Sobolev spaces and Bochner Laplacian on complex projective varieties and stratified pseudomanifolds
SSobolev spaces and Bochner Laplacian on complex pro jective varieties and stratifiedpseudomanifolds
Francesco BeiInstitute f¨ur Mathematik, Humboldt Universit¨at zu Berlin,E-mail addresses: [email protected] [email protected]
Abstract
Let V ⊂ CP n be an irreducible complex projective variety of complex dimension v and let g be the K¨ahlermetric on reg( V ), the regular part of V , induced by the Fubini Study metric of CP n . In [31] Li and Tianproved that W , (reg( V ) , g ) = W , (reg( V ) , g ), that the natural inclusion W , (reg( V ) , g ) (cid:44) → L (reg( V ) , g )is a compact operator and that the heat operator associated to the Friedrich extension of the scalar Laplacian∆ : C ∞ c (reg( V )) → C ∞ c (reg( V )), that is e − t ∆ F : L (reg( V ) , g ) → L (reg( V ) , g ), is a trace class operator.The goal of this paper is to provide an extension of the above result to the case of Sobolev spaces ofsections and symmetric Schr¨odinger type operators with potential bounded from below where the underlingRiemannian manifold is the regular part of a complex projective variety endowed with the Fubini-Studymetric or the regular part of a stratified pseudomanifold endowed with an iterated edge metric. Keywords : Projective variety, Fubini Study metric, Stratified pseudomanifold, Iterated edge metric, Sobolevspace, Bochner Laplacian, Heat kernel.
Mathematics subject classification : 58J35, 58J10, 35P15, 31C12.
Contents
Introduction
Complex projective varieties endowed with the Fubini-Study metric as well as stratified pseudomanifolds withan iterated edge metric are important examples of singular spaces with a rich interplay between topological1 a r X i v : . [ m a t h . DG ] M a r nd analytic questions. An important topic in this setting is certainly provided by the heat operator andits properties. Many papers during the last thirty years have been devoted to explore this subject. With-out any goal of completeness we can mention here the seminal paper of Cheeger [20], where the study of theheat kernel on stratified pseudomanifolds has been initiated, [15], [16], [17], [33], [34] where the heat oper-ator on manifolds with conical singularities and on manifolds with edges is studied, [14], [31] [35] and [36]that deal with the heat operator on complex projective varieties and so on. In particular in [31], generaliz-ing the results established in [35] and [36], Li and Tian proved, without any assumptions on the singularitiesof V , that W , (reg( V ) , g ) = W , (reg( V ) , g ) (in other words the L -Stokes Theorem holds for functions),that the natural inclusion W , (reg( V ) , g ) (cid:44) → L (reg( V ) , g ) is a compact operator and that the heat oper-ator associated to the Friedrich extension of the scalar Laplacian ∆ : C ∞ c (reg( V )) → C ∞ c (reg( V )), that is e − t ∆ F : L (reg( V ) , g ) → L (reg( V ) , g ), is a trace class operator.In this paper we are interested to extend the result of Li and Tian to the case of Sobolev spaces of sections andto symmetric Schr¨odinger type operators with potential bounded form below where the underling Riemannianmanifold is the regular part of a complex projective variety endowed with the Fubini-Study metric or the regularpart of a stratified pseudomanifold with an iterated edge metric.Let us go more into the details explaining the structure of the paper. The first section is devoted to thebackground material. We recall briefly the definition of L p spaces, maximal and minimal extension of a dif-ferential operator and the notion of Sobolev space associated to a connection. In particular, given an openand possibly incomplete Riemannian manifold ( M, g ) with a vector bundle E endowed with a metric h , we willconsider the spaces W , ( M, E ) and W , ( M, E ). The former is the space of sections s ∈ L ( M, E ) such that ∇ s , applied in the distributional sense, lies in L ( M, T ∗ M ⊗ E ). The latter is defined as the graph closure of ∇ : L ( M, E ) → L ( M, T ∗ M ⊗ E ) with core domain C ∞ c ( M, E ), the space of smooth sections with compactsupport. In the second section we recall Kato’s inequality and then we provide some results about the domina-tion of semigroups. In particular, under some additional assumptions, we give a proof of the domination of theheat semigroups on a possibly incomplete Riemannian manifold, which is based on Kato’s inequality. The thirdsection contains some general results concerning Sobolev spaces of sections and symmetric Schr¨odinger operatorwith potential bounded from below. The fourth section concerns irreducible complex projective varieties. Themain result of its first part is the following theorem:
Theorem 0.1.
Let V ⊂ CP n be an irreducible complex projective variety of complex dimension v . Let E be avector bundle over reg( V ) and let h be a metric on E , Hermitian if E is a complex vector bundle, Riemannianif E is a real vector bundle. Let g be the K¨ahler metric on reg( V ) induced by the Fubini-Study metric of CP n .Finally let ∇ : C ∞ (reg( V ) , E ) → C ∞ (reg( V ) , T ∗ reg( V ) ⊗ E ) be a metric connection. We have the followingproperties: • W , (reg( V ) , E ) = W , (reg( V ) , E ) . • Assume that v > . Then there exists a continuous inclusion W , (reg( V ) , E ) (cid:44) → L vv − (reg( V ) , E ) . • Assume that v > . Then the inclusion W , (reg( V ) , E ) (cid:44) → L (reg( V ) , E ) is a compact operator. The proof of this theorem lies essentially on a combination of Kato’s inequality, Sobolev inequality andthe existence of a suitable sequence of cut-off functions. Moreover from Theorem 0.1 we have the followingapplication: for a large class of first order differential operators D : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , F ), seeTheorem 3.2 for the definition, which includes for instance the de Rham differential d k , the Dolbeault operator ∂ p,q and Dirac type operators, we have the following inclusion: D ( D max ) ∩ L ∞ (reg( V ) , E ) ⊂ D ( D min ) . (1)In the second part of the fourth section we consider Schr¨ondinger type operators P : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ), P := ∇ t ◦ ∇ + L , which are positive and formally self-adjoint. We study some propertiesof e − tP F , the heat operator associated to the Friedrich extension of P . Our mean result is the following: Theorem 0.2.
Let V , E , g , h , and ∇ be as in Theorem 0.1. Let P := ∇ t ◦ ∇ + L , P : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) be a Schr¨odinger type operator with L ∈ C ∞ (reg( V ) , End( E )) . Assume that: • P is symmetric and positive. There is a constant c ∈ R such that, for each s ∈ C ∞ (reg( V ) , E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . Consider P F : L (reg( V ) , E ) → L (reg( V ) , E ) and ∆ F : L (reg( V ) , g ) → L (reg( V ) , g ) respectively theFriedrich extension of P and the Friedrich extension of ∆ : C ∞ c (reg( V )) → C ∞ c (reg( V )) . Then the heatoperator associated to P F e − tP F : L (reg( V ) , E ) −→ L (reg( V ) , E ) is a trace class operator and its trace satisfies the following inequality: Tr( e − tP F ) ≤ me − tc Tr( e − t ∆ F ) (2) where m is the rank of the vector bundle E . This theorem is proved applying the results about the domination of semigroups recalled in the secondsection. In the remaining part of the forth section we discuss some corollaries of Theorem 0.2. In particular weget an asymptotic inequality for the eigenvalues of P F and an estimate for the trace Tr( e − tP F ) when t ∈ (0 , ∇ t ◦ ∇ ) F : L (reg( V ) , E ) → L (reg( V ) , E ), the Friedrichextension of the Bochner Laplacian ∇ t ◦ ∇ : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ). Finally, another applicationof Theorem 0.2, is provided by the extension of Cor. 5.5 of [31] to our setting. More precisely we prove thefollowing result: Theorem 0.3.
There exists a positive constant γ = γ ( d, n, m ) , that is γ depends only on the dimension ofthe ambient space CP n , on the degree d and on the rank m , such that for every irreducible complex projectivevariety V ⊂ CP n of degree d , for every vector bundle E on reg( V ) of rank m endowed with an arbitrary metric h and for every Schr¨odinger type operator P : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) as in Theorem 0.2 with L ≥ ,the ( md ) -th eingenvalue of P F , that is λ md , satisfies the following inequality: < γ ≤ λ md . (3)The fifth section contains applications to stratified pseudomanifolds. We start recalling the basic definitionsand properties and then we prove analogous results to those proved in the fourth section. More precisely wehave the following theorem: Theorem 0.4.
Let X be a compact, smoothly Thom-Mather stratified pseudomanifold of dimension m . Con-sider on reg( X ) an iterated edge metric g . Let E be a vector bundle over reg( X ) and let h be a metricon E , Riemannian if E is a real vector bundle, Hermitian if E is a complex vector bundle. Finally let ∇ : C ∞ (reg( X ) , E ) → C ∞ (reg( V ) , T ∗ reg( X ) ⊗ E ) be a metric connection. We have the following proper-ties: • W , (reg( X ) , E ) = W , (reg( X ) , E ) . • Assume that m > . Then there exists a continuous inclusion W , (reg( X ) , E ) (cid:44) → L mm − (reg( X ) , E ) . • Assume that m > . Then the inclusion W , (reg( X ) , E ) (cid:44) → L (reg( X ) , E ) is a compact operator. Similarly to (1), using Theorem (0.4), we derive the following conclusion: for a large class of first orderdifferential operators D : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , F ), see Theorem 3.2 for the definition, which includesfor instance the de Rham differential d k and Dirac type operators, we have the following inclusion: D ( D max ) ∩ L ∞ (reg( X ) , E ) ⊂ D ( D min ) . (4)Furthermore we prove the following theorem: Theorem 0.5.
Let X , E , g , h , and ∇ be as in Theorem 0.4. Let P := ∇ t ◦ ∇ + L, P : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ) be a Schr¨odinger type operator with L ∈ C ∞ (reg( X ) , End( E )) . Assume that: • P is symmetric and positive. • There is a constant c ∈ R such that, for each s ∈ C ∞ (reg( X ) , E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . et P F : L (reg( X ) , E ) → L (reg( X ) , E ) be the Friedrich extension of P and let ∆ F : L (reg( X ) , g ) → L (reg( X ) , g ) be the Friedrich extension of ∆ : C ∞ c (reg( X )) → C ∞ c (reg( X )) . Then the heat operator associatedto P F e − tP F : L (reg( X ) , E ) −→ L (reg( X ) , E ) is a trace class operator and its trace satisfies the following inequality: Tr( e − tP F ) ≤ re − tc Tr( e − t ∆ F ) (5) where r is the rank of the vector bundle E . Finally, in the last part of the fifth section, using Theorem 0.5, we derive some consequences for the operator P F , such as discreteness, an asymptotic inequality for its eigenvalues and an estimate for the trace Tr( e − tP F )when t ∈ (0 , ∇ t ◦ ∇ ) F : L (reg( X ) , E ) → L (reg( X ) , E ), that is the Friedrich extension of the Bochner Laplacian ∇ t ◦ ∇ : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ). Acknowledgments.
I wish to thank Pierre Albin, Jochen Br¨uning, Simone Cecchini, Eric Leichtnam, PaoloPiazza and Jean Ruppenthal for interesting comments and useful discussions. This research has been financiallysupported by the SFB 647 : Raum-Zeit-Materie.
The aim of this section is to recall briefly some basic notions about L p -spaces, Sobolev spaces and differentialoperators and then to prove some propositions that we will use often in the rest of the paper. We refer to [3],[8], [25], or the appendix in [44] for a thorough discussion about this background material. Let ( M, g ) be anopen and possibly incomplete Riemannian manifold of dimension m . Let E be a vector bundle over M of rank k and let h be a metric on E , Hermitian if E is a complex vector bundle, Riemannian if E is a real vector bundle.Let dvol g be the one-density associated to g . We consider M endowed with the Riemannian measure as in [25]pag. 59 or [8] pag. 29. A section s of E is said measurable if, for any trivialization ( U, φ ) of E , φ ( s | U ) is givenby a k -tuple of measurable functions. Given a measurable section s let | s | h be defined as | s | h := ( h ( s, s )) / .Then for every p , 1 ≤ p < ∞ we can define L p ( M, E ) as the space of measurable sections s such that (cid:107) s (cid:107) L p ( M,E ) := ( (cid:90) M | s | ph dvol g ) /p < ∞ . For each p ∈ [1 , ∞ ) we have a Banach space, for each p ∈ (1 , ∞ ) we get a reflexive Banach space and in thecase p = 2 we have a Hilbert space whose inner product is (cid:104) s, t (cid:105) L ( M,E ) := (cid:90) M h ( s, t ) dvol g . Moreover C ∞ c ( M, E ), the space of smooth sections with compact support, is dense in L p ( M, E ) for p ∈ [1 , ∞ ) . Finally L ∞ ( M, E ) is defined as the space of measurable sections whose essential supp is bounded, that is thespace of measurable sections s such that | s | h is bounded almost everywhere. Also in this case we get a Banachspace. Clearly all the spaces we defined so far depend on M , E , h and g but in order to have a lighter notationwe prefer to write L p ( M, E ) instead of L p ( M, E, h, g ). In the case E is the trivial bundle M × R we will write L p ( M, g ) while for the k-th exterior power of the cotangent bundle, that is Λ k T ∗ M , we will write as usual L p Ω k ( M, g ).Let now F be another vector bundle over M endowed with a metric ρ . Let P : C ∞ c ( M, E ) −→ C ∞ c ( M, F ) be adifferential operator of order d ∈ N . Then the formal adjoint of PP t : C ∞ c ( M, F ) −→ C ∞ c ( M, E )is the differential operator uniquely characterized by the following property: for each u ∈ C ∞ c ( M, E ) and foreach v ∈ C ∞ c ( M, F ) we have (cid:90) M h ( u, P t v ) dvol g = (cid:90) M ρ ( P u, v ) dvol g .
4e can look at P as an unbounded, densely defined and closable operator acting between L ( M, E ) and L ( M, F ). In general P admits several different closed extensions all included between the minimal and the max-imal one. We recall now their definitions. The domain of the maximal extension of P : L ( M, E ) −→ L ( M, F )is defined as D ( P max ) := { s ∈ L ( M, E ) : there is v ∈ L ( M, F ) such that (cid:90) M h ( s, P t φ ) dvol g = (6)= (cid:90) M ρ ( v, φ ) dvol g for each φ ∈ C ∞ c ( M, F ) } . In this case we put P max s = v. In other words the maximal extension of P is the one defined in the distributional sense .The domain of the minimal extension of P : L ( M, E ) −→ L ( M, F ) is defined as D ( P min ) := { s ∈ L ( M, E ) such that there is a sequence { s i } ∈ C ∞ c ( M, E ) with s i → s (7)in L ( M, E ) and
P s i → w in L ( M, F ) to some w ∈ L ( M, F ) } . We put P min s = w. Briefly the minimal extension of P is the closure of C ∞ c ( M, E ) under the graph norm (cid:107) s (cid:107) L ( M,E ) + (cid:107) P s (cid:107) L ( M,F ) .It is immediate to check that P ∗ max = P t min and that P ∗ min = P t max (8)that is P t max / min : L ( M, F ) → L ( M, E ) is the Hilbert space adjoint of P min / max respectively. Moreover wehave the following two L -orthogonal decomposition for L ( M, E ) L ( M, E ) = ker( P min / max ) ⊕ im( P t max / min ) . (9)We have the following properties that we will use often later: Proposition 1.1.
Let ( M, g ) , E and F be as above. Let P : C ∞ c ( M, E ) → C ∞ c ( M, F ) be a differential operatorsuch that P t ◦ P : C ∞ c ( M, E ) → C ∞ c ( M, E ) is elliptic. Let P : L ( M, E ) → L ( M, F ) be a closed extension of P .In particular P might be P max or P min . Let P ∗ be the Hilbert space adjoint of P . Then C ∞ ( M, E ) ∩ D ( P ∗ ◦ P ) is dense in D ( P ) with respect to the graph norm of P . In particular we have that C ∞ ( M, E ) ∩ D ( P max / min ) isdense in D ( P max / min ) with respect to the graph norm of P max / min .Proof. We start pointing out that, if D ( P ) is the domain of P and D ( P ∗ ) is the domain of P ∗ , then D ( P ∗ ◦ P )is given by { s ∈ D ( P ) : P s ∈ D ( P ∗ ) } . Consider now P ∗ ◦ P : L ( M, E ) → L ( M, E ). Then, according to [12]pag. 98, we have that (cid:84) k ∈ N D (( P ∗ ◦ P ) k ) is dense in D ( P ∗ ◦ P ) with respect to the graph norm of P ∗ ◦ P . Byelliptic regularity we have that (cid:84) k ∈ N D (( P ∗ ◦ P ) k ) ⊂ C ∞ ( M, E ) and therefore C ∞ ( M, E ) ∩ D ( P ∗ ◦ P ) is densein D ( P ∗ ◦ P ) with respect to the graph norm of P ∗ ◦ P . Thus, in order to complete the proof, we have to showthat: • The inclusion D ( P ∗ ◦ P ) (cid:44) → D ( P ) is continuous where each space is endowed with the corresponding graphnorm. • D ( P ∗ ◦ P ) is dense in D ( P ) with respect to the graph norm of P .The first point it is a consequence of the following inequality: for each s ∈ D ( P ∗ ◦ P ) (cid:107) P s (cid:107) L ( M,F ) = (cid:104) s, P ∗ ( P ( s )) (cid:105) L ( M,E ) ≤
12 ( (cid:107) s (cid:107) L ( M,E ) + (cid:107) P ∗ ( P ( s )) (cid:107) L ( M,E ) )and therefore (cid:107) s (cid:107) L ( M,E ) + (cid:107) P s (cid:107) L ( M,E ) ≤
32 ( (cid:107) s (cid:107) L ( M,E ) + (cid:107) P ∗ ( P ( s )) (cid:107) L ( M,E ) ) . For the second point we can argue in this way: let v ∈ D ( P ) and assume that for each s ∈ D ( P ∗ ◦ P ) wehave (cid:104) v, s (cid:105) L ( M,E ) + (cid:104) P v, P s (cid:105) L ( M,F ) = 0. This is equivalent to say that (cid:104) v, s + P ∗ ( P ( s )) (cid:105) L ( M,E ) = 0. ButId +( P ∗ ◦ P ), where Id is the identity operator, has dense image because it is an injective and self-adjointoperator. We can therefore conclude that v = 0 and this completes the proof.5 roposition 1.2. Let ( M, g ) , E and F be as above. Let P : C ∞ c ( M, E ) → C ∞ c ( M, F ) be a first order differentialoperator. Let s ∈ D ( P max ) . Assume that there is an open subset U ⊂ M with compact closure such that s | M \ U = 0 . Then s ∈ D ( P min ) .Proof. The statement follows by Lemma 2.1 in [23].Now, in the remaining part of this section, we recall the notion of
Sobolev space associated to a metricconnection.
Consider again the bundle E endowed with the metric h . Let ∇ : C ∞ ( M, E ) −→ C ∞ ( M, T ∗ M ⊗ E )be a metric connection, that is a connection which satisfies the following property: for each s, v ∈ C ∞ ( M, E )we have d ( h ( s, v )) = h ( ∇ s, v ) + h ( s, ∇ v ). Clearly h and g induce in a natural way a metric on T ∗ M ⊗ E thatwe label by ˜ h . Let ∇ t : C ∞ c ( M, T ∗ M ⊗ E ) −→ C ∞ c ( M, E ) be the formal adjoint of ∇ with respect to ˜ h and g .Then the Sobolev space W , ( M, E ) is defined in the following way: W , ( M, E ) := { s ∈ L ( M, E ) : there is v ∈ L ( M, T ∗ M ⊗ E ) such that (cid:90) M h ( s, ∇ t φ ) dvol g = (10)= (cid:90) M ˜ h ( v, φ ) dvol g for each φ ∈ C ∞ c ( M, T ∗ M ⊗ E ) } Using (6) we have W , ( M, E ) = D ( ∇ max ). Moreover we have also the Sobolev space W , ( M, E ) whosedefinition is the following: W , ( M, E ) := { s ∈ L ( M, E ) such that there is a sequence { s i } ∈ C ∞ c ( M, E ) with s i → s (11)in L ( M, E ) and ∇ s i → w in L ( M, T ∗ M ⊗ E ) to some w ∈ L ( M, T ∗ M ⊗ E ) } . Analogously to the previous case, using (7), we have W , ( M, E ) = D ( ∇ min ). As is this well known W , ( M, E )and W , ( M, E ) are two Hilbert spaces. We adopt the same convention for the notation we used before. Insteadof writing W , ( M, E, g, h ) or W , ( M, E, g, h ) we will simply write W , ( M, E ) and W , ( M, E ). For the trivialbundle M × R we will write W , ( M, g ) and W , ( M, g ). We recall the following result:
Proposition 1.3.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold of dimension m . Let E be a vector bundle over M endowed with a metric h . Let U ⊂ M be an open subset with compact closure.Consider the spaces L ( U, E | U ) and W , ( U, E | U ) where U is endowed with the metric g | U . Then the naturalinclusion W , ( U, E | U ) (cid:44) → L ( U, E | U ) (12) is a compact operator . Therefore the map i : W , ( U, E | U ) → L ( M, E ) given by i ( f ) = (cid:26) f on U on M \ U (13) is an injective and compact operator.Proof. See for example [32] pag. 349 or [44] pag. 179 for (12). Now (13) follows immediately by the followingdecomposition: W , ( U, E | U ) (cid:44) → L ( U, E | U ) i → L ( M, E ) . Consider now d k : Ω kc ( M ) → Ω k +1 c ( M ), the de Rham differential acting on the space of smooth k -formswith compact support. Given a Riemannian metric g on M , we will label by (cid:104) , (cid:105) g k and by | | g k respectivelythe metric and the pointwise norm induced by g on Λ k T ∗ M for each k = 0 , ..., m where m = dim( M ). In thecase k = 1, with a little abuse of notation, we will simply write (cid:104) , (cid:105) g and | | g instead of (cid:104) , (cid:105) g and | | g . Finallywe will label by ˜ g k the metric that g induces on T ∗ M ⊗ Λ k T ∗ M . Following (6) and (7) we will denote by d k, max / min : L Ω k ( M, g ) → L Ω k +1 ( M, g ) respectively the maximal and the minimal extension of d k acting onthe space of L k -forms. Proposition 1.4.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold of dimension m . Let E be a vector bundle over M endowed with a metric h . Let s ∈ W , ( M, E ) ∩ C ∞ ( M, E ) and let f ∈ D ( d , min ) with compact support. Then f s ∈ W , ( M, E ) and we have ∇ min ( f s ) = η ⊗ s + f ∇ s where η = d , min f . roof. Let U be an open subset of M such that U is compact and supp( f ) ⊂ U . Let { φ j } j ∈ N a sequences ofsmooth functions with compact support such that φ j converges to f in D ( d , min ) with respect to the graphnorm. Consider now a smooth function with compact support γ such that γ | supp( f ) = 1 and γ | M \ U = 0. Let { ψ j } j ∈ N be the sequence of smooth functions with compact support defined as ψ j = γφ j . Then it is immediateto check that also ψ j converges to f in D ( d , min ) with respect to the graph norm. Finally consider the sequence { ψ j s } j ∈ N . We first note that η ⊗ s + f ∇ s ∈ L ( M, T ∗ M ⊗ E ) because (cid:107) η ⊗ s + f ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107) η (cid:107) L Ω ( M,g ) (cid:107) s | U (cid:107) L ∞ ( U,E ) + (cid:107) f (cid:107) L ( M,g ) (cid:107)∇ s | U (cid:107) L ∞ ( U,T ∗ U ⊗ E ) . Now in order to complete the proof we have to show that { ψ j s } j ∈ N converges to f s in the graph norm of ∇ min We have (cid:107) ψ j s − f s (cid:107) L ( M,E ) ≤ (cid:107) s | U (cid:107) L ∞ ( U,E | U ) (cid:107) ψ j − f (cid:107) L ( M,g ) and therefore lim j →∞ (cid:107) ψ j s − f s (cid:107) L ( M,E ) = 0. In the same way (cid:107) ψ j ∇ s − f ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107)∇ s | U (cid:107) L ∞ ( U,T ∗ U ⊗ E | U ) (cid:107) ψ j − f (cid:107) L ( M,g ) and therefore lim j →∞ (cid:107) ψ j ∇ s − f ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) = 0. Again (cid:107) d ψ j ⊗ s − η ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107) s | U (cid:107) L ∞ ( U,E | U ) (cid:107) d ψ j − η (cid:107) L Ω ( M,g ) and therefore lim j →∞ (cid:107) d ψ j ⊗ s − η ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) = 0. So we can conclude that ψ j s converges to f s in D ( ∇ min ) with respect to the graph norm and that ∇ min ( f s ) = η ⊗ s + f ∇ s .We conclude this section with the following proposition. Proposition 1.5.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Let E be a vectorbundle over M , h a metric on E and let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E ) be a metric connection. Considerthe Sobolev space W , ( M, E ) . Then C ∞ ( M, E ) ∩ L ∞ ( M, E ) ∩ W , ( M, E ) is dense in W , ( M, E ) .Proof. We give the proof in the case E is a complex vector bundle endowed with a Hermitian metric h . When E is real the proof is the same with the obvious modifications. According to Prop. 1.1 it is enough to show that C ∞ ( M, E ) ∩ L ∞ ( M, E ) ∩ W , ( M, E ) is dense in C ∞ ( M, E ) ∩ W , ( M, E ). Let s ∈ C ∞ ( M, E ) ∩ W , ( M, E )and define s n := s ( | s | h n + 1) . (14)Clearly s n ∈ C ∞ ( M, E ). Moreover it is immediate to note that ( | s | h n + 1) − ∈ L ∞ ( M, g ). Therefore, by thefact that s ∈ L ( M, E ), we can conclude that s n ∈ L ( M, E ) ∩ C ∞ ( M, E ). Finally we have | s n | h = | s | h ( | s | h n + 1) ≤ n . In this way we get that s n ∈ C ∞ ( M, E ) ∩ L ∞ ( M, E ) ∩ L ( M, E ). Now consider ∇ s n . We have: ∇ s n = −
12 ( | s | h n + 1) − n Re( h ( ∇ s, s )) ⊗ s + ( | s | h n + 1) − ∇ s where Re( h ( ∇ s, s )) is the real part of h ( ∇ s, s ). First of all we want to show that ∇ s n ∈ L ( M, T ∗ M ⊗ E ).By the assumptions ∇ s ∈ L ( M, T ∗ M ⊗ E ). As remarked above we have ( | s | h n + 1) − ∈ L ∞ ( M, g ). Thereforewe can conclude that ( | s | h n + 1) − ∇ s ∈ L ( M, T ∗ M ⊗ E ) . For − ( | s | h n + 1) − n Re( h ( ∇ s, s )) ⊗ s we argue asfollows. First of all we note that | −
12 ( | s | h n + 1) − n Re( h ( ∇ s, s )) ⊗ s | ˜ h ≤ n ( | s | h n + 1) − |∇ s | ˜ h | s | h . (15)It is clear that ( | s | h n + 1) − | s | h ∈ L ∞ ( M, g ) and | s | h ∈ L ( M, g ). Therefore ( | s | h n + 1) − | s | h ∈ L ( M, g ).Moreover ( | s | h n + 1) − | s | h ∈ L ∞ ( M, g ). In fact we have( | s | h n + 1) − | s | h ≤ | s | h ( | s | h n ) + 1 ≤ n. (16)7his shows that ( | s | h n + 1) − | s | h ∈ L ∞ ( M, g ) ∩ L ( M, g ). By the fact that |∇ s | ˜ h ∈ L ( M, g ) we can concludethat − n ( | s | h n + 1) − |∇ s | ˜ h | s | h ∈ L ( M, g ) . According to (15) this implies that − n ( | s | h n + 1) − Re( h ( ∇ s, s )) ⊗ s ∈ L ( M, T ∗ M ⊗ E )and in conclusion we proved that ∇ s n ∈ L ( M, T ∗ M ⊗ E ). Finally the last step is to show that s n convergesto s in the graph norm of ∇ . For (cid:107) s − s n (cid:107) L ( M,E ) we have (cid:107) s − s n (cid:107) L ( M,E ) = (cid:90) M (1 − ( | s | h n + 1) − ) | s | h dvol g and using the Lebesgue dominate convergence theorem we get lim n →∞ (cid:107) s − s n (cid:107) L ( M,E ) = 0. Now we show thatlim n →∞ (cid:107)∇ s − ∇ s n (cid:107) L ( M,T ∗ M ⊗ E ) = 0. We have (cid:107)∇ s −∇ s n (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107)− n ( | s | h n +1) − Re( h ( ∇ s, s )) ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) + (cid:107)∇ s − ( | s | h n +1) − ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) . For (cid:107)∇ s − ( | s | h n + 1) − ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) we have (cid:107)∇ s − ( | s | h n + 1) − ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) = (cid:90) M (1 − ( | s | h n + 1) − ) |∇ s | h dvol g and, again by the Lebesgue dominate convergence theorem, we can conclude thatlim n →∞ (cid:107)∇ s − ( | s | h n + 1) − ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) = 0 . For the remaining term, using (15), we have (cid:107) − n ( | s | h n + 1) − Re( h ( ∇ s, s )) ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) = (cid:90) M | − n ( | s | h n + 1) − Re( h ( ∇ s, s )) ⊗ s | h dvol g ≤≤ (cid:90) M n ( | s | h n + 1) − |∇ s | h | s | h dvol g . Using (16) we have | s | h ( | s | h n + 1) − = ( | s | h ( | s | h n + 1) − ) ≤ n and this in turn implies that 1 n |∇ s | h | s | h ( | s | h n + 1) − ≤ |∇ s | h . So we are again in the position to apply the Lebesgue dominate convergence theorem and thus we can concludethat0 ≤ lim n →∞ (cid:107) − n ( | s | h n + 1) − Re( h ( ∇ s, s )) ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:90) M lim n →∞ n ( | s | h n + 1) − |∇ s | h | s | h dvol g = 0 . In conclusion we proved that s n converges to s in the graph norm of ∇ and this complete the proof. In this section we recall the Kato’s inequality and then, following the lines of [27] and [28], we discuss its relationwith the theory of domination of semigroups. Unlike [27] and [28] we are interested to apply this tool in anincomplete setting and this requires a more careful analysis because in general the Laplacian ∆ , with coredomain given by the smooth functions with compact support, is not longer an essentially self-adjoint operator.8 .1 Kato’s inequality Consider again an open and possibly incomplete Riemaniann manifold (
M, g ). Let E be a vector bundle over M and let h be a metric on E , Hermitian whether E is complex , Riemannian whether E is real. Finally let ˜ h be the natural metric that h and g induce on T ∗ M ⊗ E . Proposition 2.1.
Let M , g , E , h and ˜ h be as described above. Let ∇ : C ∞ ( M, E ) −→ C ∞ ( M, T ∗ M ⊗ E ) be a metric connection and let s ∈ C ∞ ( M, E ) . Let Z ⊂ M be the zero set of s . Then on M \ Z we have thefollowing pointwise inequality: | d ( | s | h ) | g ≤ |∇ s | ˜ h . (17) If s ∈ W , ( M, E ) ∩ C ∞ ( M, E ) then | s | h ∈ D ( d , max ) and we have (cid:107) d , max | s | h (cid:107) L Ω ( M,g ) ≤ (cid:107)∇ s (cid:107) L ( M,T ∗ M ⊗ E ) (18) In particular, if ( E, h ) is a complex vector bundle endowed with a Hermitian metric, d , max ( | s | h ) satisfies: d , max ( | s | h ) = (cid:26) Re( h ( ∇ s, s )) | s | − h on M \ Z on Z (19) while if ( E, h ) is a real vector bundle endowed with a Riemannian metric, d , max ( | s | h ) satisfies: d , max ( | s | h ) = (cid:26) h ( ∇ s, s ) | s | − h on M \ Z on Z (20) Proof.
As for the previous proof we treat only the complex case. The proof for the real case is completelyanalogous with the obvious modifications. The proof of (17) is based on the following observations: On M \ Z we have | d ( | s | h ) | g = 2 | s | h | d ( | s | h ) | g . On the other hand | d ( | s | h ) | g = 2 | Re( h ( ∇ s, s )) | g ≤ |∇ s | ˜ h | s | h . Therefore(17) holds. For (18) and (19) we argue as in [7] VI.31. Consider φ ∈ Ω c ( M ) and let (cid:15) n := n . Then (cid:90) M | s | h ( d t φ ) dvol g = lim n →∞ (cid:90) M ( | s | h + (cid:15) n ) ( d t φ ) dvol g = lim n →∞ (cid:90) M (cid:104) d (( | s | h + (cid:15) n ) ) , φ (cid:105) g dvol g = lim n →∞ (cid:90) M (cid:104) ( | s | h + (cid:15) n ) − Re( h ( ∇ s, s )) , φ (cid:105) g dvol g = (cid:90) M (cid:104) η, φ (cid:105) g dvol g where η is defined as in (19). In particular for the pointwise norms we have | η | g ≤ |∇ s | ˜ h . Therefore, if s ∈ W , ( M, E ) ∩ C ∞ ( M, E ), we can conclude that | s | h ∈ D ( d , max ) and that (18) and (19) hold. Corollary 2.1.
Under the assumptions of Prop. 2.1. • If s ∈ W , ( M, E ) then | s | h ∈ D ( d , max ) and we have (cid:107) d , max | s | h (cid:107) L Ω ( M,g ) ≤ (cid:107)∇ max s (cid:107) L ( M,T ∗ M ⊗ E ) , • If s ∈ W , ( M, E ) then | s | h ∈ D ( d , min ) and we have (cid:107) d , min | s | h (cid:107) L Ω ( M,g ) ≤ (cid:107)∇ min s (cid:107) L ( M,T ∗ M ⊗ E ) .Proof. Let s ∈ W , ( M, E ). According to Prop. 1.5 there is a sequence { φ n } n ∈ N ⊂ W , ( M, E ) ∩ C ∞ ( M, E )such that lim n →∞ φ n = s in W , ( M, E ). We have lim n →∞ | φ n | h = | s | h in L ( M, g ) and, using (18) and thefact that ∇ φ n → ∇ max s in L ( M, T ∗ M ⊗ E ), we get (cid:107) d , max | φ n | h (cid:107) L Ω ( M,g ) ≤ τ for some positive τ ∈ R . Thisimplies the existence of a subsequence { φ (cid:48) n } n ∈ N ⊂ { φ n } n ∈ N such that { d , max ( | φ (cid:48) n | h ) } n ∈ N converges weakly in L Ω ( M, g ) to some β ∈ L Ω ( M, g ), see for instance [21] pag. 132. Now let ω ∈ Ω c ( M ). We have (cid:104)| s | h , d t ω (cid:105) L ( M,g ) = lim n →∞ (cid:104)| φ (cid:48) n | h , d t ω (cid:105) L ( M,g ) = lim n →∞ (cid:104) d , max | φ (cid:48) n | h , ω (cid:105) L Ω ( M,g ) = (cid:104) β, ω (cid:105) L Ω ( M,g ) . Therefore, according to (6), we proved that | s | h ∈ D ( d , max ) and d , max | s | h = β .Now to estimate (cid:107) d , max | s | h (cid:107) L Ω ( M,g ) , using (18), we have (cid:107) d , max | s | h (cid:107) L Ω ( M,g ) = (cid:107) β (cid:107) L Ω ( M,g ) = lim n →∞ (cid:104) d , max | φ (cid:48) n | h , β (cid:105) L Ω ( M,g ) ≤ lim n →∞ (cid:107) d , max | φ (cid:48) n | h (cid:107) L Ω ( M,g ) (cid:107) β (cid:107) L Ω ( M,g ) ≤ lim n →∞ (cid:107)∇ φ (cid:48) n (cid:107) L ( M,T ∗ M ⊗ E ) (cid:107) β (cid:107) L Ω ( M,g ) = (cid:107)∇ max s (cid:107) L ( M,T ∗ M ⊗ E ) (cid:107) β (cid:107) L Ω ( M,g ) . Hence the first point is proved.For the second point we argue in a similar manner. Let s ∈ W , ( M, E ) and let { ψ n } n ∈ N be a sequence ofsmooth sections with compact support such that lim n →∞ ψ n = s in W , ( M, E ). As in the previous case wehave lim n →∞ | ψ n | h = | s | h in L ( M, g ) and, using (18) and the fact that ∇ ψ n → ∇ min s in L ( M, T ∗ M ⊗ E ),9e get (cid:107) d , max | ψ n | h (cid:107) L Ω ( M,g ) ≤ τ (cid:48) for some positive τ (cid:48) ∈ R . This implies the existence of a subsequence { ψ (cid:48) n } n ∈ N ⊂ { ψ n } n ∈ N such that { d , max ( | ψ (cid:48) n | h ) } n ∈ N converges weakly in L Ω ( M, g ) to some γ ∈ L Ω ( M, g ).Moreover we observe that | ψ n | h ∈ D ( d , min ) because | ψ n | h ∈ D ( d , max ) and it has compact support, see Prop.1.2. Now let ω ∈ D ( d t , max ). We have (cid:104)| s | h , d t , max ω (cid:105) L ( M,g ) = lim n →∞ (cid:104)| ψ (cid:48) n | h , d t , max ω (cid:105) L ( M,g ) = lim n →∞ (cid:104) d , min | ψ (cid:48) n | h , ω (cid:105) L Ω ( M,g ) = (cid:104) γ, ω (cid:105) L Ω ( M,g ) . Therefore, for each ω ∈ D ( d t , max ), we have (cid:104)| s | h , d t , max ω (cid:105) L ( M,g ) ≤ (cid:107) γ (cid:107) L Ω ( M,g ) (cid:107) ω (cid:107) L Ω ( M,g ) and this showsthat | s | h ∈ D (( d t , max ) ∗ ) that is | s | h ∈ D ( d , min ). Finally, by the previous point, we have (cid:107) d , min ( | s | h ) (cid:107) L Ω ( M,g ) = (cid:107) d , max ( | s | h ) (cid:107) L Ω ( M,g ) ≤ (cid:107)∇ max s (cid:107) L ( M,T ∗ M ⊗ E ) = (cid:107)∇ min s (cid:107) L ( M,T ∗ M ⊗ E ) and the proposition is thus established. In this subsection we give a very brief account about some results on quadratic forms and the Friedrich extensionof a positive and symmetric operator. We follow the Appendix C.1 in [32] and we refer to it for the proofs.For a thorough treatment of the subject we refer to [37] and [38]. Let H be a Hilbert space with inner product (cid:104) , (cid:105) . Let B : H → H be a linear, unbounded and densely defined operator. Assume that B is symmetricand positive, that is B is extended by its adjoint B ∗ and (cid:104) Bu, u (cid:105) ≥ u ∈ D ( B ). The quadratic formassociated to B , usually labeled by Q B , is by definition Q B ( u, v ) := (cid:104) Bu, v (cid:105) . Let (cid:104) , (cid:105) B be the inner productgiven by (cid:104) , (cid:105) + Q B and let D ( Q B ) be the completion of D ( B ) through (cid:104) , (cid:105) B . It is immediate to check thatthe identity Id : D ( B ) → D ( B ) extends as a bounded and injective map i Q B : D ( Q B ) → H . Therefore in whatfollows we will identify D ( Q B ) with its image in H through i Q B which is given by { u ∈ H : there exists { u n } n ∈ N ⊂ D ( B ) such that (cid:104) u n − u, u n − u (cid:105) → (cid:104) u n − u m , u n − u m (cid:105) B → m, n → ∞} . Now we define the
Friedrich extension of B , labeled by B F , as the operator whose domain is given by { u ∈ D ( Q B ) : there exists v ∈ H with Q B ( u, w ) = (cid:104) v, w (cid:105) for any w ∈ D ( Q B ) } and we put B F u := v . Defined in this way B F is a positive and self-adjoint operator. Moreover the aboveconstruction is equivalent to require that D ( B F ) = { u ∈ D ( B ∗ ) : there exists { u n } ⊂ D ( B ) such that (cid:104) u − u n , u − u n (cid:105) → (cid:104) B ( u n − u m ) , u n − u m (cid:105) → n, m → ∞} and B F ( u ) = B ∗ ( u ), that is in a concise way D ( B F ) := D ( Q B ) ∩ D ( B ∗ ) and B F u := B ∗ u for u ∈ D ( B F ). We conclude this reminder with the following result: Proposition 2.2.
Let
E, F be two vector bundles over an open and possibly incomplete Riemannian manifold ( M, g ) . Let h E and h F be two metrics on E and F respectively. Let D : C ∞ c ( M, E ) → C ∞ c ( M, F ) be anunbounded and densely defined differential operator. Let D t : C ∞ c ( M, F ) → C ∞ c ( M, E ) be its formal adjoint.Then for D t ◦ D : L ( M, E ) → L ( M, E ) we have:1. ( D t ◦ D ) F = ( D t ) max ◦ D min . D ( Q D t ◦ D ) = D ( D min ) and Q D t ◦ D ( u, v ) = (cid:104) D min u, D min v (cid:105) L ( M,E ) for each u, v ∈ D ( Q D t ◦ D ) . Proof.
Both statements follow immediately from the definitions and the constructions given above. Moreoverthe first point is also proved in [13], pag. 447.
We refer to [8] and to [25] for the background on the heat operator.
Theorem 2.1.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Let E be a vector bundleon M and let h be a metric on E . Let ∇ be a metric connection, let ∇ t be its formal adjoint and let P : C ∞ c ( M, E ) → C ∞ c ( M, E ) , P = ( ∇ t ◦ ∇ ) + L (21) be a Schr¨odinger type operator with L ∈ End( E ) such that P is symmetric and positive. • There is a constant c ∈ R such that, for each s ∈ C ∞ ( M, E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . Let P F be the Friedrich extension of P and let ∆ F be the Friedrich extension of the Laplacian acting on smoothfunctions with compact support ∆ : C ∞ c ( M ) → C ∞ c ( M ) . Then, for the respective heat operators e − tP F and e − t ∆ F , we have the following domination of semigroups: | e − tP F s | h ≤ e − tc e − ∆ F | s | h (22) for each s ∈ L ( M, E ) . This theorem is proved in [26], Theorem 2.13. Here we provide a different proof, under some additionalassumptions, in the spirit of [27]. Our additional assumptions are: • D ( d , max ) = D ( d , min ) on ( M, g ). • vol g ( M ) < ∞ .Clearly the second assumption is satisfied in our cases of interest, that is M is the regular part of a complexprojective variety V ⊂ CP n and g is the K¨ahler metric induced by the Fubini-Study metric on CP n or M isthe regular part of a smoothly Thom-Mather stratified pseudomanifold endowed with an iterated edge metric.Moreover, as we will see in Prop. 4.2 and in Prop. 5.2 , also the first assumption is fulfilled in our cases ofinterest. We give the proof assuming that E is a Hermitian vector bundle. In the real case the proof is analogouswith the obvious modifications. We divide the proof through several propositions. In order to state the firstresult we recall from [39] pag. 201 the following notion: Let ( X, µ ) be a σ -finite measure space. A function f ∈ L ( X, µ ) is called positive if it is non negative almost everywhere and is not the zero function. A boundedoperator A : L ( X, µ ) → L ( X, µ ) is called positivity preserving if Af is positive whenever f is positive. Proposition 2.3.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Let ∆ F be the Friedrichextension of the Laplacian acting on smooth functions with compact support ∆ : C ∞ c ( M ) → C ∞ c ( M ) . Considerthe heat operator e − t ∆ F : L ( M, g ) −→ L ( M, g ) . Then e − t ∆ F is positivity preserving for all t > .Proof. According to the Beurling-Deny criterion, see [39] pag. 209 or Theorem 3 in the appendix of [7], thestatement is equivalent to prove that if f ∈ D ( Q ∆ F ) then | f | ∈ D ( Q ∆ F ) and Q ∆ F ( | f | , | f | ) ≤ Q ∆ F ( f, f ). ByProp. 2.2 this condition becomes: for each f ∈ D ( d , min ) we have | f | ∈ D ( d , min ) and (cid:104) d , min | f | , d , min | f |(cid:105) L Ω ( M,g ) ≤ (cid:104) d , min f, d , min f (cid:105) L Ω ( M,g ) . Finally this last inequality is a consequence of Prop. 2.1 and Cor. 2.1.
Proposition 2.4.
Under the assumption of Prop. 2.3. For each λ > the operator (∆ F + λ ) − : L ( M, g ) → L ( M, g ) is positivity preserving.Proof. It is a consequence of the following formula combined with Prop. 2.3:(∆ F + λ ) − = (cid:90) ∞ e − λt e − t ∆ F dt, λ > . See for instance [7], Prop. 2 in the appendix.
Proposition 2.5.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Let P : C ∞ c ( M, E ) → C ∞ c ( M, E ) be as in the statement of Theorem 2.1. Given s ∈ C ∞ ( M, E ) and (cid:15) > let us define | s | h,(cid:15) as | s | h,(cid:15) := ( | s | h + (cid:15) ) . • Assume that vol g ( M ) < ∞ . Let s ∈ D ( P max ) ∩ C ∞ ( M, E ) . Then | s | h,(cid:15) ∈ D (∆ , max ) ∩ C ∞ ( M ) . • Assume that vol g ( M ) < ∞ and that D ( d , min ) = D ( d , max ) . Let s ∈ D ( P F ) ∩ C ∞ ( M, E ) . Then | s | h,(cid:15) ∈ D (∆ F ) ∩ C ∞ ( M ) . roof. According to [28] pag. 30-31, we haveRe( h ( P s, s )) = Re( h (( ∇ t ◦ ∇ ) s + Ls, s )) = Re( h (( ∇ t ◦ ∇ ) s, s )) + h ( Ls, s ) ≥ | s | h,(cid:15) ∆ | s | h,(cid:15) + c | s | h . Therefore | ∆ | s | h,(cid:15) | ≤ | P s | h | s | h | s | h,(cid:15) + | c || s | h | s | h | s | h,(cid:15) . In this way we can conclude that | ∆ | s | h,(cid:15) | ∈ L ( M, g ) because | P s | h ∈ L ( M, g ), | s | h ∈ L ( M, g ) and | s | h | s | h,(cid:15) ∈ L ∞ ( M, g ). By the fact that vol g ( M ) < ∞ we have that | s | h,(cid:15) ∈ L ( M, g ) and thus the proof of the first pointis complete. For the second point we can argue in this way: let m ≥ | c | + 1. Then, for each φ ∈ C ∞ c ( M, E ), wehave m ( (cid:104) φ, φ (cid:105) L ( M,E ) + (cid:104) P φ, φ (cid:105) L ( M,E ) ) ≥ m (cid:104) φ, φ (cid:105) L ( M,E ) + (cid:104) P φ, φ (cid:105) L ( M,E ) == m (cid:104) φ, φ (cid:105) L ( M,E ) + (cid:104) ( ∇ t ◦ ∇ ) φ, φ (cid:105) L ( M,E ) + (cid:104) Lφ, φ (cid:105) L ( M,E ) ≥≥ m (cid:104) φ, φ (cid:105) L ( M,E ) + (cid:104) ( ∇ t ◦ ∇ ) φ, φ (cid:105) L ( M,E ) + c (cid:104) φ, φ (cid:105) L ( M,E ) ≥ (cid:104) ( ∇ t ◦ ∇ ) φ, φ (cid:105) L ( M,E ) + (cid:104) φ, φ (cid:105) L ( M,E ) . Therefore, if Q P is the quadratic form associated to P and Q ∇ t ◦∇ is the quadratic form associated to ∇ t ◦ ∇ ,we proved that (cid:104) , (cid:105) L ( M,E ) + Q P ≥ m ( (cid:104) , (cid:105) L ( M,E ) + Q ∇ t ◦∇ ) . (23)This implies immediately that the identity Id : C ∞ c ( M, E ) → C ∞ c ( M, E ) induces a linear, bounded and injectivemap i : D ( Q P ) → D ( Q ∇ t ◦∇ ) . Now if we take s ∈ D ( P F ) ∩ C ∞ ( M, E ) we know that s ∈ D ( Q P ) ∩ D ( P max ). Bythe fact that s ∈ D ( P max ), as a consequence of the first point of this proposition, we get | s | h,(cid:15) ∈ D (∆ , max ). Bythe fact that s ∈ D ( Q P ), using (23) and Prop. 2.2, we get s ∈ D ( ∇ min ) and Q ∇ t ◦∇ ( s, s ) = (cid:104)∇ min s, ∇ min s (cid:105) L ( M,T ∗ M ⊗ E ) . Now, using Kato’s inequality in Prop. 2.1 and the fact that vol g ( M ) < ∞ , we have | s | h,(cid:15) ∈ D ( d , max ). Finallythe assumption D ( d , max ) = D ( d , min ) implies that | s | h,(cid:15) ∈ D ( d , min ). Therefore | s | h,(cid:15) ∈ D ( d , min ) ∩ D (∆ , max ).This in turn implies immediately that | s | h,(cid:15) ∈ D ( d t , max ◦ d , min ) that is | s | h,(cid:15) ∈ D (∆ F ) according to Prop.2.2. Proposition 2.6.
Under the assumptions of Theorem 2.1. Let D be defined as D := D ( P F ) ∩ C ∞ ( M, E ) .Then for all λ > , all s ∈ D , all f ∈ C ∞ c ( M ) such that f ≥ , there exists u ∈ L ( M, E ) such that1. | u | h = (∆ F + λ ) − f .2. (cid:104) s, u (cid:105) L ( M,E ) = (cid:104)| s | h , | u | h (cid:105) L ( M,g ) . Re (cid:104) P F s, u (cid:105) L ( M,E ) ≥ (cid:104)| s | h , (∆ F + c ) | u | h (cid:105) L ( M,g ) Proof.
For the first two points we follow the construction given in [27] pag. 895. Let g = (∆ F + λ ) − f . Let e be a measurable section of E such that h p ( e ( p ) , e ( p )) = 1 for each p ∈ M . Define sign( s ) assign( s ) := (cid:26) s | s | h s ( p ) (cid:54) = 0 e s ( p ) = 0 (24)Now define u := g sign( s ). It follows immediately that (cid:104) s, u (cid:105) L ( M,E ) = (cid:104)| s | h , | u | h (cid:105) L ( M,g ) and that | u | h = g . Inparticular the last equality follows by Prop. 2.4. This proves the first two points of the proposition. About thethird point, by the fact that s is smooth, we will write simply P s instead of P F s . As explained in the proofof Prop. 2.5, we have Re( h ( P s, s )) ≥ | s | h,(cid:15) ∆ | s | h,(cid:15) + c | s | h , that is Re( h ( P s, s/ | s | h,(cid:15) )) ≥ ∆ | s | h,(cid:15) + c | s | h / | s | h,(cid:15) .This implies that Re( h ( P s, | u | h s | s | h,(cid:15) )) ≥ (∆ | s | h,(cid:15) ) | u | h + c | s | h | s | h,(cid:15) | u | h . We can integrate because
P s ∈ L ( M, E ), | s | h , | u | h ∈ L ( M, g ), s/ | s | h,(cid:15) ∈ L ∞ ( M, E ) and ∆ | s | h,(cid:15) ∈ L ( M, g ) . In this way we get (cid:90) M Re( h ( P s, | u | h s | s | h,(cid:15) )) dvol g ≥ (cid:90) M (∆ | s | h,(cid:15) ) | u | h dvol g + (cid:90) M c | s | h | s | h,(cid:15) | u | h dvol g (cid:104) P s, | u | h s | s | h,(cid:15) (cid:105) L ( M,E ) ≥ (cid:104) ∆ | s | h,(cid:15) , | u | h (cid:105) L ( M,g ) + (cid:104) c | s | h | s | h,(cid:15) , | u | h (cid:105) L ( M,g ) . (25)For the right hand side of (25) we know that | u | h ∈ D (∆ F ) because | u | h = (∆ F + λ ) − f and λ >
0. Moreover,by Prop. 2.5, we also know that | s | h,(cid:15) ∈ D (∆ F ). Therefore, on the right hand side of (25), integration by partis allowed. This lead us to the expressionRe (cid:104) P s, | u | h s | s | h,(cid:15) (cid:105) L ( M,E ) ≥ (cid:104)| s | h,(cid:15) , ∆ F | u | h (cid:105) L ( M,g ) + (cid:104) | s | h | s | h,(cid:15) , c | u | h (cid:105) L ( M,g ) that is (cid:90) M Re( h ( P s, | u | h s | s | h,(cid:15) )) dvol g ≥ (cid:90) M ( | s | h,(cid:15) ∆ F | u | h + c | s | h | s | h,(cid:15) | u | h ) dvol g . (26)Keeping in mind (24) and applying the Lebesgue’s dominate convergence theorem, (26) becomes (cid:90) M Re( h ( P s, u )) dvol g ≥ (cid:90) M | s | h (∆ F | u | h + c | u | h ) dvol g that is Re (cid:104) P s, u (cid:105) L ( M,E ) ≥ (cid:104)| s | h , (∆ F + c ) | u | h (cid:105) L ( M,g ) . Finally we have the last proposition.
Proposition 2.7.
Under the assumptions of Prop. 2.6. For each µ > max { , − c } and for each β ∈ L ( M, E ) we have (∆ F + c + µ ) − | β | h ≥ | ( P F + µ ) − β | h (27) This in turn implies that e − t (∆ F + c ) | β | h ≥ | e − tP F β | h (28) and therefore e − tc e − t ∆ F | β | h ≥ | e − tP F β | h (29) Proof.
Let D = D ( P F ) ∩ C ∞ ( M, E ), s ∈ D , f ∈ C ∞ c ( M ), f ≥ µ > − c and u ∈ L ( M, E ) such that | u | h = (∆ F + µ + c ) − f . Then, by Prop. 2.6, we know thatRe (cid:104) P F s, u (cid:105) L ( M,g ) ≥ (cid:104)| s | h , (∆ F + c ) | u | h (cid:105) L ( M,g ) . By the second point of Prop. 2.6, for each µ ≥
0, we still haveRe (cid:104) ( µ + P F ) s, u (cid:105) L ( M,E ) ≥ (cid:104)| s | h , (∆ F + c + µ ) | u | h (cid:105) L ( M,g ) . The previous inequality, requiring µ > max { , − c } , produces (cid:104) ( | µ + P F ) s | h , | u | h (cid:105) L ( M,g ) ≥ (cid:104)| s | h , f (cid:105) L ( M,g ) . (30)Now, if we put β := ( P F + µ ) s , (30) becomes (cid:104) ( | β | h , | u | h (cid:105) L ( M,g ) ≥ (cid:104)| ( P F + µ ) − β | h , f (cid:105) L ( M,g ) . Finally, keepingin mind that | u | h = (∆ F + µ + c ) − f , we get (cid:104) (∆ F + µ + c ) − | β | h , f (cid:105) L ( M,g ) ≥ (cid:104)| ( P F + µ ) − β | h , f (cid:105) L ( M,g ) (31)and therefore (∆ F + µ + c ) − | β | h ≥ | ( P F + µ ) − β | h (32)because f , according to Prop. 2.6, is any non negative function lying in C ∞ c ( M ). As s ∈ D , which by Prop. 1.1is dense in D ( P F ) with the graph norm of P F , we have that β runs over a dense subset in L ( M, E ) and thus(27) follows by the continuity of the resolvent and the map | | h : L ( M, E ) → L ( M, g ). The second statement,that is (28), follows by a general result of functional analysis, see for example [27] pag. 897 or Corollary 15 inthe appendix of [7]. Finally (29) follows by (28) applying the Trotter’s product formula. See for example, [28]pag. 31 or [37] pag. 295-297. 13
Some general results
This section is made of two subsections. The first collects some results concerning Sobolev spaces of sections.The second one concerns Schroedinger operators with potential bounded from below.
We start with the following proposition.
Proposition 3.1.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Assume that thereexists a sequence of Lipschitz functions with compact support { φ j } j ∈ N such that • ≤ φ j ≤ for each j . • φ j → pointwise. • lim j →∞ (cid:107) d , min φ j (cid:107) L Ω ( M,g ) = 0 .Let E be a vector bundle over M and let h be a metric on E , Riemannian if E is a real vector bundle, Hermitianif E is a complex vector bundle. Finally let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E ) be a metric connection. Thenwe have W , ( M, E ) = W , ( M, E ) . Proof.
We start pointing out that by Theorem 11.3 in [25], the fact that φ j has compact support for every j ∈ N and Prop. 1.2, we get that { φ j } j ∈ N ⊂ D ( d , min ) so that the third point in the statement is well defined.According to Prop. 1.5, in order to prove the first point, it is enough to show that C ∞ ( M, E ) ∩ L ∞ ( M, E ) ∩ W , ( M, E ) ⊂ W , ( M, E ) . Let η j := d , min φ j . Let s ∈ C ∞ ( M, E ) ∩ L ∞ ( M, E ) ∩ W , ( M, E ). Then, by Prop. 1.4, we have φ j s ∈ W , ( M, E ). Moreoverlim j →∞ (cid:107) s − φ j s (cid:107) L ( M,E ) = lim j →∞ (cid:90) M (1 − φ j ) | s | h dvol g = (cid:90) M lim j →∞ (1 − φ j ) | s | h dvol g = 0 (33)by the Lebesgue dominate convergence theorem.If now we consider ∇ min ( φ j s ) then, by Prop. 1.4, we have ∇ min ( φ j s ) = η j ⊗ s + φ j ∇ s and therefore (cid:107)∇ s − ∇ min ( φ j s ) (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107)∇ s − φ j ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) + (cid:107) η j ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) . For the first term we havelim j →∞ (cid:107)∇ s − φ j ∇ s (cid:107) L ( M,T ∗ M ⊗ E ) = lim j →∞ (cid:90) M (1 − φ j ) |∇ s | h dvol g = 0again by the Lebesgue dominate convergence theorem. For (cid:107) η j ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) we have: (cid:107) η j ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) ≤ (cid:107) η j (cid:107) L Ω ( M,g ) (cid:107) s (cid:107) L ∞ ( M,E ) and therefore lim j →∞ (cid:107) η j ⊗ s (cid:107) L ( M,T ∗ M ⊗ E ) = 0because lim j →∞ (cid:107) η j (cid:107) L Ω ( M,g ) = 0. So we established thatlim j →∞ (cid:107)∇ s − ∇ ( φ j s ) (cid:107) L ( M,T ∗ M ⊗ E ) = 0 . (34)By (33) and (34) we showed that s ∈ W , ( M, E ) and this completes the proof.As a consequence of Prop. (3.1) we have the following result.14 roposition 3.2.
Let ( M, g ) and { φ j } j ∈ N be as in Prop. 3.1. Let E and F be two vector bundles over M endowed respectively with metrics h and ρ , Riemannian if E and F are real vector bundles, Hermitian if E and F are complex vector bundles. Finally let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E ) be a metric connection.Consider a first order differential operator of this type: D := θ ◦ ∇ : C ∞ c ( M, E ) → C ∞ c ( M, F ) (35) where θ ∈ C ∞ ( M, Hom( T ∗ M ⊗ E, F )) . Assume that θ extends as a bounded operator θ : L ( M, T ∗ M ⊗ E ) → L ( M, F ) . Then we have the following inclusion: D ( D max ) ∩ L ∞ ( M, E ) ⊂ D ( D min ) . (36) In particular (36) holds when D is the de Rham differential d k : Ω kc ( M ) → Ω k +1 c ( M ) , a Dirac operator D : C ∞ c ( M, E ) → C ∞ c ( M, E ) or the Dolbeault operator ∂ p,q : Ω p,qc ( M ) → Ω p,q +1 c ( M ) in the case M is a complexmanifold.Proof. The first statement, that is (36), follows arguing as in the proof of Prop. 3.1 and then using the continuityof θ : L ( M, T ∗ M ⊗ E ) → L ( M, F ). The fact that (36) holds for the de Rham differential d k : Ω kc ( M ) → Ω k +1 c ( M ), for a Dirac operator D : C ∞ c ( M, E ) → C ∞ c ( M, E ) or for the Dolbeault operator ∂ p,q : Ω p,qc ( M ) → Ω p,q +1 c ( M ) is a straightforward verification.We have now the following proposition. Proposition 3.3.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold. Assume that for some v ∈ R with v > we have a continuous inclusion W , ( M, g ) (cid:44) → L vv − ( M, g ) . Let E be a vector bundle over M and let h be a metric on E , Riemannian if E is a real vector bundle, Hermitian if E is a complex vector bundle.Finally let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E ) be a metric connection. Then we have the following properties: • We have a continuous inclusion W , ( M, E ) (cid:44) → L vv − ( M, E ) . • If furthermore M has finite volume then the inclusion W , ( M, E ) (cid:44) → L ( M, E ) is a compact operator.Proof. Using Cor. 2.1, we get the continuous inclusion W , ( M, E ) (cid:44) → L vv − ( M, E ) . Now we prove the second statement. Let { s i } i ∈ N be a bounded sequence in W , ( M, E ). By Prop. 1.1 we canassume that each s i is smooth. Let B ∈ R be a positive number such that (cid:107) s i (cid:107) L ( M,E ) + (cid:107)∇ s i (cid:107) L ( M,T ∗ M ⊗ E ) ≤ B (37)Let { K i } i ∈ N be an exhausting sequence of compact subset of M , that is K i ⊂ int( K i +1 ), int( K i ) is the interiorof K i and (cid:83) i ∈ N K i = M . Let { χ i } i ∈ N ⊂ C ∞ c ( M ) be a sequence of smooth functions with compact supportsuch that • ≤ χ i ≤ • χ i | K i = 1, • χ i | M \ K i +1 = 0Then, according to Prop. 1.4, { χ s i } i ∈ N is a bounded sequence in W , ( U, E ) where U is any open subset of M with compact closure such that supp( χ ) ⊂ U . Therefore, applying Prop. 1.3, we get the existence of asubsequence of { s i } i ∈ N , that we label { s i, } i ∈ N , such that { χ s i, } i ∈ N converges in L ( M, E ) to some element ˜ s .Now consider the sequence { χ s i, } i ∈ N . Arguing as in the previous case we get the existence of a subsequenceof { s i, } i ∈ N , that we label by { s i, } i ∈ N , such that { χ s i, } i ∈ N converges in L ( M, E ) to some ˜ s ∈ L ( M, E ).Iterating this construction we get a countable family of sequences {{ χ s i, } i ∈ N , { χ s i, } i ∈ N , ..., { χ n s i,n } i ∈ N , ... } (38)such that, for each n ∈ N , { s i,n +1 } i ∈ N is a subsequence of { s i,n } i ∈ N and such that { χ n s i,n } i ∈ N converges in L ( M, E ) to some element ˜ s n ∈ L ( M, E ). Now we want to prove that the sequence { ˜ s n } n ∈ N is a Cauchysequence in L ( M, E ). In order to prove this claim let k and m be two natural numbers with m > k > { χ m s i,m } i ∈ N converges to ˜ s m in L ( M, E ) and that { χ k s i,k } i ∈ N converges to ˜ s k in L ( M, E ). By the fact that { s i,m } i ∈ N is a subsequence of { s i,k } i ∈ N we also knowthat { χ k s i,m } i ∈ N converges to ˜ s k in L ( M, E ). Therefore we can find a number j ∈ N such that for each i ≥ j we have max {(cid:107) ˜ s m − χ m s i,m (cid:107) L ( M,E ) , (cid:107) ˜ s k − χ k s i,m (cid:107) L ( M,E ) } ≤ m (39)This implies that (cid:107) ˜ s m − ˜ s k (cid:107) L ( M,E ) ≤ (40) ≤ (cid:107) ˜ s m − χ m s i,m (cid:107) L ( M,E ) + (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) + (cid:107) χ k s i,m − ˜ s k (cid:107) L ( M,E ) ≤≤ m + (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) . In this way, in order to conclude that { ˜ s n } n ∈ N is a Cauchy sequence in L ( M, E ), we have to estimate (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) . Let T k be defined as T k := M \ K k . The fact that vol g ( M ) < ∞ implies immediately thatlim k →∞ vol g ( T k ) = 0. Then, for (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) , we have (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) = (cid:90) M ( χ m − χ k ) | s i,m | h dvol g . Now, in virtue of the first point of the theorem, we know that | s i,m | h ∈ L vv − ( M, g ). Clearly ( χ m − χ k ) ∈ L z ( M, g ) for each z ∈ [1 , ∞ ]. Moreover ( χ m − χ k ) = 0 on M \ T k and ( χ m − χ k ) ≤ T k . Therefore wecan apply the H¨older inequality, see [3] pag. 88, and we get (cid:90) M ( χ m − χ k ) | s i,m | h dvol g ≤ (cid:107) ( χ m − χ k ) (cid:107) L v ( M,g )) (cid:107)| s i,m | h (cid:107) L vv − ( M,g ) ≤ (41) ≤ (cid:107) (cid:107) L v ( T k ,g | Tk )) (cid:107)| s i,m | h (cid:107) L vv − ( M,g ) ≤ BC (vol g ( T k )) v and lim k →∞ (vol g ( T k )) v = 0where B is the same constant of (37) and C is the same constant of the continuous inclusion W , ( M, E ) (cid:44) → L vv − ( M, E ). In this way, for (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) , we get the following inequality (cid:107) χ m s i,m − χ k s i,m (cid:107) L ( M,E ) ≤ ( BC ) (vol g ( T k )) v (42)which in turn, by (40), implies that (cid:107) ˜ s m − ˜ s k (cid:107) L ( M,E ) ≤ m + ( BC ) (vol g ( T k )) v . Therefore for each δ > m ∈ N such that for each m > k > m we have (cid:107) ˜ s m − ˜ s k (cid:107) L ( M,E ) ≤ m + ( BC ) (vol g ( T k )) v ≤ δ. (43)In this way we proved that { ˜ s n } n ∈ N is a Cauchy sequence in L ( M, E ) and this implies that there exists anaccumulation point s ∈ L ( M, E ) for { ˜ s n } n ∈ N . Finally, in order to conclude the proof, we have to show that s isan accumulation point in L ( M, E ) for the original sequence { s i } i ∈ N . Let γ >
0. Then we can find an element˜ s m ∈ { ˜ s n } n ∈ N such that (cid:107) s − ˜ s m (cid:107) L ( M,E ) ≤ γ . Consider now the countable family of sequences defined in (38) {{ χ s i, } i ∈ N , { χ s i, } i ∈ N , ..., { χ n s i,n } i ∈ N , ... } . (44)We recall that, for each n ∈ N , { χ n s i,n } converges in L ( M, E ) to ˜ s n , that { s i,n +1 } i ∈ N is a subsequence of { s i,n } i ∈ N and that { s i, } i ∈ N is a subsequence of { s i } i ∈ N . Then we can find a positive integer number i ∈ N such that (cid:107) ˜ s m − χ m s i,m (cid:107) L ( M,E ) ≤ γ for each i > i . Now if we consider (cid:107) χ m s i,m − s i,m (cid:107) L ( M,E ) then, arguingas in (41)–(42), we get (cid:107) χ m s i,m − s i,m (cid:107) L ( M,E ) ≤ ( BC ) (vol g ( T m )) v and ( BC ) (vol g ( T m )) v < γ when m issufficiently big. Ultimately for m and i sufficiently big we have (cid:107) s − s i,m (cid:107) L ( M,E ) ≤≤ (cid:107) s − ˜ s m (cid:107) L ( M,E ) + (cid:107) ˜ s m − χ m s i,m (cid:107) L ( M,E ) + (cid:107) χ m s i,m − s i,m (cid:107) L ( M,E ) ≤≤ γ + ( BC ) (vol g ( T m )) v ≤ γ. This shows that s is an accumulation point in L ( M, E ) for the original sequence { s i } i ∈ N because s i,m ∈ { s i } i ∈ N and therefore the proof is completed. 16 emark 3.1. We can reformulate the statement of Prop. 3.1 saying that D ( ∇ max ) = D ( ∇ min ) . Analogouslywe can reformulate the statement of Prop. 3.3 saying that there exists a continuous inclusion D ( ∇ min ) (cid:44) → L vv − ( M, E ) and that the natural inclusion D ( ∇ min ) (cid:44) → L ( M, E ) is a compact operator where D ( ∇ min ) isendowed with the corresponding graph norm. We conclude this section with the following corollary:
Corollary 3.1.
Consider an open and possibly incomplete Riemannian manifold ( M, g ) . Let E be a vectorbundle over M endowed with a metric h , Riemannian if E is a real vector bundle, Hermitian if E is a complexvector bundle. Finally let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E ) be a metric connection. Assume that the naturalinclusion W , ( M, E ) (cid:44) → L ( M, E ) is a compact operator. Then the image of ∇ min , im( ∇ min ) , is a closedsubset of L ( M, T ∗ M ⊗ E ) .Proof. Consider the operator ∇ t max ◦∇ min : L ( M, E ) → L ( M, E ) whose domain is given by D ( ∇ t max ◦∇ min ) := { s ∈ D ( ∇ min ) : ∇ min s ∈ D ( ∇ t max ) } . Arguing as in the proof of Prop. 1.1 we get the following inequality (cid:107)∇ min s (cid:107) L ( M,T ∗ M ⊗ E ) ≤
12 ( (cid:107) s (cid:107) L ( M,E ) + (cid:107)∇ t max ( ∇ min s ) (cid:107) L ( M,E ) )for each s ∈ D ( ∇ t max ◦ ∇ min ). Therefore we can conclude that the natural inclusion D ( ∇ t max ◦ ∇ min ) (cid:44) → D ( ∇ min ) (45)is a continuous operator where each domain is endowed with the corresponding graph norm. In this way, usingthe assumption on W , ( M, g ) (cid:44) → L ( M, g ), we get that the natural inclusion D ( ∇ t max ◦ ∇ min ) (cid:44) → L ( M, E ) (46)is a compact operator where D ( ∇ t max ◦∇ min ) is endowed with its graph norm. As it is well know this is equivalentto say that ∇ t max ◦∇ min is a discrete operator and this in turn implies in particular that it is a Fredholm operatoron its domain endowed with the graph norm. Therefore we can conclude that im( ∇ t max ◦ ∇ min ) is closed in L ( M, E ). Now consider the two following orthogonal decompositions of L ( M, E ): L ( M, E ) = ker( ∇ min ) ⊕ im( ∇ t max ) and L ( M, E ) = ker( ∇ t max ◦ ∇ min ) ⊕ im( ∇ t max ◦ ∇ min ) . Clearly ker( ∇ t max ◦ ∇ min ) = ker( ∇ min ). Therefore we have the following chain of inclusions:im( ∇ t max ◦ ∇ min ) ⊂ im( ∇ t max ) ⊂ im( ∇ t max ) = im( ∇ t max ◦ ∇ min ) = im( ∇ t max ◦ ∇ min )which in particular implies that im( ∇ t max ) = im( ∇ t max ) and therefore, taking the adjoint, im( ∇ min ) = im( ∇ min ). Proposition 3.4.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold of finite volume anddimension m > . Assume that we have a continuous inclusion W , ( M, g ) → L mm − ( M, g ) . Let ∆ F : L ( M, g ) (cid:44) → L ( M, g ) be the Friedrich extension of the Laplacian ∆ : C ∞ c ( M ) → C ∞ c ( M ) . Then the heatoperator e − t ∆ F : L ( M, g ) → L ( M, g ) is a trace class operator and we have the following inequalities for itstrace: Tr( e − t ∆ F ) ≤ C vol g ( M ) t − m (47) for t ∈ (0 , .Proof. As showed in [1] pag. 1062 or in [45] Theorem 2.1 the continuous inclusion W , ( M, g ) (cid:44) → L mm − ( M, g )is equivalent to the following property: k ∆ ( t, x, y ) ≤ Ct − m (48)for each ( x, y ) ∈ M × M , t ∈ (0 ,
1) and where k ∆ ( t, x, y ) is the smooth kernel of the heat operator e − t ∆ F and C is a positive constant. This implies immediately that tr( e − t ∆ F ) ≤ Ct − m for t ∈ (0 , (cid:90) M × M ( k ∆ ( t, x, y )) dvol g ( x ) dvol g ( v ) < ∞ t ∈ (0 ,
1) and this in turn implies that e − t ∆ F : L ( M, g ) → L ( M, g ) is a Hilbert-Schimdt operator foreach t ∈ (0 , t = t + t we get e − t ∆ F = e − t ∆ F ◦ e − t ∆ F and this tellsus that e − t ∆ F : L ( M, g ) → L ( M, g ) is a trace class operator for each t ∈ (0 , t > t = n tn with n > t , we have e − t ∆ F = e − tn ∆ F ◦ ... ◦ e − tn ∆ F ( n -times) and this allows us to concludethat e − t ∆ F : L ( M, g ) → L ( M, g ) is a trace class operator. Finally we have Tr( e − t ∆ F ) = (cid:82) M k ∆ ( t, x, x ) dvol g and therefore for each t ∈ (0 ,
1) we findTr( e − t ∆ F ) = (cid:90) M k ∆ ( t, x, x ) dvol g ≤ (cid:90) M Ct − m dvol g = C vol g ( M ) t − m . As in the previous proposition consider again a possibly incomplete Riemannian manifold (
M, g ) of finitevolume and dimension m . Let E be a vector bundle over M endowed with a metric h , Riemannian if E is areal vector bundle, Hermitian if E is a complex vector bundle. Finally let ∇ : C ∞ ( M, E ) → C ∞ ( M, T ∗ M ⊗ E )be a metric connection. Our goal is to study some properties of Schr¨odinger type operators, that is operatorsof type ∇ t ◦ ∇ + L (49)where ∇ t : C ∞ c ( M, T ∗ M ⊗ E ) → C ∞ c ( M, E ) is the formal adjoint of ∇ and L ∈ C ∞ ( M, End( E )). In particularwe are interested in (49) acting on L ( M, E ) with C ∞ c ( M, E ) as core domain. One of the reasons, as it iswell known, is provided by the fact that the square of the typical first order differential operators arising indifferential geometry, for instance the Gauss-Bonnet operator d + δ , the Hodge-Dolbeault operator ∂ + ∂ t andthe Spin-Dirac operator ð , are Schr¨odinger type operators. Proposition 3.5.
Let ( M, g ) be an open and possibly incomplete Riemannian manifold of finite volume anddimension m > . Assume that we have a continuous inclusion W , ( M, g ) → L mm − ( M, g ) . Let V , E , g , h ,and ∇ be as described above. Let P := ∇ t ◦ ∇ + L, P : C ∞ c ( M, E ) → C ∞ c ( M, E ) be a Schr¨odinger type operator with L ∈ C ∞ ( M, End( E )) . Assume that: • P is symmetric and positive. • There is a constant c ∈ R such that, for each s ∈ C ∞ ( M, E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . Let P F : L ( M, E ) → L ( M, E ) be the Friedrich extension of P and let ∆ F : L ( M, g ) → L ( M, g ) be theFriedrich extension of ∆ : C ∞ c ( M ) → C ∞ c ( M ) . Then the heat operator associated to P F e − tP F : L ( M, E ) −→ L ( M, E ) is a trace class operator and its trace satisfies the following inequality: Tr( e − tP F ) ≤ re − tc Tr( e − t ∆ F ) . (50) where r is the rank of the vector bundle E .Proof. According to Theorem 2.1 we know that | e − tP F s | h ≤ e − tc e − ∆ F | s | h (51)for each s ∈ L ( M, E ) . Let k P ( t, x, y ) be the smooth kernel of the heat operator e − tP F and analogously let k ∆ ( t, x, y ) be the smooth kernel of the heat operator e − t ∆ F . Therefore, for each pair ( x, y ) ∈ M × M , k P ( t, x, y ) ∈ Hom( E y , E x ) and analogously k ∆ ( t, x, y ) ∈ Hom( R , R ) that is k ∆ ( t, x, y ) ∈ C ∞ ( M × M ). Let uslabel by (cid:107) (cid:107) h, op the pointwise operator norm for the linear operators acting between ( E y , h y ) and ( E x , h x ). Asexplained in [28] pag. 32, the inequality (51) implies that for the pointwise operator norm (cid:107) k P ( t, x, y ) (cid:107) h, op thefollowing inequality holds: (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ e − tc k ∆ ( t, x, y ) .
18n particular for x = y we have (cid:107) k P ( t, x, x ) (cid:107) h, op ≤ e − tc k ∆ ( t, x, x ) . (52)This implies immediately the following inequality for the pointwise tracestr( k P ( t, x, x )) ≤ re − tc k ∆ ( t, x, x ) . (53)By Prop. 3.4 we know that e − t ∆ F : L ( M, g ) → L ( M, g ) is a trace class operator. Therefore, using (52)and (53), we get that also e − tP F : L ( M, E ) → L ( M, E ) is a trace class operator and its trace satisfies theinequality Tr( e − tP F ) ≤ re − tc Tr( e − t ∆ F ) . (54) Corollary 3.2.
Under the assumptions of Prop. 3.5. For each t ∈ (0 , we have the following inequalities forthe pointwise trace and for the heat trace of e − tP F respectively: tr( k P ( t, x, x )) ≤ Cre − tc t − m Tr( e − tP F ) ≤ Cre − tc vol g ( M ) t − m where C is the same constant of (47) . The operator P F : L ( M, E ) → L ( M, E ) is a discrete operator. If welabel its eigenvalues with ≤ λ ≤ λ ≤ ... ≤ λ n ≤ ... then there exists a positive constant K such that we have the following asymptotic inequality λ j ≥ Kj m + c (55) as j → ∞ .Proof. The inequality for the pointwise trace as well as that for the heat trace of e − tP F follow by Prop. 3.4 andProp. 3.5. By the fact that e − tP F is a trace class operator we get immediately that P F is a discrete operator.Finally, by (50) and the first point of this corollary, we know that (cid:80) j e − tλ j ≤ Cre − tc vol g ( M ) t − m . This isequivalent to say that (cid:88) j ∈ N e − tµ j ≤ Cr vol g ( M ) t − m where µ j := λ j − c . Now the thesis follows applying a classical argument from Tauberian theory, see for instance[43] pag. 107. Proposition 3.6.
Under the assumptions of Theorem 3.5. Let k P ( t, x, y ) and (cid:107) k P ( t, x, y ) (cid:107) h, op be as in theproof of Theorem 3.5. Then the following inequality holds for < t < : (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ Ce − tc t − m . (56) where C is the same positive constant of (47) . This implies that1. e − tP F is a ultracontractive operator for each < t < . This means, see [42], that for each < t < there exists C t > such that (cid:107) e − tP F s (cid:107) L ∞ ( M,E ) ≤ C t (cid:107) s (cid:107) L ( M,E ) for each s ∈ L ( M, E ) . In particular, for each < t < , e − tP F : L ( M, E ) → L ∞ ( M, E ) is continuous.2. If s is an eigensection of P F : L ( M, E ) → L ( M, E ) then s ∈ L ∞ ( M, E ) .Proof. As pointed out in the proof of Prop. 3.5 we have (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ e − ct k ∆ ( t, x, y ). By the assumptionswe know that there is a continuous inclusion W , ( M, g ) (cid:44) → L mm − ( M, g ). As recalled in the proof of Prop. 3.4this is equivalent to say that, for some positive constant C , k ∆ ( t, x, y ) ≤ Ct − m , < t < . Combining together the previous inequalities we have, for 0 < t < (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ Ce − tc t − m and this establishes (56). Finally the remaining two properties follow immediately using (56).19 Applications to irreducible complex projective varieties
This section concerns irreducible complex projective varieties V ⊂ CP n . This means that V is the zero set of afamily of homogeneous polynomials such that it is not possible to decompose V as V = V ∪ V with V ⊂ V , V ⊂ V , V (cid:54) = V , V (cid:54) = V and such that V and V are the zero set of other two families of homogeneouspolynomials. Using the language of Zariski topology this means that V is a Zariski closed subset of CP n andit is not possible to decompose V as V = V ∪ V with V ⊂ V , V ⊂ V , V (cid:54) = V , V (cid:54) = V where V and V are other two Zariski closed subsets of CP n . Our reference for this topic is [24]. Given an irreducible complexprojective variety V ⊂ CP n we will label by sing( V ) the singular locus of V and by reg( V ) := V \ sing( V )the regular part of V . The regular part of V , reg( V ), becomes a K¨ahler manifold when we endow it with theK¨ahler metric induced by the Fubini-Study metric of CP n . In particular we get an open and incomplete K¨ahlermanifold when sing( V ) (cid:54) = ∅ .Now we state a proposition which provides the existence of a suitable sequence of cut-off functions. A similarresult is contained in [31] pag. 871 and in [45], Theorem 3.1 and Theorem 3.2. First we recall the followingproperty. Proposition 4.1.
Let M be a complex manifold and let h and g be two Hermitian metrics on M such that g ≥ h .Then for each η ∈ Ω c ( M ) we have (cid:107) η (cid:107) L Ω ( M,h ) ≤ (cid:107) η (cid:107) L Ω ( M,g ) . Therefore the identity map Id : Ω c ( M ) → Ω c ( M ) extends as a continuous inclusion L Ω ( M, g ) (cid:44) → L Ω ( M, h ) so that for each φ ∈ L Ω ( M, g ) we have (cid:107) φ (cid:107) L Ω ( M,h ) ≤ (cid:107) φ (cid:107) L Ω ( M,g ) .Proof. The proof lies on a careful calculation of linear algebra. It is carried out, for instance, in [22] pag.146.
Proposition 4.2.
Let V ⊂ CP n be an irreducible complex projective variety of complex dimension v and let g be the K¨ahler metric on reg( V ) induced by the Fubini-Study metric of CP n . Then there exists a sequence ofLipschitz functions { φ j } j ∈ N with compact support in reg( V ) such that • ≤ φ j ≤ for each j . • φ j → pointwise. • φ j ∈ D ( d , min ) for each j ∈ N and lim j →∞ (cid:107) d , min φ j (cid:107) L Ω (reg( V ) ,g ) = 0 .In particular ∈ D ( d , min ) .Proof. Let π : ˜ V −→ V be a resolution of singularities (which exists thanks to the fundamental work of Hironaka,see [29]). We recall that π : ˜ V −→ V is a holomorphic and surjective map such that π | ˜ V \ E : ˜ V \ E −→ V \ sing( V )is a biholomorphism where E = π − (sing( V )) is the exceptional set. Moreover we can assume that E is adivisor with only normal crossings, that is, the irreducible components of E are regular and meet complextransversely. In particular, and this is what we need for our purpose, ˜ V \ π − (reg( V )) is a finite union ofcompact complex submanifolds, that is ˜ V \ π − (reg( V )) = ∪ mi =1 S i for some m ∈ N . Therefore for the realcodimension of S i we have cod R ( S i ) ≥ i = 1 , ..., m . Consider now a Hermitian metric h on ˜ V . Let usdefine V (cid:48) := π − (reg( V )) and let h (cid:48) be defined us h (cid:48) := h | V (cid:48) . As a first step we want to show that on ( V (cid:48) , h (cid:48) )there is a sequence of Lipschitz functions with compact support { ψ j } j ∈ N which satisfies the three propertiesstated in this proposition. To this aim we adapt to our case the strategy used in [18] and in [31]. Define M i := ˜ V \ S i . Let r i be the distance function to S i induced by h . Let (cid:15) n := n and let (cid:15) (cid:48) n := e − (cid:15) n = e − n .Then we define ψ j,M i as ψ j,M i := r i ≥ (cid:15) n ( r i (cid:15) n ) (cid:15) n (cid:15) (cid:48) n ≤ r i ≤ (cid:15) n ( (cid:15) (cid:48) n (cid:15) n ) (cid:15) n ( r i (cid:15) (cid:48) n − (cid:15) (cid:48) n ≤ r i ≤ (cid:15) (cid:48) n ≤ r i ≤ (cid:15) (cid:48) n (57)By (57) we get easily that each ψ j,M i is a Lipschitz function with compact support. Therefore, combiningtogether Theorem 11.3 in [25], Prop. 1.2 and the fact that M i has finite volume, we get that { ψ j,M i } j ∈ N ⊂ ( d , min ) on ( M i , h | M i ). Clearly we have 0 ≤ ψ j,M i ≤ j →∞ ψ j,M i = 1 pointwise. Moreover, accordingto [18], we have lim j →∞ (cid:107) d , min ψ j,M i (cid:107) L Ω ( M i ,h | Mi ) = 0 . (58)Here, we only recall that the previous limit is based on an estimate of the volume of a tubular neighborhoodof S i and that for this estimate the lower bound on the real codimension of S i plays a fundamental role. Nowdefine ψ j := m (cid:89) i =1 ψ j,M i . For each j ∈ N , ψ j is defined as a product of a finite number of non negative Lipschitz functions with compactsupport and bounded above by 1. Therefore ψ j is in turn a non negative Lipschitz function which compactsupport and bounded above by 1. Thus, arguing as above, we can conclude that { ψ j } j ∈ N ⊂ D ( d , min ) on( V (cid:48) , h (cid:48) ). Clearly for each ψ j we have 0 ≤ ψ j ≤ j →∞ ψ j = 1 pointwise. Now we have to show thatlim j →∞ (cid:104) d , min ψ j , d , min ψ j (cid:105) L Ω ( V (cid:48) ,h (cid:48) ) = 0 . (59)We have d , min ψ j = (cid:80) mi =1 γ i d , min ψ j,M i where γ i is given by the product ψ j,M ...ψ j,M i − ψ j,M i +1 ...ψ j,M m . Bythe fact that 0 ≤ γ i ≤ j →∞ (cid:104) d , min ψ j,M p , d , min ψ j,M q (cid:105) L Ω ( V (cid:48) ,h (cid:48) ) = 0 for each p, q ∈ { , ..., m } . This follows because (cid:104) d , min ψ j,M p , d , min ψ j,M q (cid:105) L Ω ( V (cid:48) ,h (cid:48) ) ≤ (cid:107) d , min ψ j,M p (cid:107) L Ω ( V (cid:48) ,h (cid:48) ) (cid:107) d , min ψ j,M q (cid:107) L Ω ( V (cid:48) ,h (cid:48) ) , and, by (58), we have lim j →∞ (cid:107) d , min ψ j,M p (cid:107) L Ω ( V (cid:48) ,h (cid:48) ) = 0 , and lim j →∞ (cid:107) d , min ψ j,M q (cid:107) L Ω ( V (cid:48) ,h (cid:48) ) = 0 . This allow us to conclude that on ( V (cid:48) , h (cid:48) ) there is a sequence of Lipschitz functions with compact support { ψ j } j ∈ N which satisfies the three properties stated in this proposition. Now let ˜ g be the K¨ahler metric on V (cid:48) defined as π ∗ g . We can look at ˜ g as the pull-back of the Fubini-Study metric on CP n through the map π : ˜ V −→ CP n . By the fact that dπ , the differential of π , degenerates on ˜ V \ V (cid:48) we get that ˜ g ≤ Ch (cid:48) , for somepositive real constant C >
0. Now, as an immediate application of Prop. 4.1, we can conclude that the sequence { ψ j } j ∈ N satisfies the three properties stated in this proposition also with respect to the K¨ahler manifold ( V (cid:48) , ˜ g ).Finally, by the fact that π | V (cid:48) : ( V (cid:48) , ˜ g ) −→ (reg( V ) , g ) is an isometry, defining φ j := ψ j ◦ ( π | V (cid:48) ) − we obtain ourdesired sequence on (reg( V ) , g ).Now we have the main result of this section. Theorem 4.1.
Let V ⊂ CP n be an irreducible complex projective variety of complex dimension v . Let E be avector bundle over reg( V ) and let h be a metric on E , Riemannian if E is a real vector bundle, Hermitian if E is a complex vector bundle. Let g be the K¨ahler metric on reg( V ) induced by the Fubini-Study metric of CP n .Finally let ∇ : C ∞ (reg( V ) , E ) → C ∞ (reg( V ) , T ∗ reg( V ) ⊗ E ) be a metric connection. We have the followingproperties: • W , (reg( V ) , E ) = W , (reg( V ) , E ) . • Assume that v > . Then there exists a continuous inclusion W , (reg( V ) , E ) (cid:44) → L vv − (reg( V ) , E ) . • Assume that v > . Then the inclusion W , (reg( V ) , E ) (cid:44) → L (reg( V ) , E ) is a compact operator.Proof. The first point follows by Prop. 4.2 and by Prop. 3.1. The continuous inclusion W , (reg( V ) , g ) (cid:44) → L vv − (reg( V ) , g ) is established in [31] pag. 874 or in [45] pag. 113. Now, by the first point of this theorem (orby [31] Theorem 4.1 or by [45] Cor. 3.1), we know that W , (reg( V ) , g ) = W , (reg( V ) , g ) and therefore wehave the continuous inclusion W , (reg( V ) , g ) (cid:44) → L vv − (reg( V ) , g ). Now, using Prop. 2.1, we get the continuousinclusion C ∞ (reg( V ) , E ) ∩ W , (reg( V ) , E ) (cid:44) → L vv − (reg( V ) , E ) . C ∞ (reg( V ) , E ) ∩ W , (reg( V ) , E ) in W , (reg( V ) , E ), see Prop. 1.2, the continuousinclusion W , (reg( V ) , E ) (cid:44) → L vv − (reg( V ) , E ) is established. Finally the third point is a consequence of thesecond point and Prop. 3.3. Remark 4.1.
We can reformulate the statement of Theorem 4.1 saying that D ( ∇ max ) = D ( ∇ min ) , there existsa continuous inclusion D ( ∇ max ) (cid:44) → L vv − (reg( V ) , E ) and that the natural inclusion D ( ∇ max ) (cid:44) → L (reg( V ) , E ) is a compact operator where D ( ∇ max ) is endowed with the corresponding graph norm. Corollary 4.1.
Under the assumptions of Theorem 4.1. Then im( ∇ min ) = im( ∇ max ) is a closed subspace of L (reg( V ) , T ∗ reg( V ) ⊗ E ) .Proof. According to Prop. 4.1 we know that ∇ max = ∇ min and therefore im( ∇ max ) = im( ∇ min ). Now thethesis follows by Cor. 3.1.We conclude this section with the following proposition. The case of the Dolbeault operator is alreadytreated in [41]. Proposition 4.3.
Let (reg( V ) , g ) be as in Theorem 4.1. Let E and F be two vector bundles over reg( V ) endowed respectively with metrics h and ρ , Riemannian if E and F are real vector bundles, Hermitian if E and F are complex vector bundles. Finally let ∇ : C ∞ (reg( V ) , E ) → C ∞ ( M, T ∗ reg( V ) ⊗ E ) be a metric connection.Consider a first order differential operator of this type: D := θ ◦ ∇ : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , F ) (60) where θ ∈ C ∞ (reg( V ) , Hom( T ∗ reg( V ) ⊗ E, F )) . Assume that θ extends as a bounded operator θ : L (reg( V ) , T ∗ reg( V ) ⊗ E ) → L (reg( V ) , F ) . Then we have the following inclusion: D ( D max ) ∩ L ∞ (reg( V ) , E ) ⊂ D ( D min ) . (61) In particular (61) holds when D is the de Rham differential d k : Ω kc (reg( V )) → Ω k +1 c (reg( V )) , a Dirac operator D : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) or the Dolbeault operator ∂ p,q : Ω p,qc (reg( V )) → Ω p,q +1 c (reg( V )) .Proof. This follows applying Theorem 4.1 and Prop. 3.2.
As in the previous section consider again an irreducible complex projective variety V ⊂ CP n of complexdimension v . Let reg( V ) be its regular part and let E be a vector bundle over reg( V ) endowed with a metric h ,Riemannian if E is a real vector bundle, ermitian if E is a complex vector bundle. Finally let g be the K¨ahlermetric on reg( V ) induced by the Fubini-Study metric of CP n and let ∇ : C ∞ (reg( V ) , E ) → C ∞ (reg( V ) , T ∗ M ⊗ E ) be a metric connection. In this section we consider again some Schr¨odinger type operators ∇ t ◦ ∇ + L (62)where ∇ t : C ∞ c (reg( V ) , T ∗ M ⊗ E ) → C ∞ c (reg( V ) , E ) is the formal adjoint of ∇ and L ∈ C ∞ (reg( V ) , End( E )). Theorem 4.2.
Let V , E , g , h , and ∇ be as described above. Let P := ∇ t ◦ ∇ + L, P : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) be a Schr¨odinger type operator with L ∈ C ∞ (reg( V ) , End( E )) . Assume that: • P is symmetric and positive. • There is a constant c ∈ R such that, for each s ∈ C ∞ (reg( V ) , E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . et P F : L (reg( V ) , E ) → L (reg( V ) , E ) be the Friedrich extension of P and let ∆ F : L (reg( V ) , g ) → L (reg( V ) , g ) be the Friedrich extension of ∆ : C ∞ c (reg( V )) → C ∞ c (reg( V )) . Then the heat operator asso-ciated to P F e − tP F : L (reg( V ) , E ) −→ L (reg( V ) , E ) is a trace class operator and its trace satisfies the following inequality: Tr( e − tP F ) ≤ me − tc Tr( e − t ∆ F ) . (63) where m is the rank of the vector bundle E .Proof. This follows by Prop. 3.5 and by Theorem 4.1.
Corollary 4.2.
Under the assumptions of Theorem 4.2. The operator P F : L (reg( V ) , E ) → L (reg( V ) , E ) isa discrete operator. Moreover, for t ∈ (0 , , we have the following inequalities: tr( k P ( t, x, x )) ≤ me − tc (4 πt ) − v (1 + 4 v ( v + 1)6 t + O ( t )) (64)Tr( e − tP F ) ≤ me − tc (4 πt ) − v (cid:18) vol g (reg( V ))(1 + 4 v ( v + 1)6 t ) + O ( t ) (cid:19) . (65) Let { λ j } be the sequence of eigenvalues of P F : L (reg( V ) , E ) → L (reg( V ) , E ) . Then we have the followingasymptotic inequality: λ j ≥ (cid:18) (2 π ) v jω v m vol g (reg( V )) (cid:19) v + c (66) as j → ∞ where ω v is the volume of the unit v -ball in R v .Proof. The fact that P F : L (reg( V ) , E ) → L (reg( V ) , E ) is a discrete operator is a consequence of Theorem4.2. Inequalities (64), (65) and (66) follow by Theorem 4.2 and Corollary 5.4 in [31].An important case of the previous corollary is given by ( ∇ t ◦ ∇ ) F , the Friedrich extension of the BochnerLaplacian ∇ t ◦ ∇ : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ). As before we label by k ∇ t ◦∇ ( t, x, y ) and by k ∆ ( t, x, y )the smooth kernel of the heat operators e − t ( ∇ t ◦∇ ) F and e − t ∆ F respectively. Corollary 4.3.
Let
V, E, h and g as in the statement of Theorem 4.2. Consider the Bochner Laplacian ∇ t ◦ ∇ : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) . Let ( ∇ t ◦ ∇ ) F : L (reg( V ) , E ) → L (reg( V ) , E ) (67) be its Friedrich extension. Then e − t ( ∇ t ◦∇ ) F : L (reg( V ) , E ) −→ L (reg( V ) , E ) (68) is a trace class operator; its pointwise trace and its trace satisfy respectively the following inequalities: tr( k ∇ t ◦∇ ( t, x, x )) ≤ m (4 πt ) − v (1 + 4 v ( v + 1)6 t + O ( t )) (69)Tr( e − t ( ∇ t ◦∇ ) F ) ≤ m (4 πt ) − v (cid:18) vol g (reg( V ))(1 + 4 v ( v + 1)6 t ) + O ( t ) (cid:19) (70) for t ∈ (0 , . Furthermore (67) is a discrete operator and its sequence of eigenvalues, { λ j } , satisfies thefollowing asymptotic inequality: λ j ≥ (cid:18) (2 π ) v jω v m vol g (reg( V )) (cid:19) v (71) as j → ∞ where ω v is the volume of the unit v -ball in R v . Finally a core domain for (67) is given by { s ∈ C ∞ (reg( V ) , E ) ∩ L (reg( V ) , E ) , ∇ s ∈ L (reg( V ) , T ∗ reg( V ) ⊗ E ) , ∇ t ( ∇ s ) ∈ L (reg( V ) , E ) } . (72) The last statement is equivalent to say that ∇ t ◦ ∇ , with domain given by (72) , is essentially self-adjoint. roof. The assertion (68)–(71) follow by Theorem 4.2 and Corollary 4.2. For (72) we have ( ∇ t ◦ ∇ ) F = ∇ t max ◦ ∇ min by Prop. 2.2. Moreover, by Prop. 1.1, we know that C ∞ (reg( V ) , E ) ∩ D ( ∇ t max ◦ ∇ min ) is densein D ( ∇ t max ◦ ∇ min ) with respect to its graph norm and by Theorem 4.1 we know that ∇ max = ∇ min andtherefore ∇ t max = ∇ t min . All together these propositions imply immediately (72). Finally we point out thatthe discreteness of (67) follows already by Theorem 4.1 when v >
1. In fact we know that the inclusion D ( ∇ max ) (cid:44) → L (reg( V ) , E ) is compact where D ( ∇ max ) is endowed with its graph norm. Moreover, as showedin the proof of Prop. 1.1, the inclusion D (( ∇ t ◦ ∇ ) F ) (cid:44) → D ( ∇ max ) is continuous where again each domain isendowed with its graph norm. Therefore the inclusion D (( ∇ t ◦ ∇ ) F ) (cid:44) → L (reg( V ) , E ) is a compact operatorand this is well known to be equivalent to the discreteness of ( ∇ t ◦ ∇ ) F : L (reg( V ) , E ) → L (reg( V ) , E ). Proposition 4.4.
Under the assumptions of Theorem 4.2. Assume that the complex dimension of V satisfies v > . Let k P ( t, x, y ) and (cid:107) k P ( t, x, y ) (cid:107) h, op be as in the proof of Theorem 3.5. Then the following inequalityholds for < t < : (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ Ce − tc t − v . (73) This implies that1. e − tP F is a ultracontractive operator for each < t < . This means, see [42], that for each < t < there exists C t > such that (cid:107) e − tP F s (cid:107) L ∞ (reg( V ) ,E ) ≤ C t (cid:107) s (cid:107) L (reg( V ) ,E ) for each s ∈ L (reg( V ) , E ) . In particular, for each < t < , e − tP F : L (reg( V ) , E ) → L ∞ (reg( V ) , E ) iscontinuous.2. If s is an eigensection of P F : L (reg( V ) , E ) → L (reg( V ) , E ) then s ∈ L ∞ (reg( V ) , E ) .Proof. This follows by Prop. 3.6 and by Theorem 4.2.With the next result we extend Cor. 5.5 of [31] to our setting. We refer to [24] for the notion of degree of acomplex projective variety.
Theorem 4.3.
There exists a positive constant γ = γ ( d, n, m ) , that is γ depends only on the dimension ofthe ambient space CP n , on the degree d and on the rank m , such that for every irreducible complex projectivevariety V ⊂ CP n of degree d , for every vector bundle E on reg( V ) of rank m endowed with an arbitrary metric h and for every Schr¨odinger type operator P : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) as in Theorem 4.2 with L ≥ ,the ( md ) -th eingenvalue of P F , that is λ md , satisfies the following inequality: < γ ≤ λ md . (74) Proof.
As first step we show that λ md (cid:54) = 0. By (23) we have a continuous inclusion D ( P F ) (cid:44) → D ( ∇ min ) whereeach domain is endowed with the corresponding graph norm. Consider now s ∈ ker( P F ). Then, by Prop.4.4 and by the fact that P is elliptic, we have s ∈ L ∞ (reg( V ) , E ) ∩ C ∞ (reg( V ) , E ). We want to show that s ∈ ker( ∇ min ). Let { φ k } k ∈ N be a sequence as in Prop. 4.2. Then we have:0 = P F s = (cid:104) P F s, s (cid:105) L (reg( V ) ,E ) = lim k →∞ (cid:104) P F s, φ k s (cid:105) L (reg( V ) ,E ) = lim k →∞ (cid:104)∇ t ( ∇ s ) + Ls, φ k s (cid:105) L (reg( V ) ,E ) (75)= lim k →∞ (cid:104)∇ s, ∇ ( φ k s ) (cid:105) L (reg( V ) ,E ) + lim k →∞ (cid:104) L ( φ k s ) , φ k s (cid:105) L (reg( V ) ,E ) ≥ lim k →∞ (cid:104)∇ s, ∇ ( φ k s ) (cid:105) L (reg( V ) ,E ) = lim k →∞ (cid:104)∇ s, φ k ∇ s (cid:105) L (reg( V ) ,E ) + lim k →∞ (cid:104)∇ s, φ k ( d , min φ k ) ⊗ s (cid:105) L (reg( V ) ,E ) = (cid:107)∇ s (cid:107) L (reg( V ) ,T ∗ reg( V ) ⊗ E ) . In the computations of the limits above we used the Dominate convergence Theorem to deduce thatlim k →∞ (cid:104)∇ s, φ k ∇ s (cid:105) L (reg( V ) ,T ∗ reg( V ) ⊗ E ) = (cid:107)∇ s (cid:107) L (reg( V ) ,T ∗ reg( V ) ⊗ E ) and the inequality (cid:104)∇ s, φ k ( d , min φ k ) ⊗ s ) (cid:105) L (reg( V ) ,T ∗ reg( V ) ⊗ E ) ≤ (cid:107)∇ s (cid:107) L (reg( V ) ,T ∗ reg( V ) ⊗ E ) (cid:107) s (cid:107) L ∞ (reg( V ) ,E ) (cid:107) d , min φ k (cid:107) L Ω (reg( V ) ,g ) to deduce that lim k →∞ (cid:104)∇ s, φ k ( d , min φ k ) ⊗ s (cid:105) L (reg( V ) ,E ) = 0 .
24y (75) we can thus conclude that ∇ s = 0. This tells us that ker( P F ) is made of parallel sections and this inturn implies that dim(ker( P F )) ≤ m . Hence for the eigenvalues of P F we have0 ≤ λ ≤ λ ≤ ... ≤ λ m − ≤ λ m − < λ m ≤ ... and so we proved that λ md (cid:54) = 0.Now we proceed adapting to our context the proof given in [31], Cor. 5.5. Let g F S be the Fubini-Study metricon CP n . Let us label by ˜ k ( t, x, y ) the heat kernel of the unique self-adjoint extension of the Laplacian ∆ acting on C ∞ ( CP n ) that, with a little abuse of notation, we still label by ∆ : L ( CP n , g F S ) → L ( CP n , g F S ).If we denote by r : CP n × CP n → R the distance function induced by the Fubini-Study metric then weknow that ˜ k ( t, x, y ) = ˜ k ( t, r ( x, y )), see [31]. Therefore, by Theorem 4.2 and Theorem 2.1 in [31], we get thattr( e − tP F ) ≤ me − tc ˜ k ( t, e − t ∆ : L ( CP n , g F S ) → L ( CP n , g F S ), we havevol
F S ( CP n )˜ k ( t,
0) = (cid:90) CP n ˜ k ( t,
0) dvol g FS = Tr( e − t ∆ ) = ∞ (cid:88) j =0 e − tµ j ≤ (1 + b ( n ) t − n ) (76)where b ( n ) is a positive constant depending only on n , µ j is the ( j + 1)-th eigenvalue of ∆ : L ( CP n , g F S ) → L ( CP n , g F S ) and vol
F S ( CP n ) is the volume of CP n with respect to the Fubini-Study metric. In particular thelast inequality in (76) comes from the asymptotic expansion of ˜ k ( t,
0) and holds for t ∈ (0 , e − tP F ) and ˜ k ( t,
0) over reg( V ), we have:Tr( e − tP F ) ≤ me − tc vol g (reg( V ))˜ k ( t,
0) = me − tc vol g (reg( V ))vol F S ( CP n ) (1 + ∞ (cid:88) j =1 e − tµ j ) ≤ me − tc vol g (reg( V ))vol F S ( CP n ) (1 + b ( n ) t − n )and finally, using the fact that vol g (reg( V )) = d vol F S ( CP n ) , we get q + ∞ (cid:88) i = q e − tλ j ≤ me − tc d (1 + b ( n ) t − n ) (77)where λ j is the ( j + 1)-th eigenvalue of P F and q := dim(ker( P F )). Clearly we have λ md ≥ λ j for md ≥ j . Thisin turn implies that e − tλ md ≤ e − tλ j for every j ≤ md and therefore ( md − q + 1) e − tλ md ≤ (cid:80) mdj = q e − tλ j . In thisway for every 0 < α < λ md , performing the substitution t = αλ md , we get from (77) the following inequality q + ( md − q + 1) e − α ≤ me − tc d (cid:32) b ( n ) (cid:18) αλ md (cid:19) − n (cid:33) ≤ md (cid:32) b ( n ) (cid:18) αλ md (cid:19) − n (cid:33) (78)and therefore α (cid:18) q − md + ( md − q + 1) e − α mdb ( n ) (cid:19) n ≤ λ md which in turn implies α (cid:18) − md + ( md + 1) e − α mdb ( n ) (cid:19) n ≤ λ md . (79)Clearly, choosing α sufficiently small, we have − md + ( md + 1) e − α > γ := α (cid:18) − md + ( md + 1) e − α mdb ( n ) (cid:19) n . As an immediate consequence of Theorem 4.3 we have the following corollary. In the case N = C we getCor. 5.5 in [31]. Corollary 4.4.
There exists a positive constant γ (cid:48) = γ (cid:48) ( d, n ) , that is γ (cid:48) depends only on the dimension of theambient space CP n and on the degree d , such that for every irreducible complex projective variety V ⊂ CP n of degree d , for every Hermitian line bundle ( N, h ) on reg( V ) and for every Schr¨odinger type operator P : C ∞ c (reg( V ) , N ) → C ∞ c (reg( V ) , N ) as in Theorem 4.2 with L ≥ we have the following uniform lower boundfor λ d , the d -th eigenvalue of P F : < γ (cid:48) ≤ λ d . (80) see for instance [31] pag. 876 or [45] pag. 97 Corollary 4.5.
Let V , E , g and h be as in Theorem 4.1. Assume that E is a Clifford module. Let D = ˜ c ◦ ∇ D : C ∞ c (reg( V ) , E ) → C ∞ c (reg( V ) , E ) (81) be a Dirac operator where ∇ : C ∞ (reg( V ) , E ) → C ∞ (reg( V ) , T ∗ reg( V ) ⊗ E ) is a metric connection and ˜ c ∈ Hom( T ∗ reg( V ) ⊗ E, E ) is the bundle homomorphism induced by the Clifford multiplication. Let D be theDirac Laplacian and let L be the endomorphism of E arising in the Weitzenb¨ock decomposition formula, see[40]pag. 43–44, D = ∇ t ◦ ∇ + L. (82) Assume that there is a constant c ∈ R such that h ( Lφ, φ ) ≥ ch ( φ, φ ) for each φ ∈ C ∞ c (reg( V ) , E ) . ThenTheorem 4.2, Corollary 4.2 and Prop. 4.4 hold for e − tD , F where D , F is the Friedrich extension of D . Inparticular e − tD , F : L (reg( V ) , E ) → L (reg( V ) , E ) is a trace class operator and D , F : L (reg( V ) , E ) → L (reg( V ) , E ) is a discrete operator. As application of the previous theorem we have the following corollary.
Corollary 4.6.
Let V and g be as in Theorem 4.1. Assume that reg( V ) is a spin manifold and assume moreoverthat s g , the scalar curvature of g , satisfies s g ≥ c for some c ∈ R . Let Σ be the spinor bundle on reg( V ) and let ð : C ∞ c (reg( V ) , Σ) → C ∞ c (reg( V ) , Σ) (83) be the associated spin Dirac operator. Then Theorem 4.2, Corollary 4.2 and Prop. 4.4 hold for e − t ð , F : L (reg( V ) , Σ) → L (reg( V ) , Σ) . (84) In particular (84) is a trace class operator and ð : L (reg( V ) , Σ) → L (reg( V ) , Σ) is a discrete operator.Proof. It is a consequence of the Lichnerowicz formula, ð = ∇ t ◦ ∇ + s g , see for instance [30] pag. 160. Corollary 4.7.
Let V ⊂ CP n be an irreducible complex projective variety of complex dimension v . Let g bethe K¨ahler metric on reg( V ) induced by the Fubini Study metric of CP n . Let k ∈ { , ..., v } and consider theBochner-Weitzenb¨ock identity for the Laplacian ∆ k : Ω kc (reg( V )) → Ω kc (reg( V )) , see [30] pag. 155 or [40] pag.43–44, ∆ k = ∇ tk ◦ ∇ k + L k , (85) where ∇ k : Ω k (reg( V )) → C ∞ (reg( V ) , T ∗ reg( V ) ⊗ Λ k T ∗ reg( V )) is the metric connection induced by the LeviCivita conneection. Assume that there is a constant c such that (cid:104) L k η, η (cid:105) g k ≥ c (cid:104) η, η (cid:105) g k for each η ∈ Ω kc (reg( V )) .Let ∆ F k be the Friedrich extension of ∆ k and let e − t ∆ F k : L Ω k (reg( V ) , g ) → L Ω k (reg( V ) , g ) (86) be the heat operator associated to ∆ F k . Then (86) is a trace class operator. In particular Theorem 4.2, Cor. 4.2and Prop. 4.4 hold for (86) .Consider now the Hodge-Kodaira Laplacian ∆ p,q,∂ : Ω p,qc (reg( V )) → Ω p,qc (reg( V )) such that p + q = k . Let ∆ F p,q,∂ be the Friedrich extension of ∆ p,q,∂ and let e − t ∆ F p,q,∂ : L Ω p,q (reg( V ) , g ) → L Ω p,q (reg( V ) , g ) (87) be the heat operator associated to ∆ F p,q,∂ . Then (87) is a trace class operator. As in the previous case Theorem4.2, Cor. 4.2 and Prop. 4.4 hold for (87) .Proof. The first part of the theorem, that is the one concerning with ∆ F k , is an immediate application ofTheorem 4.2, Cor. 4.2 and Prop. 4.4. The second part follows by the fact that ∆ k = 2 (cid:76) p + q = k ∆ p,q,∂ and that,see for instance [4] pag. 169, ∆ F k = 2 (cid:76) p + q = k ∆ F p,q,∂ . 26 Applications to stratified pseudomanifolds
This last section contains applications concerning Thom-Mather stratified pseudomanifolds. We start recallingbriefly the basic definitions and properties. We first recall that, given a topological space Z , C ( Z ) stands forthe cone over Z that is Z × [0 , / ∼ where ( p, t ) ∼ ( q, r ) if and only if r = t = 0. Definition 5.1.
A smoothly Thom-Mather stratified pseudomanifold X of dimension m is a metrizable, locallycompact, second countable space which admits a locally finite decomposition into a union of locally closed strata G = { Y α } , where each Y α is a smooth, open and connected manifold, with dimension depending on the index α .We assume the following:(i) If Y α , Y β ∈ G and Y α ∩ Y β (cid:54) = ∅ then Y α ⊂ Y β (ii) Each stratum Y is endowed with a set of control data T Y , π Y and ρ Y ; here T Y is a neighborhood of Y in X which retracts onto Y , π Y : T Y → Y is a fixed continuous retraction and ρ Y : T Y → [0 , is acontinuous function in this tubular neighborhood such that ρ − Y (0) = Y . Furthermore, we require that if Z ∈ G and Z ∩ T Y (cid:54) = ∅ then ( π Y , ρ Y ) : T Y ∩ Z → Y × [0 , is a proper smooth submersion.(iii) If W, Y, Z ∈ G , and if p ∈ T Y ∩ T Z ∩ W and π Z ( p ) ∈ T Y ∩ Z then π Y ( π Z ( p )) = π Y ( p ) and ρ Y ( π Z ( p )) = ρ Y ( p ) .(iv) If Y, Z ∈ G , then Y ∩ Z (cid:54) = ∅ ⇔ T Y ∩ Z (cid:54) = ∅ , T Y ∩ T Z (cid:54) = ∅ ⇔ Y ⊂ Z, Y = Z or Z ⊂ Y . (v) For each Y ∈ G , the restriction π Y : T Y → Y is a locally trivial fibration with fibre the cone C ( L Y ) over some other stratified space L Y (called the link over Y ), with atlas U Y = { ( φ, U ) } where each φ is a trivialization π − Y ( U ) → U × C ( L Y ) , and the transition functions are stratified isomorphisms whichpreserve the rays of each conic fibre as well as the radial variable ρ Y itself, hence are suspensions ofisomorphisms of each link L Y which vary smoothly with the variable y ∈ U .(vi) For each j let X j be the union of all strata of dimension less or equal than j , then X m − = X m − and X \ X m − dense in X The depth of a stratum Y is largest integer k such that there is a chain of strata Y = Y k , ..., Y such that Y j ⊂ Y j − for 1 ≤ j ≤ k. A stratum of maximal depth is always a closed subset of X . The maximal depth ofany stratum in X is called the depth of X as stratified spaces. Consider the filtration X = X m ⊃ X m − = X m − ⊃ X m − ⊃ ... ⊃ X . (88)We refer to the open subset X \ X m − of a smoothly Thom-Mather-stratified pseudomanifold X as its regularset, and the union of all other strata as the singular set,reg( X ) := X \ sing( X ) where sing( X ) := (cid:91) Y ∈ G , depth( Y ) > Y. Given two Thom-Mather smoothly stratified pseudomanifolds X and X (cid:48) , a stratified isomorphism between themis a homeomorphism F : X → X (cid:48) which carries the open strata of X to the open strata of X (cid:48) diffeomorphically,and such that π (cid:48) F ( Y ) ◦ F = F ◦ π Y , ρ (cid:48) F ( Y ) ◦ F = ρ Y for all Y ∈ G ( X ). For more details, properties and commentswe refer to [2], [10], [11] and the bibliography cited there. Here we point out that a large class of topologicalspace such as irreducible complex analytic spaces or quotient of manifolds through a proper Lie group actionbelong to this class of spaces.As a next step we introduce the class of smooth Riemmanian metrics on reg( X ) which we are interested in.The definition is given by induction on the depth of X . We label by ˆ c := ( c , ..., c m ) a ( m − Definition 5.2.
Let X be a smoothly Thom-Mather-stratified pseudomanifold and let g be a Riemannian metricon reg( X ) . If depth( X ) = 0 , that is X is a smooth manifold, a ˆ c -iterated edge metric is understood to be anysmooth Riemannian metric on X . Suppose now that depth( X ) = k and that the definition of ˆ c -iterated edgemetric is given in the case depth( X ) ≤ k − ; then we call a smooth Riemannian metric g on reg( X ) a ˆ c -iteratededge metric if it satisfies the following properties: Let Y be a stratum of X such that Y ⊂ X i \ X i − ; by Def. 5.1 for each q ∈ Y there exist an openneighbourhood U of q in Y such that φ : π − Y ( U ) −→ U × C ( L Y ) is a stratified isomorphism; in particular, φ : π − Y ( U ) ∩ reg( X ) −→ U × reg( C ( L Y )) is a smooth diffeomorphism. Then, for each q ∈ Y , there exists one of these trivializations ( φ, U ) suchthat g restricted on π − Y ( U ) ∩ reg( X ) satisfies the following properties: ( φ − ) ∗ ( g | π − Y ( U ) ∩ reg( X ) ) ∼ dr + h U + r c m − i g L Y (89) where m is the dimension of X , h U is a Riemannian metric defined over U and g L Y is a ( c , ..., c m − i − ) -iterated edge metric on reg( L Y ) , dr + h U + r c m − i g L Y is a Riemannian metric of product type on U × reg( C ( L Y )) and with ∼ we mean quasi-isometric. We remark that in (89) the neighborhood U can be chosen sufficiently small so that it is diffeomorphic to(0 , i and h U it is quasi-isometric to the Euclidean metric restricted on (0 , i . Moreover we point out thatwith this kind of Riemannian metrics we have vol g (reg( X )) < ∞ in case X is compact. There is the followingnontrivial existence result: Proposition 5.1.
Let X be a smoothly Thom-Mather stratified pseudomanifold of dimension m . For any ( m − -tuple of positive numbers ˆ c = ( c , ..., c m ) , there exists a smooth Riemannian metric on reg( X ) which isa ˆ c -iterated edge metric.Proof. See [10] or [2] in the case ˆ c = (1 , ..., , ..., c = (1 , ...,
1) we will call this kind of metrics simply iterated edge metric .The importance of this class of metrics lies on its deep connection with the topology of X . In fact, as pointed outby Cheeger in his seminal paper [19] (see also [5] and the bibliography cited there for further developments ) the L -cohomology of reg( X ) associated to an iterated edge metric is isomorphic to the intersection cohomology of X associated with a perversity depending only on ˆ c . In other words the L -cohomology of these kind of metrics(which a priori is an object that lives only on reg( X )) provides non trivial topological informations of the wholespace X .Now we have the following proposition which assure the existence of a suitable sequence of cut-off functions. Proposition 5.2.
Let X be a compact, smoothly Thom-Mather stratified pseudomanifold of dimension m .Consider on reg( X ) an iterated edge metric g . Then there exists a sequence of Lipschitz functions with compactsupport contained in reg( X ) , { φ j } j ∈ N , such that • ≤ φ j ≤ for each j . • φ j → pointwise. • φ j ∈ D ( d , min ) for each j ∈ N and lim j →∞ (cid:107) d , min φ j (cid:107) L Ω (reg( V ) ,g ) = 0 .In particular ∈ D ( d , min ) .Proof. See [6].
Theorem 5.1.
Let X be a compact, smoothly Thom-Mather stratified pseudomanifold of dimension m . Con-sider on reg( X ) an iterated edge metric g . Let E be a vector bundle over reg( X ) and let h be a metricon E , Riemannian if E is a real vector bundle, Hermitian if E is a complex vector bundle. Finally let ∇ : C ∞ (reg( X ) , E ) → C ∞ (reg( V ) , T ∗ reg( X ) ⊗ E ) be a metric connection. We have the following proper-ties: • W , (reg( X ) , E ) = W , (reg( X ) , E ) . • Assume that m > . Then there exists a continuous inclusion W , (reg( X ) , E ) (cid:44) → L mm − (reg( X ) , E ) . • Assume that m > . Then the inclusion W , (reg( X ) , E ) (cid:44) → L (reg( X ) , E ) is a compact operator. roof. The first point follows by Prop. 5.2 and by Prop. 3.1. The continuous inclusion W , (reg( X ) , g ) (cid:44) → L mm − (reg( M ) , g ) is established in [1] Prop. 2.2. By the first point of this theorem we know that W , (reg( X ) , g ) = W , (reg( X ) , g ) and therefore we have the continuous inclusion W , (reg( X ) , g ) (cid:44) → L mm − (reg( X ) , g ). Now, us-ing Prop. 2.1, we get the continuous inclusion C ∞ (reg( X ) , E ) ∩ W , (reg( X ) , E ) (cid:44) → L mm − (reg( X ) , E ) . Finally, by the density of C ∞ (reg( X ) , E ) ∩ W , (reg( X ) , E ) in W , (reg( X ) , E ), see Prop. 1.2, the continuousinclusion W , (reg( X ) , E ) (cid:44) → L mm − (reg( X ) , E ) is established. Finally the third point is a consequence of thesecond point and Prop. 3.3. Corollary 5.1.
Under the assumptions of Theorem 5.1. Then im( ∇ min ) = im( ∇ max ) is a closed subspace of L (reg( X ) , T ∗ reg( X ) ⊗ E ) .Proof. According to Prop. 5.1 we know that ∇ max = ∇ min and therefore im( ∇ max ) = im( ∇ min ). Now thethesis follows by Cor. 3.1.We have also the following application. Proposition 5.3.
Let (reg( X ) , g ) be as in Theorem 5.1. Let E and F be two vector bundles over reg( X ) endowed respectively with metrics h and ρ , Riemannian if E and F are real vector bundles, Hermitian if E and F are complex vector bundles. Finally let ∇ : C ∞ (reg( X ) , E ) → C ∞ (reg( X ) , T ∗ reg( X ) ⊗ E ) be a metricconnection. Consider a first order differential operator of this type: D := θ ◦ ∇ : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , F ) (90) where θ ∈ C ∞ (reg( X ) , Hom( T ∗ reg( X ) ⊗ E, F )) . Assume that θ extends as a bounded operator θ : L (reg( X ) , T ∗ reg( X ) ⊗ E ) → L (reg( X ) , F ) . Then we have the following inclusion: D ( D max ) ∩ L ∞ (reg( X ) , E ) ⊂ D ( D min ) . (91) In particular (91) holds when D is the de Rham differential d k : Ω kc (reg( X )) → Ω k +1 c (reg( X )) or a Dirac typeoperator D : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ) .Proof. This follows by Theorem 5.1 and Prop. 3.2.Finally consider again the setting of Theorem 5.1. The remaining part of this section collects applications tosome Schr¨odinger type operators ∇ t ◦ ∇ + L (92)where ∇ : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , T ∗ M ⊗ E ) is a metric connection, ∇ t : C ∞ c (reg( X ) , T ∗ M ⊗ E ) → C ∞ c (reg( X ) , E ) is the formal adjoint of ∇ and L ∈ C ∞ (reg( X ) , End( E )) is a bundle homomorphism. Theorem 5.2.
Let X , E , g , h , and ∇ be as described above. Let P := ∇ t ◦ ∇ + L, P : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ) be a Schr¨odinger type operator with L ∈ C ∞ (reg( X ) , End( E )) . Assume that: • P is symmetric and positive. • There is a constant c ∈ R such that, for each s ∈ C ∞ (reg( X ) , E ) , we have h ( Ls, s ) ≥ ch ( s, s ) . Let P F : L (reg( X ) , E ) → L (reg( X ) , E ) be the Friedrich extension of P and let ∆ F : L (reg( X ) , g ) → L (reg( X ) , g ) be the Friedrich extension of ∆ : C ∞ c (reg( X )) → C ∞ c (reg( X )) . Then the heat operator associatedto P F e − tP F : L (reg( X ) , E ) −→ L (reg( X ) , E ) is a trace class operator and its trace satisfies the following inequality: Tr( e − tP F ) ≤ re − tc Tr( e − t ∆ F ) (93) where r is the rank of the vector bundle E . roof. This follows by Prop. 3.5 and by Theorem 5.1.Analogously to the previous section we have now the following corollaries.
Corollary 5.2.
Under the assumptions of Theorem 5.2. For each t ∈ (0 , we have the following inequalitiesfor the pointwise trace and for the heat trace of e − tP F respectively: tr( k P ( t, x, x )) ≤ rCe − tc t − m Tr( e − tP F ) ≤ rCe − tc vol g (reg( X )) t − m where C is the positive constant arising from Prop. 3.4. The operator P F : L (reg( X ) , E ) → L (reg( X ) , E ) isa discrete operator. If we label its eigenvalues with ≤ λ ≤ λ ≤ ... ≤ λ n ≤ ... then there exists a positive constant K such that we have the following asymptotic inequality λ j ≥ Kj m + c as j → ∞ .Proof. This follows by Cor. 3.2.
Corollary 5.3.
Let
X, E, h and g as in the statement of Theorem 5.2. Consider the Bochner Laplacian ∇ t ◦ ∇ : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ) . Let ( ∇ t ◦ ∇ ) F : L (reg( X ) , E ) → L (reg( X ) , E ) (94) be its Friedrich extension. Then e − t ( ∇ t ◦∇ ) F : L (reg( X ) , E ) −→ L (reg( X ) , E ) (95) is a trace class operator; for every t ∈ (0 , its pointwise trace and its trace satisfy the following inequalities: tr( k ∇ t ◦∇ ( t, x, x )) ≤ rCt − m (96)Tr( e − t ( ∇ t ◦∇ ) F ) ≤ rC vol g (reg( X )) t − m (97) where C is the positive constant arising from Prop. 3.4. (94) is a discrete operator and its sequence of eigen-values, { λ j } , satisfies the following asymptotic inequality: λ j ≥ Kj m (98) as j → ∞ where K is a positive constant. Finally a core domain for (94) is given by { s ∈ C ∞ (reg( X ) , E ) ∩ L (reg( X ) , E ) : ∇ s ∈ L (reg( X ) , T ∗ reg( X ) ⊗ E ) and ∇ t ( ∇ s ) ∈ L (reg( X ) , E ) } . (99) The last statement is equivalent to say that ∇ t ◦ ∇ , with domain given by (99) , is essentially self-adjoint.Proof. The proof of (99) is analogous to that we have provided for (72). The remaining points are consequencesof Cor. 5.2.
Proposition 5.4.
Under the assumptions of Theorem 5.2. Assume that m > . Let k P ( t, x, y ) and (cid:107) k P ( t, x, y ) (cid:107) h, op be as in the proof of Theorem 3.5. Then the following inequality holds for < t < : (cid:107) k P ( t, x, y ) (cid:107) h, op ≤ Ce − tc t − m . (100) This implies that: . e − tP F is a ultracontractive operator for each < t < . This means, see [42], that for each < t < there exists C t > such that (cid:107) e − tP F s (cid:107) L ∞ (reg( X ) ,E ) ≤ C t (cid:107) s (cid:107) L (reg( X ) ,E ) for each s ∈ L (reg( X ) , E ) . In particular, for each < t < , e − tP F : L (reg( V ) , E ) → L ∞ (reg( X ) , E ) iscontinuous.2. If s is an eigensection of P F : L (reg( X ) , E ) → L (reg( X ) , E ) then s ∈ L ∞ (reg( X ) , E ) .Proof. This follows by Prop. 3.6 and by Theorem 5.2.Analogously to the previous section we have the following applications to Dirac operators.
Corollary 5.4.
Let X , E , g and h be as in Theorem 5.1. Assume that E is a Clifford module. Let D = ˜ c ◦ ∇ D : C ∞ c (reg( X ) , E ) → C ∞ c (reg( X ) , E ) (101) be a Dirac operator where ∇ : C ∞ (reg( X ) , E ) → C ∞ (reg( X ) , T ∗ reg( X ) ⊗ E ) is a metric connection and ˜ c ∈ Hom( T ∗ reg( X ) ⊗ E, E ) is the bundle homomorphism induced by the Clifford multiplication. Let D be theDirac Laplacian and let L be the endomorphism of E arising in the Weitzenb¨ock decomposition formula, see[40]pag. 43–44, D = ∇ t ◦ ∇ + L. (102) Assume that there is a constant c ∈ R such that h ( Lφ, φ ) ≥ ch ( φ, φ ) for each φ ∈ C ∞ c (reg( V ) , E ) . ThenTheorem 5.2, Corollary 5.2 and Prop. 5.4 hold for e − tD , F where D , F is the Friedrich extension of D . Inparticular e − tD , F : L (reg( X ) , E ) → L (reg( X ) , E ) is a trace class operator and D , F : L (reg( X ) , E ) → L (reg( X ) , E ) is a discrete operator. Corollary 5.5.
Let X and g be as in Theorem 5.1. Assume that reg( X ) is a spin manifold and assumemoreover that s g , the scalar curvature of g , satisfies s g ≥ c for some c ∈ R . Let Σ be the spinor bundle on reg( X ) and let ð : C ∞ c (reg( X ) , Σ) → C ∞ c (reg( X ) , Σ) (103) be the associated spin Dirac operator. Then Theorem 5.2, Corollary 5.2 and Prop. 5.4 hold for e − t ð , F : L (reg( X ) , Σ) → L (reg( X ) , Σ) . (104) In particular (104) is a trace class operator and ð : L (reg( X ) , Σ) → L (reg( X ) , Σ) is a discrete operator. Corollary 5.6.
Let X and g be as in Theorem 5.1. Let k ∈ { , ..., m } and consider the Bochner-Weitzenb¨ockidentity for the Laplacian ∆ k : Ω kc (reg( X )) → Ω kc (reg( X )) , see [30] pag. 155 or [40] pag. 43–44, ∆ k = ∇ tk ◦ ∇ k + L k , (105) where ∇ k : Ω k (reg( X )) → C ∞ (reg( X ) , T ∗ reg( X ) ⊗ Λ k T ∗ reg( X )) is the metric connection induced by the LeviCivita connection. Assume that there is a constant c such that (cid:104) L k η, η (cid:105) g k ≥ c (cid:104) η, η (cid:105) g k for each η ∈ Ω kc (reg( X )) .Let ∆ F k be the Friedrich extension of ∆ k and let e − t ∆ F k : L Ω k (reg( X ) , g ) → L Ω k (reg( X ) , g ) (106) be the heat operator associated to ∆ F k . Then (106) is a trace class operator. In particular Theorem 5.2, Cor.5.2 and Prop. 5.4 hold for (86) . eferences [1] K. Akutagawa, G. Carron, R. Mazzeo. The Yamabe problem on stratified spaces. Geom. Funct. Anal.
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