ρ λκ k κ ∗ ( π κ u ) k BC sκ is a norm for it. Other choices of singularity data and localization systems lead to equivalent norms. P r o o f. (1) Assertion (i) and the claim that BC s,λ is a Banach space and |||·||| s, ∞ ; λ a norm are clear.(2) Let ( e ρ, e K ) be a singularity datum and (cid:8) ( e π e κ , e χ e κ ) ; e κ ∈ e K (cid:9) a localization system subordinate to e K . Suppose j ∈ N and w ∈ BC j e κ . Then κ ∗ e κ ∗ ( e χw ) = ( e χw ) ◦ ( e κ ◦ κ − ) = ( e κ ◦ κ − ) ∗ ( e χw ) . Thus it follows from Leibniz’ rule, (3.21), and (2.2)(iii) that k κ ∗ e κ ∗ ( e χw ) k BC jκ ≤ c ( j ) k w k BC j e κ , (8 . that is, (cid:0) w κ ∗ e κ ∗ ( e χw ) (cid:1) ∈ L ( BC j e κ , BC jκ ) , j ∈ N . Since BC sκ . = ( BC kκ , BC k +1 κ ) s − k, ∞ , we thus obtain (cid:0) w κ ∗ e κ ∗ ( e χw ) (cid:1) ∈ L ( BC s e κ , BC sκ ) . (8 . (3) Using P e κ e π e κ = we find κ ∗ ( ρ λκ π κ u ) = κ ∗ (cid:16) ρ λκ π κ X e κ e π e κ u (cid:17) = ( κ ∗ π κ ) X e κ ∈ N ( κ ) ( ρ κ / e ρ e κ ) λ (cid:0) κ ∗ e κ ∗ ( e κ ∗ e π e κ ) (cid:1) e ρ λ e κ (cid:0) κ ∗ e κ ∗ (cid:0)e κ ∗ ( e π e κ e χu ) (cid:1)(cid:1) . (8 . From (2.1)(vi) and (2.2)(ii) it follows ρ κ ∼ e ρ e κ for κ ∈ K and e κ ∈ N ( κ ) . Thus we infer from (3.21), (8.2), and(8.3) that k ρ λκ κ ∗ ( π κ u ) k BC sκ ≤ c X e κ ∈ N ( κ ) k e ρ λ e κ e κ ∗ ( e π e κ u ) k BC s e κ , κ ∈ K . This implies that the norm associated with ( e ρ, e K ) and the corresponding localization system is stronger than theoriginal one. Thus the last part of the assertion follows by interchanging the roles of the singularity data.We fix now any one of the equivalent norms for BC s,λ . Then (cid:2) BC s,λ ( V ) ; s ≥ (cid:3) is the weighted H¨olderscale of ( σ, τ ) -tensor fields on M . Remark 8.2
We expect BC s,λ . = ( BC k,λ , BC k +1 ,λ ) s − k, ∞ , k < s < k + 1 , k ∈ N . (8 . However, we cannot prove this relation since we do not know whether (cid:0) ℓ ∞ ( BC k ) , ℓ ∞ ( BC k +1 ) (cid:1) s − k, ∞ . = ℓ ∞ (cid:0) ( BC k , BC k +1 ) s − k, ∞ (cid:1) . Thus we leave (8.4) as an open problem. (cid:3)
We denote by C s,λ ( V ) the closure of D ( M, V ) in BC s,λ ( V ) for s ≥ . Then [ C s,λ ( V ) ; s ≥ is calledweighted small H¨older scale . The small H¨older space C s,λ should not be confused with the little H¨olderspace bc s,λ which is the closure of BC s +1 ,λ in BC s,λ . Of course, bc s,λ = C s,λ if M is compact. Theorem 8.3
Suppose s ≥ . Then ψ λ ∞ is a retraction from c (cid:0) C s ( X , E ) (cid:1) onto C s,λ ( V ) , and ϕ λ ∞ is acoretraction. P r o o f. Since D ( X κ , E ) is dense in C s ,κ := C s ( X κ , E ) , it follows that D ( X , E ) is dense in c (cid:0) C s ( X , E ) (cid:1) .By Theorems 5.1 and 8.1 the diagram D ( X , E ) D ( M, V ) c ( C s ) C s,λ ( V ) ℓ ∞ ( BC s ) BC s,λ ( V ) ψ λ ∞ ψ λ ∞ dd ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ❄ ❄ is commuting. From this we read off that we can insert the missing vertical arrow. This gives the assertion. Corollary 8.4
Suppose ≤ s < s < s < ∞ . Then C s ,λ d ֒ → C s ,λ ֒ → BC s ,λ ֒ → BC s ,λ . Remarks 8.5 (a)
Let (5.2) be chosen. For q, r ∈ [1 , ∞ ] and s ∈ R denote by F sq,r ; κ the E -valued Triebel-Lizorkin spaces on X κ . Define F s,λq,r = F s,λq,r ( V ) by requiring that the diagram D ( X , E ) D ( M, V ) ℓ q ( F sq,r ) F s,λq,r ( V ) D ′ (˚ X , E ) D ′ ( ˚ M, V ) ψ λp ψ λp ψ λp ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ✲ ✄✂ ❄ ❄ ❄ be commuting. Then F s,λq,r ( V ) is a Banach space, a weighted Triebel-Lizorkin space of ( σ, τ ) -tensor fields on M ,and u
7→ k ϕ λq u k ℓ q ( F sq,r ) is a norm for it. The topology of F s,λq,r is independent of the particular choice of the singularity datum and thelocalization system. If M = ( R m , g m ) and T ( M ) = [[ ]] , then we recover F sq,r ( R m ) .P r o o f . The first part of the assertion follows by obvious modifications of the proof of Theorem 8.1 using thefact that BC k ( R m ) is a point-wise multiplier space for F sq,r ( R m ) , provided k = k ( s, q, r ) is sufficiently large(cf. Theorem 6.1 in W. Yuan, W. Sickel, and D. Yang [41] or, if q < ∞ , Theorem 4.2.2 in [40]). The last part isa consequence of the invariance of F sq,r ( R m ) under diffeomorphisms (see Theorem 6.7 in [41]). (b) It is clear that we can replace in the above construction the Triebel-Lizorkin spaces F sq,r ( R m ) by anyscale of spaces for which a BC k -point-wise multiplier and the diffeomorphism theorem are valid. Thus, dueto Theorems 6.1 and 6.7 in [41], we can replace F sq,r ( R m ) by the scales F s,τq,r ( R m ) and B s,τq,r ( R m ) of Triebel-Lizorkin and Besov type (see [41] for precise definitions). However, this has to be done with care. In fact, wecould take, in particular, a scale B sp,q ( R m ) with q = p . But then, due to Remark 7.6(b), the spaces B s,λp,q ( V ) constructed this way do not coincide with the Besov spaces obtained in Remark 7.6(b) by interpolation. (cid:3) Suppose σ i , τ i ∈ N for i = 0 , , . Then V σ τ × V σ τ → V σ τ , ( v , v ) v • v (9 . is called vector bundle multiplication if it is (fiber-wise) bilinear and satisfies | v • v | g ≤ c | v | g | v | g , v i ∈ V σ i τ i , i = 1 , . Examples 9.1 (a)
The duality pairing h· , ·i : V στ × V τσ → V is a multiplication. (b) The map V στ × V στ → V , ( u, v ) ( u | v ) g is a multiplication. (c) The tensor product ⊗ : V σ τ × V σ τ → V σ + σ τ + τ is a multiplication. (d) Assume ≤ i ≤ σ and ≤ j ≤ τ . We denote by C ij : V στ → V σ − τ − , a C ij a the contraction withrespect to positions i and j . Then | C ij a | g ≤ | a | g for a ∈ V στ .Suppose ≤ i ≤ σ + σ and ≤ j ≤ τ + τ . Then C ij : V σ τ × V σ τ → V σ + σ − τ + τ − , ( a, b ) C ij ( a ⊗ b ) is a multiplication, a contraction . (cid:3) In the following, we call the point-wise extension of (9.1) point-wise multiplication induced by (9.1) anddenote it again by • . Theorem 9.2
Let (9.1) be one of the multiplications of Examples 9.1. Suppose ≤ s ≤ t , λ , λ ∈ R , and λ = λ + λ . Then point-wise multiplication induced by (9.1) is a continuous bilinear map from BC t,λ ( V σ τ ) × H s,λ p ( V σ τ ) into H s,λ p ( V σ τ ) if either s = t ∈ N or t > s , from BC t,λ ( V σ τ ) × B s,λ p ( V σ τ ) into B s,λ p ( V σ τ ) if < s < t , and from BC s,λ ( V σ τ ) × BC s,λ ( V σ τ ) into BC s,λ ( V σ τ ) . P r o o f. Suppose s > if F = B . Let assumption (5.2) be satisfied. Then, given u ∈ BC t,λ ( V σ τ ) and v ∈ D ( M, V σ τ ) , it follows from P e κ π e κ = and the definition of N ( κ ) that κ ∗ (cid:0) π κ ( u • v ) (cid:1) = X e κ ∈ N ( κ ) κ ∗ ( π κ u ) • κ ∗ ( π e κ v ) , κ ∈ K . (9 . Hence the point-wise multiplier properties of the H¨older spaces BC tκ = BC t ( X κ , E ) (see, for example, Theo-rem 4.7.1 in Th. Runst and W. Sickel [31] for the case t > s ; the case s = t ∈ N follows easily from Leibniz’rule) imply (cid:13)(cid:13) κ ∗ (cid:0) π κ ( u • v ) (cid:1)(cid:13)(cid:13) F sp,κ ≤ c k κ ∗ ( π κ u ) k BC tκ X e κ ∈ N ( κ ) k κ ∗ ( π e κ v ) k F sp, e κ (9 . for κ ∈ K . Note that card (cid:0) N ( κ ) (cid:1) ≤ c, κ ∈ K , (9 . by the finite multiplicity of K .It is a consequence of (2.1)(ii) and κ ∗ ( π e κ v ) = κ ∗ e κ ∗ e κ ∗ ( π e κ v ) = (cid:0) ( e κ ∗ π e κ )( e κ ∗ v ) (cid:1) ◦ ( e κ ◦ κ − ) that (cf. (8.2) and (8.3)) k κ ∗ ( π e κ v ) k F sp,κ ≤ c k e κ ∗ ( π e κ v ) k F sp, e κ , e κ ∈ N ( κ ) , κ ∈ K . (9 . Indeed, this follows from Leibniz’ rule if s ∈ N , and then, by interpolation if s / ∈ N (also see Theorem 4.3.2in [40]). Thus we obtain from (9.3)–(9.5) and the density of D ( M, V ) in F s,λ p ||| u • v ||| F s,λp ≤ c ||| u ||| BC t,λ ||| v ||| F s,λ p for u ∈ BC t,λ ( V σ τ ) and v ∈ F s,λ p ( V σ τ ) . Now the first two assertions are implied by Theorems 7.3 and 8.1.The last one is a consequence of the fact that BC s ( X κ ) is a point-wise multiplication algebra.In applications this theorem is perhaps the most useful multiplier theorem. The next theorem is an extensionof known multiplication algebra results to the present setting. Theorem 9.3
Suppose λ , λ ∈ R , λ = λ + λ , and s > m/p . Then point-wise multiplication induced by (9.1) is a continuous bilinear map from F s,λ p ( V σ τ ) × F s,λ p ( V σ τ ) into F s,λ + m/p ( V σ τ ) . P r o o f. Theorem 4.6.4 of [31] and standard extensions to the half-space case guarantee that F sp,κ is a multi-plication algebra. Hence we infer from (9.2) and(9.4) (cid:13)(cid:13) κ ∗ (cid:0) π κ ( u • v ) (cid:1)(cid:13)(cid:13) p F sp,κ ≤ c k κ ∗ ( π κ u ) k p F sp,κ X e κ ∈ N ( κ ) k e κ ( π e κ v ) k p F sp, e κ for κ ∈ K . This implies, due to (9.5), ||| u • v ||| F s,λ m/pp ≤ c ||| u ||| F s,λ p ||| v ||| F s,λ p , hence the assertion.
10 Traces
Throughout this section ∂M = ∅ . We write • V for the restriction V ∂M of V to ∂M .Since T ( ∂M ) is a subbundle of codimension of the vector bundle ( T M ) ∂M over ∂M , there exists a uniquevector field n in ( T M ) ∂M of length , orthogonal to T ( ∂M ) , and inward pointing (in any local chart meet-ing ∂M ), the inward pointing unit normal vector field on ∂M . In local coordinates, κ = ( x , . . . , x m ) , n = 1 p g | ∂U κ ∂∂x . Suppose u ∈ D ( M, V ) and k ∈ N . The trace of order k of u on ∂M , γ k u ∈ D ( ∂M, • V ) , is defined by h γ k u, a i := (cid:10) ∇ k u | ∂M, a ⊗ n ⊗ k (cid:11) , a ∈ D ( ∂M, V ′ ∂M ) . In local coordinates, where u = u ( i )( j ) ∂∂x ( i ) ⊗ dx ( j ) , we infer from (3.18), writing γ k u = ( γ k u ) ( i )( j ) ∂∂x ( i ) ⊗ dx ( j ) , that (cid:16)p g | ∂U κ (cid:17) k ( γ k u ) ( i )( j ) = (cid:18) ∂ k u ( i )( j ) ( ∂x ) k + k − X ℓ =0 b ( i )( e )( j )( e ı ) ,ℓ ∂ ℓ u ( e ı )( e ) ( ∂x ) ℓ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∂U κ , (10 . where b ( i )( e )( j )( e ı ) ,ℓ is a polynomial in the partial derivatives of the Christoffel symbols of order at most k − ℓ − . Wewrite γ = γ for the trace operator on ∂M .In the next theorem, by a universal coretraction we mean a continuous linear map which is the unique contin-uous extension of its restriction to D ( ∂M, • V ) . In this sense it is independent of s and p . Theorem 10.1
Suppose k ∈ N and s > k + 1 /p . Then γ k extends to a retraction from F s,λp ( V ) onto B s − k − /p,λ + k +1 /pp ( • V ) . It possesses a universal coretraction γ ck satisfying γ i ◦ γ ck = 0 for ≤ i ≤ k − . P r o o f. (1) Let (5.2) be chosen. It follows from Lemma 3.1(i) and (ii) and Lemma 1.4.2 in [3] that k ρ − κ √ κ ∗ g k k, ∞ + (cid:13)(cid:13) ρ κ (cid:0) √ κ ∗ g (cid:1) − (cid:13)(cid:13) k, ∞ ≤ c, κ ∈ K . (2) For t > /p we set • B t − /pp,κ := ( B t − /pp ( R m − , E ) , κ ∈ K ∂M , { } , κ ∈ K \ K ∂M , with the convention B t − /pp ( R , E ) = E . We denote by γ κ := γ ∂ H m the usual trace operator on ∂ H m if κ be-longs to K ∂M , and set γ κ := 0 if κ ∈ K \ K ∂M , where ∂ H m = { } × R m − is identified with R m − . Then weput γ k,κ := ρ kκ (cid:0)p γ κ ( κ ∗ g ) (cid:1) − k γ κ ◦ ∂ k , κ ∈ K . Note ρ κ = • ρ κ for κ ∈ K \ K ∂M .Theorems 4.6.2 and 4.6.3 of [3] imply that γ κ ◦ ∂ k is a retraction from F sp,κ onto • B s − k − /pp,κ and that thereexists a universal coretraction e γ ck,κ for it satisfying ( γ κ ◦ ∂ i ) ◦ e γ ck,κ = 0 , ≤ i ≤ k − , (10 . (setting e γ ck,κ := 0 if κ ∈ K \ K ∂M ). We put γ ck,κ := ρ − kκ (cid:0)p γ κ ( κ ∗ g ) (cid:1) k e γ ck,κ , κ ∈ K . It follows from step (1) that γ k,κ ∈ L ( F sp,κ , • B s − k − /pp,κ ) , γ ck,κ ∈ L ( • B s − k − /pp,κ , F sp,κ ) (10 . and k γ k,κ k + k γ ck,κ k ≤ c, κ ∈ K . From (10.2) and Leibniz’ rule we infer γ i,κ ◦ γ ck,κ = δ ik id , ≤ i ≤ k. (10 . (3) We use the notation of Example 2.1(e) and set ( • π • κ , • χ • κ ) := ( π κ , χ κ ) | U • κ for • κ ∈ • K . Then it is verified that (cid:8) ( • π • κ , • χ • κ ) ; • κ ∈ • K (cid:9) is a localization system subordinate to • K . We denote by • ψ λp : ℓ p ( • B s − k − /pp ) → B s − k − /p,λp ( • V ) the ‘boundary retraction’ defined analogously to ψ λp . Correspondingly, • ϕ λp is the ‘boundary coretraction’.We put T k,κ := • ρ kκ • κ ∗ ◦ γ k ◦ κ ∗ , κ ∈ K . It follows from (10.1) that T k,κ u κ = γ k,κ u κ + k − X ℓ =0 b ℓ,κ γ ℓ,κ u κ , u κ ∈ D ( X κ , E ) , (10 . where, due to (3.19) and step (1), k b ℓ,κ k k − , ∞ ≤ c for ≤ ℓ ≤ k − and κ ∈ K . Hence, using F sp,κ ֒ → F s − k + ℓp,κ ,we obtain T k,κ ∈ L ( F sp,κ , • B s − k − /pp,κ ) , k T k,κ k ≤ c, κ ∈ K . (10 . (4) For e κ ∈ N ( κ ) with κ, e κ ∈ K ∂M we set • S κ e κ := ( • κ ∗ • e κ ∗ )( • χ κ · ) . It follows from (8.1) by interpolation that,given t > , • S κ e κ ∈ L ( • B tp, e κ , • B tp,κ ) , k • S κ e κ k ≤ c ( t ) , e κ ∈ N ( κ ) , κ, e κ ∈ K ∂M . From this, (10.6), and • B s − i − /pp,κ ֒ → • B s − k − /pp,κ we infer T i,κ e κ := • S κ e κ ◦ T i, e κ ∈ L ( F sp, e κ , • B s − k − /pp,κ ) , k T i,κ e κ k ≤ c, e κ ∈ N ( κ ) , κ, e κ ∈ K ∂M , (10 . for ≤ i ≤ k .The definition of γ k implies • π • κ γ k u = γ k ( π κ u ) − k − X j =0 (cid:16) kj (cid:17) ( γ k − j π κ ) γ j ( χ κ u ) . Since χ κ u = P e κ ∈ N ( κ ) π e κ u we thus get • ϕ λ + k +1 /pp, • κ ( γ k u ) = T k,κ ( ϕ λp,κ u ) + X e κ ∈ N ( κ ) R k − ,κ e κ ( ϕ λp u ) , (10 . where R k − ,κ e κ v := k − X i =0 a i,κ e κ T i,κ e κ ( χ e κ v e κ ) , v = ( v κ ) , with a i,κ e κ := − k − X j =1 (cid:16) kj (cid:17)(cid:16) ji (cid:17) ( ρ κ /ρ e κ ) λ + j + m/p T k − j,κ ( κ ∗ π κ ) T j − i,κ e κ ( e κ ∗ π e κ ) for e κ ∈ N ( κ ) with κ, e κ ∈ K ∂M , and a i,κ e κ := 0 otherwise. It follows from (2.1)(vi), (3.21), (10.6), (10.7), and Leibniz’ rule that k a i,κ e κ k BC ℓ ( ∂ X κ ) ≤ c ( ℓ ) , κ, e κ ∈ K , ≤ i ≤ k, ℓ ∈ N . Hence, using (10.7) once more, R k − ,κ e κ ∈ L ( F sp , • B s − k − /pp,κ ) , k R k − ,κ e κ k ≤ c, κ, e κ ∈ K . (10 . Lastly, we set T k,κ v := T k,κ v κ + X e κ ∈ N ( κ ) R k − ,κ e κ ( v ) (10 . and T k v := ( T k,κ v ) . Then we deduce from (10.6), (10.9), and the finite multiplicity of K that T k ∈ L ( F sp , • B s − k − /pp ) . (10 . Moreover, (10.8) implies • ϕ λ + k +1 /pp ◦ γ k = T k ◦ ϕ λp . Hence it follows from Theorem 7.1 and (10.11) γ k = • ψ λ + k +1 /pk ◦ T k ◦ ϕ λp ∈ L (cid:0) F s,λp , • B s − k − /p,λ + k +1 /pp ( • V ) (cid:1) . (5) We set γγ ck w := ( γ ck,κ w κ ) . Then we get from (10.3) γγ ck ∈ L ( • B s − k − /pp , • F sp , ) . Note that (10.4), (10.5), and (10.10) imply T i ◦ γγ ck = δ ik id , ≤ i ≤ k. (10 . Furthermore, given v ∈ • F sp , γ k ( ψ λp v ) = X κ ρ − ( λ + m/p ) κ γ k ( π κ κ ∗ v κ )= X κ ρ − ( λ + m/p ) κ (cid:16) • π • κ γ κ ( κ ∗ v κ ) + k − X j =0 (cid:16) kj (cid:17) ( γ k − j π κ ) γ j ( κ ∗ v κ ) (cid:17) = X κ ρ − ( λ + k + m/p ) κ (cid:16) • π • κ • κ ∗ T k,κ v κ + • κ ∗ k − X j =0 (cid:16) kj (cid:17) T k − j,κ ( κ ∗ π κ ) T j,κ v κ (cid:17) . Thus we infer from (10.12) γ k ( ψ λp γγ ck w ) = X κ ρ − ( λ + k + m/p ) κ • π • κ • κ ∗ w κ = • ψ λ + k +1 /pp w for w ∈ • B s − k − /pp . Hence, by Theorem 7.1, γ ck := ψ λp ◦ γγ ck ◦ • ϕ λ + k +1 /pp ∈ L (cid:0) B s − k − /p,λ + k +1 /pp ( • V ) , F s,λp (cid:1) and γ k ◦ γ ck = id . This proves the theorem. Corollary 10.2
Suppose ≤ j < · · · < j k and s > j k + 1 /p . Then ( γ j , . . . , γ j k ) : F s,λp ( V ) → k Y i =1 B s − j i − /p,λ + j i +1 /pp ( • V ) (10 . is a retraction possessing a universal coretraction. P r o o f. For ( v , . . . , v k ) belonging to the product space in (10.13) define u i for ≤ i ≤ k inductively by u := γ cj v and u i := u i − + γ cj i ( v i − γ j i u i − ) for ≤ i ≤ k . Then γ c , given by γ c ( v , . . . , v k ) := u k , hasthe claimed properties.
11 Spaces with Vanishing Boundary Values
Throughout this section we assume ∂M = ∅ . We denote by ˚ F s,λp = ˚ F s,λp ( V ) the closure of D ( ˚ M , V ) in F s,λp .Let (5.2) be chosen. Recalling definitions (5.3) and (5.4) we put ˚ ϕ λp,κ u := ρ λ − m/p ′ κ ˚ ϕ κ u, u ∈ D ( M, V ) , and ˚ ψ λp,κ v κ := ρ − λ + m/p ′ κ ˚ ψ κ v κ , v κ ∈ D ( X κ , E ) . Furthermore, ˚ ϕ λp u := (˚ ϕ λp,κ u ) , ˚ ψ λp v := P κ ˚ ψ λp,κ v κ for u ∈ D ( M, V ) and v ∈ D ( X , E ) . Theorem 11.1
Suppose s ∈ R + \ ( N + 1 /p ) with s > if F = B . Then the diagram ˚ ϕ λp ˚ ϕ λp ˚ ψ λp ˚ ψ λp D ( ˚ M , V )˚ F s,λp ( V ) D (˚ X , E ) ℓ p (˚ F sp ) D ( ˚ M , V )˚ F s,λp ( V ) d d d ✲✲ ✲✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) ˚ ϕ λp ˚ ψ λp F s,λp ( V ) ℓ p ( F sp ) F s,λp ( V ) ✲ ✲❄ ✄(cid:0) ❄ ✄(cid:0) ❄ ✄(cid:0) is commuting and ˚ ψ λp ◦ ˚ ϕ λp = id . P r o o f. (1) It follows from (5.5) and (5.6) that the assertions concerning the first row of this diagram are validand ˚ ψ λp ◦ ˚ ϕ λp = id D ( ˚ M,V ) .(2) From Lemma 3.1(i) and (ii) and the rules for differentiating determinants we deduce √ κ ∗ g ∼ ρ mκ , k ∂ α det( κ ∗ g ) k ∞ ≤ c ( α ) ρ mκ , α ∈ N m , κ ∈ K . For α, β ∈ N m with α = β + e i , where e i is the i -th standard basis vector of R m , we get ∂ α ( √ κ ∗ g ) = ∂ β (cid:16) √ κ ∗ g ∂ i det( κ ∗ g ) (cid:17) . From this, Leibniz’ rule, and Lemma 1.4.2 in [3] we infer k√ κ ∗ g k k, ∞ ≤ c ( k ) ρ mκ , κ ∈ K , k ∈ N . This implies k ˚ ϕ κ u k W kp,κ ≤ c ( k ) ρ mκ k κ ∗ ( χ κ u ) k W kp,κ , κ ∈ K , k ∈ N . Now we obtain ˚ ϕ λp ∈ L (cid:0) W k,λp , ℓ p ( W kp ) (cid:1) for k ∈ N from (6.5) and the arguments leading from there to (6.6).Analogously we find ˚ ψ λp ∈ L (cid:0) ℓ p ( W kp ) , W k,λp (cid:1) for k ∈ N by the arguments of step (3) of the proof of Theo-rem 6.1, as well as ˚ ψ λp ◦ ˚ ϕ λp = id .(3) Since D (˚ X κ , E ) d ֒ → ˚ W kp,κ implies D (˚ X , E ) d ֒ → c c ( ˚ W kp ) , we deduce from (6.3) that D (˚ X , E ) is densein ℓ p ( ˚ W kp ) . Clearly, ˚ ψ λp (cid:0) D (˚ X , E ) (cid:1) ⊂ D ( ˚ M , V ) . Thus we infer ˚ ψ λp ∈ L (cid:0) ℓ p ( ˚ W kp ) , ˚ W k,λp (cid:1) for k ∈ N from steps(1) and (2). Similarly, we find ˚ ϕ λp (cid:0) D ( ˚ M , V ) (cid:1) ⊂ ℓ p ( ˚ W kp ) , and thus ˚ ϕ λp ∈ L (cid:0) ˚ W k,λp , ℓ p ( ˚ W kp ) (cid:1) for k ∈ N . Thisproves the theorem if s ∈ N . (4) Suppose s ∈ R + \ ( N + 1 /p ) . For < θ < set ( · , · ) θ := [ · , · ] θ if F = H , and ( · , · ) θ := ( · , · ) θ,p other-wise. Assume < s < k with k ∈ N . Then s / ∈ N + 1 /p implies ˚ F sp,κ . = ( L p,κ , ˚ F pp,κ ) s/k . Thus, cf. the proof ofTheorem 7.1, ℓ p (˚ F sp ) = (cid:0) ℓ p ( L p ) , ℓ p (˚ F kp ) (cid:1) s/k . Now we infer from step (3) that r is a retraction from ℓ p (˚ F sp ) onto ( L λp , ˚ W k,λp ) s/k . = ˚ F s,λp , since the latter inter-polation space is the closure of D ( ˚ M , E ) in ( L λp , W k,λp ) s/k . = F s,λp by the density properties of ( · , · ) θ . Corollary 11.2
Suppose ≤ s < s < ∞ and θ ∈ (0 , . If s , s , s θ / ∈ N + 1 /p , then [ ˚ H s ,λp , ˚ H s ,λp ] θ . = ˚ H s θ ,λp , ( ˚ B s ,λp , ˚ B s ,λp ) θ,p . = ˚ B s θ ,λp , provided s > in the latter case. The next theorem characterizes the spaces ˚ F s,λp by means of trace operators. Theorem 11.3 (i)
Suppose ≤ s < /p with s > if F = B . Then ˚ F s,λp = F s,λp . (ii) Assume k ∈ N and k + 1 /p < s < k + 1 + 1 /p . Set ~γ k := ( γ , . . . γ k ) . Then ˚ F s,λp = { u ∈ F s,λp ; ~γ k u = 0 } . P r o o f. (i) follows from Theorem 11.1 and the corresponding properties of these spaces on X κ .(ii) Let the assumptions of (ii) be satisfied. If u ∈ ˚ F s,λp , then it is obvious by Corollary 10.2 that ~γ k u = 0 .Conversely, suppose u ∈ F s,λp and ~γ k u = 0 . Then we infer from (10.1) that ( γ κ ◦ ∂ i ) κ ∗ ( π κ u ) = 0 for κ ∈ K and ≤ i ≤ k . Hence κ ∗ ( π κ u ) ∈ ˚ F sp,κ for κ ∈ K ∂M (cf. Theorem 2.9.4 in [37]). Consequently, ˚ ϕ λp u ∈ ℓ p (˚ F sp ) and, by Theorem 11.1, u = ˚ ψ λp (˚ ϕ λp u ) ∈ ˚ F s,λp . This proves assertion (ii). Theorem 11.4
Suppose k ∈ N and k + 1 /p < s < k + 1 + 1 /p . Put ∂ F s,λp ( • V ) := k Y i =0 B s − i − /p,λ + i +1 /pp ( • V ) . Let ~γ ck be a coretraction for ~γ k . Then F s,λp ( V ) = ˚ F s,λp ( V ) ⊕ ~γ ck ∂ F s,λp ( • V ) . P r o o f. Let X and Y be Banach spaces, r ∈ L ( X, Y ) and r c ∈ L ( Y, X ) with r ◦ r c = id . Then r c ◦ r is aprojection in L ( X ) and X = ker( r c ◦ r ) ⊕ im( r c ◦ r ) = ker( r ) ⊕ r c Y, where r c Y is the image space of Y in X , so that r c : Y → r c Y is an isometric isomorphism (cf. Lemma 4.1.5in [3] or Lemma 2.3.1 in [1]). Hence the assertion follows from Corollary 10.2 and Theorem 11.3.
12 Spaces of Negative Order
For u ∈ D ( M, V ′ ) and v ∈ D ( M, V ) we put h u, v i M := Z M h u, v i dV g . This bilinear form extends uniquely to a separating continuous bilinear form h· , ·i M : L − λp ′ ( V ′ ) × L λp ( V ) → K by which we identify the dual Banach space of L λp ( V ) with L − λp ′ ( V ′ ) , that is, L λp ( V ) ′ = L − λp ′ ( V ′ ) by means of the duality pairing h· , ·i M . (12 . It follows from Theorem 11.3(i) that D ( ˚ M , V ) d ֒ → ˚ F s,λp ( V ) d ֒ → L λp ( V ) (12 . for s ≥ , with s > if F = B . Theorem 7.4 implies that ˚ F s,λp ( V ) is reflexive, being a closed linear subspace ofa reflexive space. Thus we put, in accordance with (12.1), F − s,λp ( V ) := (cid:0) ˚ F s, − λp ′ ( V ′ ) (cid:1) ′ , s > . (12 . It is a consequence of (12.1), (12.2), and Theorem 7.1 that F s,λp ( V ) d ֒ → L λp ( V ) d ֒ → F − s,λp ( V ) d ֒ → D ( ˚ M , V ) , s > , (12 . with respect to the duality pairing h· , ·i M , that is, h u, v i F − s,λp ( V ) = h u, v i M , s > , u ∈ ˚ F s, − λp ′ ( V ′ ) , v ∈ L p ( V ) . Finally, we define B ,λp ( V ) := (cid:0) B − ,λp ( V ) , B ,λp ( V ) (cid:1) / ,p . (12 . Theorem 12.1
Suppose s ∈ R with s / ∈ − N × + 1 /p if ∂M = ∅ . Then ψ λp is a retraction from ℓ p ( F sp ) onto F s,λp ( V ) , and ϕ λp is a coretraction. P r o o f. (1) If s ≥ with s > if F = B , then this is a restatement of Theorem 7.1.(2) Suppose s < , with s / ∈ − N + 1 /p if ∂M = ∅ . Then Theorem 11.1 guarantees that ˚ ψ − λp ′ is a retractionfrom ℓ p ′ (˚ F − sp ′ ) onto F − s, − λp ′ ( V ′ ) and ˚ ϕ − λp ′ is a coretraction. Since (˚ F − sp ′ ,κ ) ′ = F sp,κ with respect to the dualitypairing h· , ·i κ := h· , ·i X κ , it follows (cid:0) ℓ p ′ (˚ F − sp ′ ) (cid:1) ′ = ℓ p ( F sp ) with respect to hh · , · ii . Using ˚ ϕ − λp ′ ,κ = ρ − λ − m/pκ ˚ ϕ κ , the proof of Theorem 5.1, and Theorem 7.1 we thus obtain ψ λp = (˚ ϕ − λp ′ ) ′ ∈ L (cid:0) ℓ p ( F sp ) , F s,λp ( V ) (cid:1) and ϕ λp = (˚ ψ − λp ′ ) ′ ∈ L (cid:0) F s,λp ( V ) , ℓ p ( F sp ) (cid:1) with ψ λp ◦ ϕ λp = id . This proves the assertion if s < .(3) If s = 0 , then the claim for B ,λp ( V ) follows by interpolation from (12.5) and steps (1) and (2). Corollary 12.2
Suppose s ∈ R and s / ∈ − N × + 1 /p if ∂M = ∅ . Then H s,λ ( V ) . = B s,λ ( V ) . It is convenient to denote by ˚ F s,λp ( V ) for each s ∈ R the closure of D ( ˚ M , V ) in F s,λp ( V ) . Then ˚ F s,λp ( V ) = F s,λp ( V ) , s < /p. In fact, this follows from Theorem 11.3(i) and (12.4).
Theorem 12.3
The Banach spaces ˚ F s,λp ( V ) and F s,λp ( V ) are reflexive for s ∈ R . Moreover, (cid:0) ˚ F s,λp ( V ) (cid:1) ′ . = ˚ F − s, − λp ′ ( V ′ ) , s ∈ R . P r o o f. This follows from Theorem 7.4, the fact that closed linear subspaces and reflexive Banach spaces arereflexive, and the duality properties of the real interpolation functor ( · , · ) / ,p (see (12.5)). Suppose ∂M = ∅ . Since F s,λp ( V ) is reflexive and densely embedded in L p ( V ) for s > , we can define for s > F − s,λp ( V ) := (cid:0) F s, − λp ′ ( V ′ ) (cid:1) ′ with respect to the duality pairing h· , ·i M . By Theorem 11.3(i) ˇ F s,λp ( V ) := F s,λp ( V ) , − /p < s < . However, if s < − /p , then ˇ F s,λp ( V ) is no longer a space of distribution sections on ˚ M , but contains distri-bution sections supported on ∂M . This is made precise by the next theorem in which we use the notations ofTheorem 11.4. Theorem 12.4
Suppose ∂M = ∅ and − k − /p < s < − k − /p with k ∈ N . Put ∂ F s,λp ( • V ) := k Y i =0 B s + i +1 − /p,λ − i − /pp ( • V ) . Then ˇ F s,λp ( V ) = F s,λp ( V ) ⊕ ( ~γ k ) ′ ∂ F s,λp ( • V ) , where ~γ k maps F − s, − λp ′ ( V ) onto Q ki =0 B − s − i − /p ′ , − λ + i +1 /p ′ p ′ ( • V ) . P r o o f. Since ∂ ( ∂M ) = ∅ the statement follows from (12.3) and Theorem 11.4 by duality (cf. Section 2of [2]).
13 Interpolation
Now we can improve on the interpolation results already noted in Corollaries 7.2 and 11.2.
Theorem 13.1
Suppose −∞ < s < s < ∞ , < θ < , and λ , λ ∈ R . (i) The following interpolation relations, (cid:2) H s ,λ p ( V ) , H s ,λ p ( V ) (cid:3) θ . = H s θ ,λ θ p ( V ) , (cid:0) B s ,λ p ( V ) , B s ,λ p ( V ) (cid:1) θ,p . = B s θ ,λ θ p ( V ) , are valid, provided s , s , s θ / ∈ − N × + 1 /p if ∂M = ∅ . (ii) Suppose ∂M = ∅ and s , s , s θ ∈ R + \ ( N + 1 /p ) . Then (cid:2) ˚ H s ,λ p ( V ) , ˚ H s ,λ p ( V ) (cid:3) θ . = ˚ H s θ ,λ θ p ( V ) , (cid:0) ˚ B s ,λ p ( V ) , ˚ B s ,λ p ( V ) (cid:1) θ,p . = ˚ B s θ ,λ θ p ( V ) . (iii) If either ∂M = ∅ or s , s , s θ / ∈ − N × + 1 /p , then (cid:0) H s ,λ p ( V ) , H s ,λ p ( V ) (cid:1) θ,p . = B s θ ,λ θ p ( V ) . (iv) Suppose ∂M = ∅ and s , s , s θ ∈ R + \ ( N + 1 /p ) . Then (cid:0) ˚ H s ,λ p ( V ) , ˚ H s ,λ p ( V ) (cid:1) θ,p . = ˚ B s θ ,λ θ p ( V ) . P r o o f. Fix (5.2)(1) Set µ := λ − λ . Denote by ρ − µκ H s p,κ the image space of the self-map u ρ − µκ u of H s p,κ so that thismap is an isometric isomorphism from H s p,κ onto ρ − µκ H s p,κ . Then Theorem 12.1 implies that the diagram ✟✟✟✟✙❍❍❍❍❥ ✲ H s ,λ p H s p,κ ρ − µκ H s p,κ u ρ − µκ u ∼ = ψ λ p,κ ψ λ p,κ (13 . is commuting. Interpolation theory guarantees (cf. formula (7) in Section 3.4.1 of [37]) [ H s p,κ , ρ − µκ H s p,κ ] θ . = ρ − θµκ H s θ p,κ , (13 . uniformly with respect to κ ∈ K . From Theorem 12.1 we infer that ψ λ p is a retraction from ℓ p ( H s p ) onto H s ,λ p and, due to (13.1), from ℓ p ( ρ − µ H s p ) onto H s ,λ p , where ρ − µ H sp := Q κ ρ µκ H sp,κ . Thus, by (13.2) and interpo-lation, ψ λ p is a retraction from ℓ p ( ρ − θµ H s θ p ) onto [ H s ,λ p , H s ,λ p ] θ . By replacing µ in (13.1) by θµ we see that ψ λ p is a retraction from ℓ p ( ρ − θµ H s θ p ) onto H s θ ,λ θ p . This implies the claim for F = H . The proof for F = B isanalogous.(2) The assertions of (ii) follow by invoking in step (1) Theorem 11.1 instead of Theorem 12.1. The remainingstatements are obtained by similar arguments from the corresponding results on X κ .
14 Embedding Theorems
Weighted Bessel potential and Besov spaces on singular manifolds enjoy embedding properties similar to theones known for the standard non-weighted spaces on R m . Theorem 14.1
Suppose s < s < s and µ < λ . (i) If ∂M = ∅ and s , s, s ∈ R + \ ( N + 1 /p ) , then ˚ H s ,λp ( V ) d ֒ → ˚ B s,λp ( V ) d ֒ → ˚ H s ,λp ( V ) . (14 . If, moreover, ρ ≤ , then ˚ F s,µp ( V ) d ֒ → ˚ F s,λp ( V ) , whereas ˚ F s,λp ( V ) d ֒ → ˚ F s,µp ( V ) if ρ ≥ . (ii) If either ∂M = ∅ or s , s, s / ∈ − N × + 1 /p , then H s ,λp ( V ) d ֒ → B s,λp ( V ) d ֒ → H s ,λp ( V ) . (14 . Furthermore, F s,µp ( V ) d ֒ → F s,λp ( V ) if ρ ≤ , whereas ρ ≥ implies F s,λp ( V ) d ֒ → F s,µp ( V ) . P r o o f. Assertions (14.1) and (14.2) follow from Theorem 13.1(ii) and (i), respectively, and the generalinterrelations of the real and complex interpolation functors.If ρ ≤ , then it is obvious that W k,µp ( V ) d ֒ → W k,λp ( V ) , ˚ W k,µp ( V ) d ֒ → ˚ W k,λp ( V ) , k ∈ N . (14 . Thus, by duality, H k,µp ( V ) d ֒ → H k,λp ( V ) , k ∈ − N × . (14 . From these embeddings we obtain, once more by interpolation, the second part of assertion (i) and assertion (ii),respectively, provided ρ ≤ . If ρ ≥ , then the embeddings in (14.3) and (14.4) are reversed. Thus the remainingstatements are also clear.The next theorem concerns embedding theorems of Sobolev type. Theorem 14.2 (i)
Suppose s < s and p , p ∈ (1 , ∞ ) satisfy s − m/p = s − m/p . Then F s ,λp ( V ) d ֒ → F s ,λ + s − s p ( V ) . (ii) Assume s ≥ t + m/p with t ≥ and s > t + m/p if t ∈ N . Then F s,λp ( V ) d ֒ → C t,λ + m/p ( V ) . P r o o f. (1) Let the assumptions of (i) be satisfied. Since s > s implies p < p , it follows from the knownembeddings F s p ,κ ֒ → F s p ,κ and from (6.2) that ℓ p ( F s p ) ֒ → ℓ p ( F s p ) . Moreover, m/p = m/p + s − s im-plies ψ λp = ψ λ + s − s p . From this and Theorem 12.1 we infer that the diagram ϕ λp ψ λ + s − s p F s ,λp F s ,λ + s − s p ℓ p ( F s p ) ℓ p ( F s p ) ✲✛❄ ✄(cid:0) ❄ ✄(cid:0) is commuting. Thus the assertions of (i) follow.(2) Let s and t satisfy the hypotheses of (ii). Then the known embeddings F sp,κ ֒ → C t ,κ guarantee c c ( F sp ) ⊂ c c ( C t ) ֒ → c ( C t ) . Thus ℓ p ( F sp ) ֒ → c ( C t ) since c ( C t ) is closed in ℓ ∞ ( C t ) . Hence, using ψ λp = ψ λ + m/p ∞ , it follows from Theo-rem 12.1 that the diagram ϕ λp ψ λ + m/p ∞ F s,λp C t,λ + m/p ℓ p ( F sp ) c ( C t ) ✲✛❄ ✄(cid:0) ❄ ✄(cid:0) is commuting. Thus claim (ii) is implied by the density of D ( M, V ) in each of the spaces.
15 Differential Forms and Exterior Derivatives
Throughout this section • M is oriented . For ≤ k ≤ m we consider the vector subbundle V k := (cid:0)V k T ∗ M, ( ·|· ) g ∗ (cid:1) of V k = T k M , the k -fold exterior product of V = T ∗ M , where V = T M = M × K . The sections of V k are the k -forms on M , that is, the differential forms of order k . We write Ω k ( M ) for the C ∞ ( M ) -module ofsmooth k -forms, and we set Ω k ( M ) := { } for k / ∈ { , , . . . , m } .We also consider the subbundle V ′ k := (cid:0)V k T M, ( ·|· ) g (cid:1) of V k = T k M . Then V ′ k = ( V k ) ′ with respect to the duality pairing h· , ·i obtained by restriction from the V k -pairing. It follows from (3.7) and the (vector bundle) conjugate linearity of g ♯ that G k : V k → V ′ k , α G k α is a vector bundle isomorphism whose inverse is G k : V ′ k → V k , v G k v. Let ω be the Riemannian volume form of M . The definition of the Hodge adjoint ∗ β ∈ Ω m − k ( M ) implies ( α | β ) g ∗ ω = α ∧ ∗ β, α, β ∈ Ω k ( M ) , (15 . (cf. Section XX.8 of [16] or Section XI.2 in [5]). By (3.8) h v, α i = h α, v i = ( α | G k v ) g ∗ , α ∈ V k , v ∈ V ′ k . Consequently, h α, v i = Z M h α, v i dV g = Z M ( α | G k v ) g ∗ ω = Z M α ∧ ∗ G k v (15 . for α ∈ Ω k ( M ) and v ∈ D ( M, V ′ k ) . Theorem 15.1
All results obtained in the preceding sections for Bessel potential and Besov spaces of ( σ, τ ) -tensor fields remain valid for the corresponding spaces of k -forms, if ( V στ , V τσ ) is replaced by ( V k , V ′ k ) . P r o o f. Obvious.Justified by this we refer in the following simply to the theorems and formulas of the preceding sections andit is understood that we mean the corresponding results for the spaces of differential forms.The exterior derivative d : Ω k ( M ) → Ω k +1 ( M ) is characterized by dα ( X , X , . . . , X k ) = X ≤ i ≤ k ( − i ∇ X i (cid:0) α ( X , . . . , b X i , . . . , X k ) (cid:1) + X ≤ i Suppose s ∈ R . (i) Assume either ∂M = ∅ or s ≥ with s > if F = B . Then d ∈ L (cid:0) F s +1 ,λp ( V k ) , F s,λp ( V k +1 ) (cid:1) (15 . and δ ∈ L (cid:0) F s +1 ,λp ( V k ) , F s,λ +2 p ( V k − ) (cid:1) . (15 . (ii) Assume ∂M = ∅ and s > − /p with s = − /p . Then d ∈ L (cid:0) ˚ F s +1 ,λp ( V k ) , F s,λp ( V k +1 ) (cid:1) (15 . and δ ∈ L (cid:0) ˚ F s +1 ,λp ( V k ) , F s,λ +2 p ( V k − ) (cid:1) . (15 . P r o o f. (1) Suppose s ≥ with s > if F = B . Then (15.8) is a consequence of (15.3) and Theorem 7.5.(2) For α ∈ Ω k ( M ) it follows from (15.1) and (15.5) that |∗ α | g ∗ ω = ∗ α ∧ ∗∗ α = ( − k ( m − k ) ∗ α ∧ α = α ∧ ∗ α = | α | g ∗ ω. Hence ρ λ +2 k − m + m − k |∗ α | g ∗ = ρ λ + k | α | g ∗ . This implies ∗ ∈ L is (cid:0) L λp ( V k ) , L λ +2 k − mp ( V m − k ) (cid:1) . (15 . From (3.11)(ii) we infer for X ∈ T M ∇ X ( α ∧ ∗ β ) = ∇ X α ∧ ∗ β + α ∧ ∇ X ( ∗ β ) , α, β ∈ Ω k ( M ) . (15 . Since ∇ X ω = 0 we obtain from (3.12) ∇ X (cid:0) ( α | β ) g ∗ ω (cid:1) = ( ∇ X α | β ) g ∗ ω + ( α |∇ X β ) g ∗ ω. Using this, (15.13), and (15.1) we deduce α ∧ ∇ X ( ∗ β ) = α ∧ ∗∇ X β for α ∈ Ω k ( M ) . Consequently, ∇ X ( ∗ β ) = ∗ ( ∇ X β ) , β ∈ Ω k ( M ) , X ∈ T M. By this and (15.12) we get ∗ ∈ L is (cid:0) W j,λp ( V k ) , W j,λ +2 k − mp ( V m − k ) (cid:1) , j ∈ N . Hence, by interpolation, ∗ ∈ L is (cid:0) F s,λp ( V k ) , F s,λ +2 k − mp ( V m − k ) (cid:1) , s ∈ R + , provided s > if F = B . Now (15.9) follows from (15.4) and step (1), provided s ≥ with s > if F = B .(3) Definition (3.7) implies | G k α | g = h G k α, G k G k α i = h α, G k α i = | α | g ∗ , α ∈ Ω k ( M ) . Thus, since ∇ commutes with g ♯ , hence with G k , ρ λ +2 k + i − k |∇ i G k α | g = ρ λ + i + k | G k ∇ i α | g = ρ λ + i + k |∇ i α | g ∗ for i ∈ N . From this we deduce G k ∈ L is (cid:0) W j,λp ( V k ) , W j,λ +2 kp ( V ′ k ) (cid:1) , ( G k ) − = G k , for j ∈ N . Thus, by interpolation, G k ∈ L is (cid:0) F s,λp ( V k ) , F s,λ +2 kp ( V ′ k ) (cid:1) , ( G k ) − = G k , (15 . for s ≥ with s > if F = B .The part of (15.9) which has already been shown and (15.14) imply A := G k − δG k ∈ L (cid:0) F s +1 , − λp ′ ( V ′ k ) , F s, − λp ′ ( V ′ k − ) (cid:1) . (15 . (4) Suppose ∂M = ∅ . Then (15.15) and Theorem 12.3 imply A ′ ∈ L (cid:0) F − s,λp ( V k − ) , F − s − ,λp ( V k ) (cid:1) for s ∈ R + with s > if F = B . From this and (15.6) we infer, by density, that A ′ is the unique continuousextension of d . This proves (15.8) for all s ∈ R with the exception s = 0 if F = B . But now we close this gap byinterpolation.(5) Suppose ∂M = ∅ and s > . Then (15.8) and (15.14) imply C := G k dG k − ∈ L (cid:0) F s +1 , − λ − p ′ ( V ′ k − ) , F s, − λp ′ ( V ′ k ) (cid:1) . Hence C ′ ∈ L (cid:0) F − s,λp ( V k ) , F − s − ,λ +2 p ( V k − ) (cid:1) . Since (15.7) shows that C ′ is the unique continuous extension of δ over F − s,λp ( V k ) we get assertion (15.9) for s < . The case F = B and s = 0 is once more covered by interpolation. Assertion (i) is thus proved.(6) Suppose ∂M = ∅ . If s ≥ , then (15.10) and (15.11) are obvious by (i). Clearly, G k maps D ( ˚ M , V k ) into D ( ˚ M , V ′ k ) . Hence (15.14) implies G k ∈ L is (cid:0) ˚ F s,λp ( V k ) , ˚ F s,λ +2 kp ( V ′ k ) (cid:1) , ( G k ) − = G k , (15 . for s ≥ with s > if F = B .Suppose − /p < s < , that is, < − s < − /p = 1 /p ′ . Then, by Theorem 11.3(i), F − s, − λp ′ ( V ′ k − ) = ˚ F − s, − λp ′ ( V ′ k − ) . From this, (15.16), and the observation of the beginning of this step we infer A ∈ L (cid:0) ˚ F − s +1 , − λp ′ ( V ′ k ) , ˚ F − s, − λp ′ ( V ′ k − ) (cid:1) . Hence, by (12.3), A ′ ∈ L (cid:0) F s,λp ( V k − ) , F s − ,λp ( V k ) (cid:1) . Thus (15.6) implies d ∈ L (cid:0) F s,λp ( V k − ) , F s − ,λp ( V k ) (cid:1) . This proves claim (15.10) if − /p < s < − /p . Now we obtain assertion (15.10) for − /p < s < by interpolation, thanks to Theorem 13.1. The proof of statement (15.11) is similar.As an immediate consequence of this theorem we see that the Hodge Laplacian ∆ Hodge := dδ + δd satisfies ∆ Hodge ∈ L (cid:0) F s +2 ,λp ( V k ) , F s,λ +2 p ( V k ) (cid:1) if either s ∈ R and ∂M = ∅ , or s ≥ with s > if F = B . If ∂M = ∅ , then ∆ Hodge ∈ L (cid:0) ˚ F s +2 ,λp ( V k ) , F s,λ +2 p ( V k ) (cid:1) , provided s > − /p with s = − /p . Note that ∆ Hodge = − ∆ M if k = 0 , where ∆ M = div grad is theLaplace-Beltrami operator of M .Finally, we apply these results to derive the mapping properties of the basic differential operators of vectoranalysis. For this we recall that the gradient and the divergence operator can be represented (taking the complexcase into account) by grad = G ◦ d : D ( M ) → D ( M, T M ) (15 . and div = − δ ◦ G : D ( M, T M ) → D ( M ) , (15 . respectively. Theorem 15.3 Suppose s ∈ R . (i) Assume either ∂M = ∅ or s ≥ with s > if F = B . Then grad ∈ L (cid:0) F s +1 ,λp ( M ) , F s,λ +2 p ( T M ) (cid:1) , div ∈ L (cid:0) F s +1 ,λp ( T M ) , F s,λp ( M ) (cid:1) . (ii) If ∂M = ∅ and s > − /p with s = − /p , then grad ∈ L (cid:0) ˚ F s +1 ,λp ( M ) , F s,λ +2 p ( T M ) (cid:1) , div ∈ L (cid:0) ˚ F s +1 ,λp ( T M ) , F s,λp ( M ) (cid:1) . P r o o f. It follows from (3.4) that h α, G β i = h G α, β i , α, β ∈ D ( M, T M ) . From this and (15.14) we obtain by duality arguments similar to the ones used in the preceding proof that G ∈ L is (cid:0) F s,λp ( T M ) , F s,λ +2 p ( T M ) (cid:1) , ( G ) − = G , for all s ∈ R if ∂M = ∅ . Similarly, (15.16) implies G ∈ L is (cid:0) ˚ F s,λp ( T M ) , ˚ F s,λ +2 p ( T M ) (cid:1) , s ∈ R . Now the assertion follows from (15.17), (15.18), and Theorem 15.2. Acknowledgements The author is grateful to a referee for calling his attention to some early Russian references and toN. Nistor for pointing out papers [7]–[10]. References [1] H. Amann. 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