Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition
aa r X i v : . [ m a t h . A P ] D ec PROPAGATION OF SINGULARITIES ON ADS SPACETIMES FORGENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHICHADAMARD CONDITION
ORAN GANNOT AND MICHA L WROCHNA
Abstract.
We consider the Klein-Gordon equation on asymptotically anti-de Sitterspacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions, andprove propagation of singularities along generalized broken bicharacteristics. Theresult is formulated in terms of conormal regularity relative to a twisted Sobolevspace. We use this to show the uniqueness, modulo regularising terms, of parametriceswith prescribed b-wavefront set. Furthermore, in the context of quantum fields,we show a similar result for two-point functions satisfying a holographic Hadamardcondition on the b-wavefront set. Introduction & main results
The Klein-Gordon equation on asymptotically anti-de Sitter (aAdS) spacetimes wasstudied in a number of works in the last several years. We refer the reader to [YG,Gal, Bac, Hol, War1, HS3, EK, HW, HS1, HS2] to mention only a few. Notably,the results include well-posedness for the Klein–Gordon equation by Vasy [Vas4] andHolzegel [Hol] in the case of Dirichlet boundary conditions, as well as well-posednessfor Neumann and Robin boundary conditions by Warnick [War1].In applications to Quantum Field Theory the main objects of interest are propaga-tors , which are singular distributions in the two spacetime variables. The key additionalingredient that is needed is a microlocal propagation of singularities theorem. In thecase of Dirichlet boundary conditions, this was established by Vasy [Vas4], and appliedin [Wro] to yield a result on distinguished parametrices largely analogous to that ofDuistermaat and H¨ormander [DH].The goal of the present paper is to provide these type of theorems in the case ofNeumann and Robin boundary conditions on the boundary ∂X of an aAdS space-time ( X, g ). These boundary conditions appear frequently in the physics literature;
Acknowledgments.
OG was partially supported by NSF grant DMS1502632. MW gratefully ac-knowledges support from the grant ANR-16-CE40-0012-01. The authors are also grateful to Andr´asVasy and Claude Warnick for helpful conversations. see e.g. [DF2, DFM]. Further motivation comes from the study of the Dirichlet-to-Neumann operator, which was recently shown to coincide with a power of the waveoperator in the case of a static metric [EdMGV].The main difficulties are two-fold. First of all, the positive commutator estimates in[Vas4] have no direct generalization outside of the Dirichlet case, for the same reasonthat the associated energy is ill-defined for Neumann and Robin conditions. Secondly,the boundary conditions must be understood in terms of weighted traces γ ± related tothe polyhomogeneous expansions of solutions to the Klein–Gordon equation near theboundary (the corresponding elliptic setting is well-understood thanks to [Maz, MV]and [Gan]).In [Vas4], Vasy proves propagation of conormal regularity relative to a scale of 0-Sobolev spaces whose weights (or lack thereof) correspond directly to the form domainof the wave operator. Unless one introduces additional weights, it is not possible topose the Neumann or Robin problems in these spaces. The use of weighted 0-Sobolevspaces in turn makes it difficult to apply the quadratic form techniques first developedin [Vas2] (and subsequently applied applied in [MVW, Vas3, Vas4]). To circumventthese problems we adopt an approach based on microlocalizing a certain twisted Sobolevspace H ( X ) ⊂ x − L ( X, dg ), introduced in the present context by Warnick [War1].We show that b-pseudodifferential operators have good mapping properties on theseSobolev spaces. This allows us to consider a b -wavefront set WF ,s b ( u ) which microlo-calizes the space H ,s loc ( X ) of conormal distributions of order s with respect to H ( X )(i.e., the subspace of H ( X ) stable under applications of at most s vector fields tan-gent to the boundary).On an aAdS spacetime ( X, g ) of dimension n ≥ P = (cid:3) g − ( n − + ν , ν > . (1.1)The condition ν > ν ± = n − ± ν denote the indicial roots of P . The definition of H ( X )(which depends on the Klein–Gordon parameter ν ) is based on stability under firstorder differential operators Q on X ◦ which are twisted in the sense that x − ν − Qx ν − is smooth up to the boundary for any boundary defining function (bdf) x . In otherwords, we work with the largest space of first-order differential operators preserving x ν − C ∞ ( X ). This is motivated by the following observation: if F ∈ x ν − C ∞ ( X ) is anyfunction satisfying F − P ( F ) ∈ x C ∞ ( X ) , (1.2)then one has P = − ( F − D z i F ) † g ij ( F D z j F − ) + F − P ( F ) , ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 3 where † refers to the dg -adjoint. Following the terminology in [HW], we call F anadmissible twisting function (the existence of admissible twisting functions is discussedin Section 4.3). Thus, modulo zeroth order terms, P is the sum of twisted derivativescomposed with their adjoints.Compared with (1.1), one gains two powers of x in the zeroth order terms, whichturns out to be crucial. This observation was first employed systematically in thestudy of AdS spacetimes by Warnick [War1], and also appeared earlier in the closelyrelated asymptotically hyperbolic setting [GNQ, CdMG]. We then associate to P aDirichlet form on H ( X ), and following the philosophy of [Vas2] carry out a positivecommutator argument at the level of quadratic forms.For x a bdf, the Dirichlet data γ − u of u is simply x − ν − u restricted to the boundary;this restriction exists as a distribution (and transforms simply under changes of bdf).The Neumann data γ + u is slightly more difficult to define; this is achieved in Sections4.4 and 4.5 for ν ∈ (0 , ν ∈ (0 , P R , corresponding to γ + u − βγ − u = 0for β ∈ C ∞ ( ∂X ) real-valued. When ν >
0, we can also consider the Dirichlet realization P D of P , corresponding to imposing γ − u = 0. We prove the following propagation ofsingularities theorem. Theorem 1 (Propagation of singularities) . Let ν ∈ (0 , . If u ∈ H ,m loc ( X ) for some m ≤ and s ∈ R ∪ { + ∞} , then WF ,s b ( u ) \ WF − ,s +1b ( P R u ) is the union of maximallyextended GBB s within the compressed characteristic set ˙ N .The same result holds for all ν > if u ∈ ˙ H ,m loc ( X ) and we consider P D instead of P R . The notions of compressed characteristic set and generalized broken bicharacteristics (or GBBs, see Definition 6.1) are defined relative to the conformally rescaled metricˆ g = x g , and are exactly the same as used to describe propagation of singularitiesfor smooth boundary value problems. Results analogous to Theorem 1 were obtainedin those settings by Melrose, Sj¨ostrand, and Taylor [Tay, MS1, Sj¨o, MS2]; cf. theworks of Lebeau [Leb] and Vasy [Vas1] for the case of manifolds with corners, andof Melrose, Vasy and Wunsch [MVW] for edge manifolds. Here, the behaviour of P at the boundary and the different nature of the boundary conditions pose particulardifficulties, which we cope with by a systematic study of continuity properties of γ ± and of the interactions of the b-pseudodifferential calculus with twisted derivatives.Theorem 1 encodes the law of reflection when a GBB from the interior (where itis just an ordinary null-bicharacteristic of g up to reparametrization) is transversallyincident upon the boundary. In the case of tangential incidence our theorem is likely ORAN GANNOT AND MICHA L WROCHNA not optimal in the sense that it does not rule out null-bicharacteristics with finite-order contact sticking to the boundary (this problem was studied in a model case byPham [Pha] for Dirichlet boundary conditions). The latter propagation phenomenonis automatically ruled out when g is a solution of the Einstein equations with negativecosmological constant, since the boundary is necessarily conformally totally geodesic(for null-geodesics, which is a conformally invariant notion).Our framework allows us to define an operatorial b-wave front set WF Opb (Λ) forcontinuous operators Λ : ˙ H − , −∞ c ( X ) → H , −∞ loc ( X ), and to study the wave front set of induced operators on the boundary , i.e. of the form γ ± Λ γ ∗± .In Quantum Field Theory, one is particularly interested in two-point functions , whichin the setting of Robin or Neumann boundary conditions are pairs of continuous op-erators Λ ± R such that P R Λ ± R = Λ ± R P R = 0 , Λ ± R ≥ , Λ + R − Λ − R = iG R , where G R is the difference between the retarded and advanced propagators (fundamen-tal solution) for P R . Following [Wro], we introduce a condition on the b-wave front setof Λ ± R which we call the holographic Hadamard condition , namely:WF Opb (Λ ± R ) ⊂ ˙ N ± × ˙ N ± . Here ˙ N ± are the future- and past-directed components of ˙ N relative to a given time-orientation (the additional global geometric hypotheses are described in Section 7.1;note that these are not needed for Theorem 1, which is purely local). We show: Theorem 2 (cf. Theorems 5 and 7) . Two-point functions Λ ± R satisfying the holographicHadamard condition exist and are unique modulo terms whose Schwartz kernels aresmooth in the interior of X . Furthermore, γ − Λ ± R γ ∗− and γ + Λ ± R γ ∗ + are unique moduloterms with smooth Schwartz kernels. This extends the results of [Wro] to Neumann and Robin boundary conditions, andthus provides the fundamental ingredients for constructing linear quantum fields andrenormalized non-linear quantities in our setting. We remark that local conditions onsingularities of two-point functions in the interior X ◦ were studied on special examplesof aAdS spacetimes by several authors [KW, BFQ, DF1]. Our holographic Hadamardcondition has, however, the advantage of giving enough information to define and studythe induced operators on the boundary, γ − Λ ± R γ ∗− and γ + Λ ± R γ ∗ + . We also stress that theproof of the existence statement in Theorem 2 crucially relies on the fact that theholographic Hadamard condition propagates well thanks to Theorem 1.Finally, we obtain a similar result on the uniqueness modulo b-regularising terms ofparametrices for P R ; see Theorem 6 for the precise statement.The paper is structured as follows. In Section 2 we recall elementary definitions andfacts on the b-pseudodifferential calculus. Section 3 introduces the weighted Sobolev ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 5 spaces and reviews continuity results for the weighted trace γ − . In Section 4 wediscuss Green’s formula for the Klein-Gordon operator and asymptotic expansions forapproximate solutions, which allow us to define the weighted trace γ + . In Section 5we introduce the Dirichlet, Neumann and Robin problems and derive some microlocalestimates. The main steps of the proof of Theorem 1 are contained in Section 6.Section 7 is devoted to propagators and their operatorial b-wave front sets, and inparticular to the proof of Theorem 2.2. b-Pseudodifferential operators Basic definitions.
In this section we briefly discuss the theory of b-pseudo-differential operators, mostly to fix the relevant notation. The presentation closelyfollows [Vas2, Sections 2, 3], where additional details and complete proofs can befound.If X is a manifold with boundary, let Ψ m b ( X ) denote the algebra of properly sup-ported b-pseudodifferential operators of order m . If k ∈ N , then Diff k b ( X ) ⊂ Ψ k b ( X ).The corresponding symbol space is S m ( b T ∗ X ), where b T ∗ X is the b-cotangent bundleover X . A priori, we consider A ∈ Ψ m b ( X ) as a continuous map A : ˙ C ∞ ( X ) → ˙ C ∞ ( X ) (2.1)which extends to a continuous endomorphism of C ∞ ( X ).The abstract sesquilinear pairing of u ∈ C −∞ ( X ) with ϕ ∈ ˙ C ∞ ( X ) (or more generallyfor the pairing between a space and its anti-dual) will be written h u, ϕ i . For theremainder of this section we fix a positive C ∞ density µ on X in order to trivialize thedensity bundle. Then u ∈ L ( X ) determines an element of C −∞ ( X ) by h u, ϕ i = Z X u · ¯ ϕ dµ As discussed below, Ψ m b ( X ) is closed under adjoints, so A ∈ Ψ m b ( X ) also extends tocontinuous endomorphisms of C −∞ ( X ) and ˙ C −∞ ( X ). The fact that the action of A on˙ C −∞ ( X ) extends that on C ∞ ( X ) comes from the fact that h Au, v i = h u, A ∗ v i for u, v ∈ C ∞ ( X ); in other words, there are no boundary terms when integrating byparts.We recall the symbol isomorphisms for b-pseudodifferential operators. There is aprincipal symbol map σ b ,m : Ψ m b ( X ) → S m ( b T ∗ X ) which descends to an isomorphism σ b ,m : (Ψ m b / Ψ m − )( X ) → ( S m /S m − )( b T ∗ X ) . The symbol map can be inverted explicitly by fixing a non-canonical quantization mapOp b : S m ( b T ∗ X ) → Ψ m b ( X ) such that σ b ,m (Op b ( A )) = a in ( S m /S m − )( b T ∗ X ). ORAN GANNOT AND MICHA L WROCHNA If A ∈ Ψ m b ( X ) and B ∈ Ψ m ′ b ( X ), then AB ∈ Ψ m + m ′ b ( X ) with principal symbol σ b ,m + m ′ ( AB ) = σ b ,m ( A ) · σ b ,m ′ ( B ) . Furthermore their commutator satisfies [
A, B ] ∈ Ψ m + m ′ − . To describe the princi-pal symbol of [ A, B ], observe that the Poisson bracket of a ∈ S m ( b T ∗ X ) and b ∈ S m ′ ( b T ∗ X ) restricted to the interior T ∗ X ◦ extends by continuity up to the bound-ary as an element of S m + m ′ − ( b T ∗ X ). In local coordinates ( x, y , . . . , y n − ) with dualb-momenta ( σ, η , . . . , η n − ), this is just the expression { a, b } = ∂ σ a · x∂ x b − x∂ x a · ∂ σ b + n − X i =1 ∂ y i a · ∂ η i b − ∂ η i a · ∂ y i b. Then σ b ,m + m ′ − ([ A, B ]) = { σ b ,m ( A ) , σ b ,m ′ ( B ) } . If A ∗ denotes the formal adjoint of A with respect to µ , then A ∗ ∈ Ψ m b ( X ), and σ b ,m ( A ∗ ) = σ b ,m ( A ). Finally, if x is a bdf,then x − s Ax s ∈ Ψ m b ( X ) for each s ∈ C , and σ b ,m ( x − s Ax s ) = σ b ,m ( A ).We will frequently use the following terminology: a continuous map ˙ C ∞ ( X ) →C −∞ ( X ) is said to be supported in U ⊂ X if its Schwartz kernel has support in U × U .Then any compactly supported operator A ∈ Ψ ( X ) defines a bounded map L ( X ) → L ( X ) . More precisely, suppose that A is supported in K for K ⊂ X compact and U ⊂ X is a neighborhood of K . Then there exists χ ∈ C ∞ c ( U ) and a compactly supported A ′ ∈ Ψ −∞ b ( X ) such that k Au k L ( X ) ≤ | σ b , ( A ) |k χu k L ( X ) + k A ′ u k L ( X ) . Since Ψ ( X ) is invariant under conjugation by powers of a bdf x , the same result istrue if L ( X ) is replaced by any weighted space x r L ( X ), where r ∈ R .2.2. Microlocalization.
We say that A ∈ Ψ m b ( X ) is elliptic at a point q ∈ b T ∗ X \ b ∈ S − m ( b T ∗ X ) such that σ b ,m ( A ) · b − ∈ S − ( b T ∗ X )in a conic neighborhood of q . The set of elliptic points of A will be written ell b ( A ) ⊂ b T ∗ X \
0. We say that A is elliptic on a conic set U ⊂ b T ∗ X \ U ⊂ ell b ( A ).Next, we define the operator b-wavefront set (or microsupport) of B ∈ Ψ m b ( X ).Following [Vas2, Section 3], it is important to give a uniform definition for boundedfamilies of operators (since b-pseudodifferential operators have conormal Schwartzkernels on a certain blow-up of X × X , there is a natural Fr´echet topology on Ψ m b ( X )which roughly corresponds to symbol seminorms. Thus it makes sense to speak ofbounded subsets of Ψ m b ( X )). ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 7 If B ⊂ Ψ m b ( X ) is bounded, we say that q / ∈ WF ′ b ( B ) if there exists A ∈ Ψ ( X ) with q ∈ ell b ( A ) such that A B is bounded in Ψ −∞ b ( X ). This agrees with the usual definitionof WF ′ b ( B ) when B = { B } consists of a single operator. For bounded families A , B wehave the usual relations WF ′ b ( A + B ) ⊂ WF ′ b ( A ) ∪ WF ′ b ( B ) , WF ′ b ( AB ) ⊂ WF ′ b ( A ) ∩ WF ′ b ( B ) . (2.2)Furthermore, for operators,WF ′ b ( A ∗ ) = WF ′ b ( A ) , WF ′ b ( x − s Ax s ) = WF ′ b ( A ) . (2.3)Next, we introduce some useful but non-standard terminology: if S ⊂ Ψ m b ( X ) is aclosed subspace, we say that a bounded linear map M : S → Ψ k b ( X ) is microlocal ifWF ′ b ( M ( A )) ⊂ WF ′ b ( A )for all A ∈ S . A typical S is the set of operators with support in a fixed compact set.Note that M necessarily preserves bounded families as well. According to (2.2), (2.3),multiplication by a fixed operator, taking adjoints, and conjugation by a bdf are allmicrolocal maps.Let A ∈ Ψ m b ( X ) and B ∈ Ψ m ′ b ( X ). If A is elliptic on WF b ( B ), then the standardsymbolic parametrix construction yields F ∈ Ψ m ′ − m b ( X ) and R, R ′ ∈ Ψ −∞ b ( X ) suchthat B = AF + R = F A + R ′ . We also need to mention the indicial family of A ∈ Ψ m b ( X ). For a fixed bdf x and v ∈ C ∞ ( X ), define b N ( A )( s ) v = x − is A ( x is u ) | ∂X , where u ∈ C ∞ ( X ) is any function restricting to v . This definition is independent ofthe choice of extension u , and depends only mildly on x . Given u ∈ x is C ∞ ( X ),( x − is Au ) | ∂X = b N ( A )( s )( x − is u | ∂X ) . The indicial family b N ( s ) is an algebra homomorphism; in particular it satisfies b N ( AB )( s ) = b N ( A )( s ) ◦ b N ( B )( s ) . Furthermore, b N ( A ∗ )( s ) = b N ( A )(¯ s ) ∗ . Here the adjoint on the left is with respect to a C ∞ density µ on X , whereas the adjoint on the right is with respect to the pullback µ | ∂X . This can also be rewritten as b N ( A ∗ )( s ) = b N ( x − s Ax s )( s ) ∗ . Using the indicial family it is possible to prove the following facts: let U be a boundarycoordinate patch with coordinates ( x, y , . . . , y n − ), and suppose A ∈ Ψ m b ( X ) has sup-port in a compact set K ⊂ U . As emphasized in [Vas2, Section 2] (see [Vas2, Lemma ORAN GANNOT AND MICHA L WROCHNA A ′ , A ′′ ∈ Ψ m b ( X ) such that D x A = A ′ D x + A ′′ . (2.4)Indeed, one can take A ′ = x − Ax and A ′′ = x − [ xD x , A ], the key here being that[ xD x , A ] ∈ x Ψ m b ( X ), as seen by analyzing its indicial family. In particular, the maps A A ′ and A A ′′ are microlocal. Since σ b ,m ( A ′ ) = σ b ,m ( A ), one deduces as in[Vas2, Lemma 2.8] that the commutator [ D x , A ] can be written in the form[ D x , A ] = A D x + A , (2.5)where A j ∈ Ψ m − j b ( X ). Here σ b ,m − ( A ) = (1 /i ) ∂ σ a and σ b ,m ( A ) = (1 /i ) ∂ x a . Again,the maps A A and A A are microlocal.3. Function spaces
Twisted derivatives.
Let X be an n -dimensional manifold with boundary. Itwill also be convenient to fix some bdf x . Motivated by [War1], we define certain twisted differential operators. Given ν ∈ R , define ν ± = n − ± ν , and then setDiff ν ( X ) = { x ν − Bx − ν − : B ∈ Diff ( X ) } . This space is independent of the choice of bdf x . The dimension-dependent shiftbetween ν and ν ± is merely due to our eventual choice of weighted L space.Of course Diff ν ( X ) ⊂ Diff ( X ◦ ), but twisted differential operators do not necessarilyhave coefficients that are smooth up to ∂X . On the other hand, if Q ∈ Diff ν ( X ), then Q : C −∞ ( X ) → C −∞ ( X )is continuous, since this is true for multiplication by any power of x . One should thinkof Diff ν ( X ) as the largest space of differential operators preserving x ν − C ∞ ( X ). This isin contrast to the much smaller space Diff ( X ), which preserves x s C ∞ ( X ) for every s .More precisely, we have the following: Lemma 3.1.
Diff ( X ) ⊂ Diff ν ( X ) ⊂ x − Diff ( X ) for each ν ∈ R .Proof. For the first inclusion, any A ∈ Diff ( X ) can be written as A = x ν − ( x − ν − Ax ν − ) x − ν − ∈ Diff ν ( X ) . For the second inclusion it suffices to work in local coordinates ( x, y , . . . , y n − ) nearthe boundary, where this is just the observation that x ( x ν − ∂ x x − ν − ) = x∂ x − ν − ∈ Diff ( X ). (cid:3) Finally, notice that Diff ν ( X ) is a C ∞ ( X )-module under both left and right multi-plication, hence also closed under commutators. Since every operator in Diff ( X ) isthe sum of a vector field and a multiplication operator, and since the space of C ∞ ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 9 vector fields is finitely generated over C ∞ ( X ), it follows that Diff ν ( X ) is also finitelygenerated.3.2. Twisted Sobolev spaces.
We now restrict to ν ∈ (0 ,
1) and define the relevantfunction spaces. Given a C ∞ density µ (to be fixed later on) on an n -dimensionalmanifold X with boundary, define µ = x − n µ and set L ( X ) := x ( n − / L ( X, µ ) = L ( X, µ ) , where x is a fixed global bdf. This space depends on µ and x , but of course the localversions L ( X ) or L ( X ) do not. We also sometimes write H ( X ) for L ( X ).The twisted Sobolev spaces corresponding to Diff ν ( X ) are defined by the followingcondition: given u ∈ C −∞ ( X ), u ∈ H ( X ) ⇐⇒ Qu ∈ L ( X ) for all Q ∈ Diff ν ( X ) . Given a compact subset K ⊂ X write H ( K ) for the set of u ∈ H ( X ) with supportin K . We also set H ( X ) = H ( X ) ∩ C −∞ c ( X ) . Note that H ( X ) is a local space in the sense that u ∈ H ( X ) implies φu ∈ H ( X )for every φ ∈ C ∞ c ( X ).As noted in the previous section, Diff ν ( X ) is finitely generated over C ∞ ( X ). Fixinga generating set Q , . . . , Q N , we equip H ( X ) with the family of seminorms u
7→ k φu k L ( X ) + N X i =1 k φQ i u k L ( X ) , φ ∈ C ∞ c ( X ) . The restriction of these seminorms to H ( K ) for K ⊂ X compact defines a Hilbertspace topology; we write k u k H ( X ) = k u k L ( X ) + N X i =1 k Q i u k L ( X ) when u has compact support. Then H ( X ) is equipped with the inductive limit topol-ogy corresponding to H ( K ) as K ranges over all compact subsets of X . Because weare assuming ν ∈ (0 , x ν − C ∞ ( X ) ⊂ L ( X ) . Since by construction Diff ν ( X ) preserves x ν − C ∞ c ( X ), it follows that x ν − C ∞ ( X ) ⊂H ( X ). Remark 3.2.
Sometimes it is useful to have a globally defined Hilbert space, at leastwhen X = R n + . We use standard coordinates ( z , . . . , z n − ) on R n + = R + × R n − , where z = x ∈ R + , and define L ( R n + ) = x ( n − / L ( R n + ) with respect to Lebesgue measure. Consider the coordinate vector fields conjugatedby x ν − , Q i = x ν − D z i x − ν − . (3.1)Along with the constant function these generate Diff ν ( R n + ). Of course in this case Q i = D z i for i = 0.We say that u ∈ H ( R n + ) if u ∈ L ( R n + ) and Q i u ∈ L ( R n + ) for each i = 0 , . . . , n − u ∈ H ( R n + ) is equivalentto u ∈ x r L ( R + ; H ( R n − )) , ( xD x + iν − ) u ∈ x r +1 L ( R + ; L ( R n − )) (3.2)when r = ( n − / L spaces are defined with respect to Lebesguemeasure. If u ∈ C −∞ ( X ) has compact support in a boundary coordinate patch U κ ,then u ∈ H ( X ) is equivalent to u ◦ κ ∈ H ( R n + ).Next, we discuss the relationship between H ( X ) and some weighted Sobolevspaces. For simplicity we first consider the case X = R n + . If u ∈ H ( R n + ), thensetting v = x − ν − u we have v ∈ x − ν − L ( R n + ) , D z i v ∈ x − ν − L ( R n + )for i = 1 , . . . , n . Weighted Sobolev spaces of this kind are well studied; see [Gri] forexample. In particular C ∞ c ( X ) is dense in the corresponding weighted space by theusual translation, truncation, and mollification arguments. Since H ( R n + ) is a localspace, this implies the density of x ν − C ∞ c ( X ) in H ( X ) for an arbitrary manifold withboundary X , hence also in H ( X ).From the identification with a weighted space as in the previous paragraph, it ispossible to show that u ∈ H ( R n + ) admits a weighted trace( x − ν − u ) | ∂X ∈ H ν ( R n + ) , (3.3)extended by continuity from x ν − C ∞ c ( R n + ). We give an alternative proof in Lemma 3.3below, since the same methods will be used later on. Lemma 3.3.
Let ν ∈ (0 , , and set r = ( n − / . If u ∈ H ( R n + ) , then the restrictionof u to any half-space { x < ε } admits an asymptotic expansion u = x ν − u − + x r +1 H ([0 , ε ); L ( R n − )) , (3.4) where u − ∈ H ν ( R n − ) . The map u γ − u := u − is continuous H ( R n + ) → H ν ( R n − ) .Proof. Given ε >
0, let φ ∈ C ∞ c ( R + ) be such that φ = 1 near { x < ε } . Replacing u with φu ∈ H ( R n + ), we can assume u has compact support. Taking the Mellin transform, itfollows from the second equation in (3.2) that M u ( s ) = ( s + iν − ) − M v ( s ) , ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 11 where v = ( xD x + iν − ) u ∈ x r +1 L ( R + ; L ( R n − )) has compact support. Thus M v ( s )is holomorphic in { Im s > r − / } .If ν ∈ (0 , s + iν − ) − M v ( s ) has precisely one pole at s = − iν − , so bystandard contour deformation arguments we obtain u = x ν − u − + x r +1 H ([0 , ε ); L ( R n − ))when x < ε . Note that u − ∈ L ( R n − ) is just a scalar multiple of M v ( − iν − ). Bythe complex interpolation method as described in [Maz], one actually has that u − ∈ H ν ( R n − ).Continuity of the map u γ − u follows by the closed graph theorem. Indeed,suppose that u j → u in H ( R n + ) and γ − u j → ˜ u in H ν ( R n − ) (hence in distributions).If φ ∈ C ∞ c ( R + ) is identically on on supp φ , then we can replace u j with φ u j (whichalso converges to u ), and hence assume u j has compact support. As noted above, γ − ( u j − u ) is a multiple of M ( v j − v )( − iν − ), where v j = ( xD x + iν − ) u j , and v j → v in x r +1 L ( R n + ). Since v j − v has compact support, it follows by theCauchy–Schwarz inequality that |M ( v j − v )( − iν − )( y ) | ≤ Z R + x ν − / | x − − r ( v j − v )( x, y ) | dx ≤ C k x − − r ( v j − v )( · , y ) k L ( R + ) for ν ∈ (0 , γ − u j → γ − u in L ( R n − ), hence also in distributions.This implies ˜ u = γ − u , completing the proof. (cid:3) If u admits a partial asymptotic expansion as in (3.4), then the coefficient u − isuniquely determined by u in the sense that u = 0 = ⇒ u − = 0 . Since elements of x ν − C ∞ c ( R n + ) certainly have an expansion as in (3.4), the map γ − agrees with the extension by continuity of the weighted restriction (3.3).Passing from R n + to an arbitrary manifold with boundary X by a partition of unity,Lemma 3.3 shows the existence of a continuous trace map γ − : H ( X ) → H ν c ( ∂X ) , hence also between the corresponding local spaces, extending u ( x − ν − u ) | ∂X when u ∈ x ν − C ∞ c ( X ). When ν = 1 /
2, the map γ − on H ( X ) depends on the choice of bdf x , but only in the mildest way: if ˜ x = ax is another bdf with a ∈ C ∞ ( X ) and a > γ − u = ( a | ∂X ) − ν − · γ − u. Here γ − and ˜ γ − are the traces defined with respect to x and ˜ x , respectively. Lemma 3.4.
There exists a continuous right inverse ρ : H ν loc ( ∂X ) → H ( X ) for γ − , which restricts to a continuous map H ν c ( ∂X ) → H ( X ) , and moreover takes C ∞ c ( ∂X ) into x ν − C ∞ c ( X ) .Proof. First we construct a continuous right inverse ρ : H ν ( R n + ) → H ( R n + ) for γ − on R n + as follows: let φ ∈ C ∞ c ([0 , ε )) be one near x = 0, and then specify the Fouriertransform of ρ ( v ) by F ( ρ ( v ))( x, ζ ) = b v ( ζ ) · φ ( h ζ i x ) x ν − . Standard manipulations using a partition of unity allows for the construction of asuitable extension ρ on a general manifold as well. (cid:3) From (3.4), the kernel of γ − on H ( X ) is justker γ − = H ( X ) ∩ x L ( X ) , and this latter space is independent of ν ∈ (0 ,
1) (cf. (3.2)). In particular, taking ν = 1 / H ( X ) ∩ x L ( X ) = x n/ H ( X ) ∩ x L ( X ) , where H ( X ) is the ordinary space of extendible Sobolev distributions on X . Inview of Hardy’s inequality in one dimension, this equality also holds at the level oftopologies. Applying Lemma 3.3 again, H ( X ) ∩ x L ( X ) = x n/ ˙ H ( X )by the well-known characterization of ˙ H ( X ) as the kernel of the usual smooth trace γ : H ( X ) → H / ( ∂X ) . Indeed, if ν = 1 /
2, then γ − agrees with γ ◦ x − n/ . If we let ˙ H ( X ) denote the closureof ˙ C ∞ c ( X ) in H ( X ), it follows thatker γ − = ˙ H ( X ) . We then let ˙ H ( X ) = ˙ H ( X ) ∩ C −∞ c ( X ) as a subspace of H ( X ). We also obtain thefollowing corollary of Hardy’s inequality: Lemma 3.5. If x ∈ C ∞ ( X ) is any bdf, then multiplication by x − defines a boundedlinear map ˙ H ( X ) → L ( X ) . We will also need a trace interpolation inequality. From [Gan, Lemma 4.2], if u ∈H ( X ), then k γ − u k L ( ∂X ) ≤ C k u k ν L ( X ) k u k − ν H ( X ) , (3.5) ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 13 where
C > u . Using Young’s inequality we concludethat for any ε > C ε > k γ − u k L ( ∂X ) ≤ ε k u k H ( X ) + C ε k u k L ( X ) . (3.6)For a different proof of (3.6) see [War2, Appendix B.2]. We will use (3.6) frequently. Remark 3.6.
In the smooth setting one gets an even stronger version of (3.5) bynoting that for u, v ∈ C ∞ c ( X ), h γu, γv i L ( ∂X ) = h R u, v i L ( X ) − h u, R v i L ( X ) (3.7)for suitable R , R ∈ Diff ( X ). Suppose that A ∈ Ψ m b ( X ) is formally self-adjoint and v = Au . Trivially estimating (3.7) by Cauchy–Schwarz yields an upper bound of theform k u k H ( X ) k Au k L ( X ) + k u k L ( X ) k Au k H ( X ) . On the other hand, one can also commute A through R to bound (3.7) (modulolower order terms) by k u k H ( X ) k Au k L ( X ) alone. One cannot obtain an analogous typeof estimate in the twisted setting from (3.5) directly, which will cause some slightcomplications later on.One can also define the spaces H ( X ) for ν ≥ H ( X ) = ˙ H ( X ) since x ν − is not square integrablefor ν ≥ Dual spaces.
For the dual spaces, we define ˙ H − ( X ) = [ H ( X )] ′ , H − ( X ) = [ ˙ H ( X )] ′ with their strong dual topologies. The duals of the spaces with compact supportsare defined by exchanging the roles of the subscripts c and loc. We use the notation˙ H − ( X ) in analogy with the smooth setting, where the dual of H ( X ) is ˙ H − ( X ) ⊂ ˙ C −∞ c ( X ), the corresponding space of supported distributions. In this slightly moregeneral setting, when ν ∈ (0 , H − ( X ) ⊂ x − ν − ˙ C −∞ c ( X )by transposition of the dense inclusion x ν − C ∞ ( X ) ⊂ H ( X ). Similarly, there is aninclusion ˙ H − ( X ) ⊂ x − ν − ˙ C −∞ ( X ). Identifying L ( X ) with its own anti-dual gives riseto inclusions L ( X ) ⊂ ˙ H − ( X ) , L ( X ) ⊂ ˙ H − ( X ) . The situation for H − ( X ) is simpler, since it can be identified with a subspace of C −∞ ( X ) by transposing the dense inclusion ˙ C ∞ c ( X ) ⊂ H ( X ). A similar commentapplies to H − ( X ) ⊂ C −∞ c ( X ). Given a compact set K ⊂ X , we write ˙ H − ( K ) and H − ( K ) for the subsets of elements with compact support in K . Interaction with the b-calculus.
In this section we discuss the interactionbetween Diff ν ( X ) and Ψ m b ( X ). Throughout, we will use the notion of a smooth twistingfunction F , meaning that F ∈ x ν − C ∞ ( X ; R ) , x − ν − F > X. Note that F itself is not a smooth function, but rather its polyhomogeneous expansionis smooth in the sense of not containing any logarithmic terms. Thus Q ∈ Diff ν ( X )if and only if it is of the form Q = F BF − for some B ∈ Diff ( X ). Of course x ν − isitself a valid smooth twisting function.We also introduce some notation for the coordinate vector fields twisted by F (cf. Re-mark 3.2). Given fixed local coordinates ( z , . . . , z n − ) on a local coordinate patch U ⊂ X (not necessarily a boundary coordinate patch) we use the notation Q i = F D z i F − . Suppose that A ∈ Ψ m b ( X ) has compact support in U . First, note that[ Q i , A ] = F ( D z i F − AF − F − AF D z i ) F − = F [ D z i , F − AF ] F − . (3.8)Because F is a smooth twisting function, F − AF ∈ Ψ m b ( X ) with the same principalsymbol as A . Now consider the special case of boundary coordinates ( x, y , . . . , y n − ),so that Q = F D x F − in the notation above. Combined with (2.5), we obtain thefollowing: Lemma 3.7.
Let A ∈ Ψ m b ( X ) have compact support in U with a = σ b ,m ( A ) . Thereexist A ∈ Ψ m − ( X ) and A ∈ Ψ m b ( X ) such that [ Q , A ] = A Q + A , where σ b ,m − ( A ) = (1 /i ) ∂ σ a and σ b ,m ( A ) = (1 /i ) ∂ x a . The maps A A and A A are microlocal. Furthermore, Q A = A ′ Q + A ′′ (3.9) for some A ′ , A ′′ ∈ Ψ m b ( X ) . The maps A A ′ and A A ′′ are microlocal. Recall in the next lemma that all pseudodifferential operators are assumed to haveproper support.
Lemma 3.8 (cf. [Vas2, Lemma 3.2, Corollary 3.4]) . Each A ∈ Ψ ( X ) defines a con-tinuous linear map H ( X ) → H ( X ) , ˙ H ( X ) → ˙ H ( X ) . By duality, A extends to a continuous map ˙ H − ( X ) → ˙ H − ( X ) , H − ( X ) → H − ( X ) . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 15
The same result holds with the roles of the subscripts loc and c reversed. Here the action of A ∈ Ψ ( X ) on H − ( X ) is by duality, namely h Af, v i = h f, A ∗ v i for v ∈ H ( X ). This is just the restriction of the action of A on the larger space C −∞ ( X ). The action of A on ˙ H − ( X ) is also by duality. Recall that for ν ∈ (0 , H − ( X ) ⊂ x − ν − ˙ C −∞ c ( X ) , and any A ∈ Ψ m b ( X ) acts on x − ν − ˙ C −∞ c ( X ) by the formula A ( x − ν − v ) = x − ν − ( x ν − Ax − ν − ) v. In this sense the action of A on ˙ H − ( X ) is given by restriction from the larger space x − ν − ˙ C −∞ ( X ).The proof of Lemma 3.8 gives even more information: if A ∈ Ψ ( X ) has compactsupport in U ⊂ X , then there exists χ ∈ C ∞ c ( U ) such that k Au k H k ( X ) ≤ C k χu k H k ( X ) (3.10)for every u ∈ H k loc ( X ) with k = ±
1, where the constant
C > A in Ψ ( X ). Similar estimates holds for ˙ H k spaces, again with constantsbounded by a seminorm of A in Ψ ( X ).In exact analogy with [Vas2, Definition 3.5], for k = 0 , ± H k ( X ) (or ˙ H k ( X )) with additional regularity as measured by operators in Ψ m b ( X ). Definition 3.9.
Let k = 0 , ± m ≥
0. Given u ∈ H k loc ( X ), we say that u ∈H k,m loc ( X ) if Au ∈ H k loc ( X ) for all A ∈ Ψ m b ( X ). When m = ∞ , we define H k, ∞ ( X ) = \ m H k,m ( X ) . The spaces ˙ H k,m ( X ) for k = ± m it suffices to check that u ∈ H k loc ( X ) and Au ∈ H k loc ( X ) for a singleelliptic A (cf. [Vas2, Remark 3.6]). The corresponding spaces with compact supportsare defined in the obvious way; for u ∈ H k,m c ( X ) with m ≥ k u k H k,m ( X ) = k u k H k ( X ) + k Au k H k ( X ) , for a choice of elliptic A , with the analogous definition for u ∈ ˙ H k,m c ( X ).It is also true that the weighted trace γ − maps H ,m loc ( X ) into H ν + m loc ( X ) (with con-tinuity following by the closed graph theorem) for m ≥
0. This follows from γ − ( Au ) = b N ( A )( − iν − )( γ − u ) (3.11)and the fact that b N ( A )( − ν − ) ∈ Ψ m ( ∂X ) is elliptic whenever A ∈ Ψ m b ( X ) is elliptic.Indeed, (3.11) holds when m = 0 by the density of x ν − C ∞ ( X ) in H ( X ), and for m > One can also give a definition of the spaces H k,m ( X ) or ˙ H k,m ( X ) for m < Definition 3.10.
Let k = ± m <
0; fix an elliptic A ∈ Ψ − m b ( X ). We let H k,m loc ( X )denote the set of u ∈ C −∞ ( X ) of the form u = u + Au , where u , u ∈ ˙ H k loc ( X ). The same definition applies to ˙ H k,m loc ( X ), where ˙ H − ,m loc ( X ) isconsidered as a subspace of x − ν − ˙ C −∞ ( X ).This definition is independent of the choice of elliptic A . Again the spaces withcompact supports are defined in the obvious way; in that case one can choose u , u with compact support (for a more detailed analysis of supports, see the discussionpreceding [Vas2, Definition 3.17]). For u ∈ H k,m c ( X ) with m < k u k H k,m ( X ) = inf {k u k H k ( X ) + k u k H k ( X ) : u = u + Au } , taking u , u with compact supports, with the analogous definition for u ∈ ˙ H k,m c ( X ).Extending Lemma 3.8, one can show that any A ∈ Ψ ( X ) defines a continuous mapbetween the spaces H k,m loc ( X ) → H k,m loc ( X ) , ˙ H k,m loc ( X ) → ˙ H k,m loc ( X )for k = 0 , ± m ∈ R (cf. [Vas4, Lemma 5.8]). Furthermore, the analogue of(3.10) holds. For instance, if A ∈ Ψ ( X ) has compact support in U ⊂ X , then thereexists χ ∈ C ∞ c ( U ) such that k Au k H k,m ( X ) ≤ C k χu k H k,m ( X ) for every u ∈ H k,m loc ( X ), where the constant C > A inΨ ( X ).We will also need to extend the definition of γ − to H ,m loc ( X ) for m <
0. It is easyto see that x ν − C ∞ ( X ) is dense in H ,m loc ( X ) since A as in Definition 3.10 preserves theformer space. Furthermore, the restriction of γ − to x ν − C ∞ ( X ) extends to a continuousmap H ,m loc ( X ) → H ν + m loc ( X ) , which follows immediately from the definition of the H ,m ( X ) norm and the fact that b N ( A )( − iν − ) ∈ Ψ − m ( ∂X ) is elliptic on ∂X . In analogy with [Vas2, Remark 3.16],given u = u + Au ∈ H ,m loc ( X ) with A ∈ Ψ − m b ( X ) elliptic and u , u ∈ H ( X ), weequivalently have γ − u = γ − u + b N ( A )( − iν − )( γ − u ) ∈ H ν + m loc ( ∂X ) . (3.12)In particular, this definition is independent of the choice of A and u , u . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 17
With these definitions ˙ H − ,m c ( X ) is identified with the dual of H , − m loc ( X ) via the L ( X ) inner product for each m ∈ R ; similarly H − ,m c ( X ) is the dual of ˙ H , − m loc ( X ).One also shows that Q ∈ Diff ν ( X ) defines a bounded map H ,m loc ( X ) → H ,m loc ( X )for each m ∈ R . This follows from the comments following Lemma 3.7. Since this isalso true for the spaces with compact support, by duality (replacing m with − m ) thetransposition Q ∗ defines a bounded map Q ∗ : H ,m loc ( X ) → ˙ H − ,m loc ( X ) . In particular, any operator of the form L = P i R ∗ i Q i with Q i , R i ∈ Diff ν ( X ) defines abounded map L : H ,m loc ( X ) → ˙ H − ,m loc ( X ) for each m ∈ R : h Lu, v i := X i h Q i u, R i v i , where u ∈ H ,m loc ( X ) and v ∈ H , − m c ( X ).There is a natural wavefront set corresponding to H k,m ( X ): Definition 3.11.
Let k = 0 , ±
1, and assume u ∈ H k,r loc ( X ) for some r ∈ R . Given q ∈ b T ∗ X \
0, we say that q / ∈ WF k,m b ( X ) if there exists A ∈ Ψ m b ( X ) such that q ∈ ell b ( A ) , Au ∈ H k loc ( X ) . When m = + ∞ , we say that q / ∈ WF k, ∞ b ( X ) if there exists A ∈ Ψ ( X ) such that q ∈ ell b ( A ) and Au ∈ H k, ∞ loc ( X ).This wavefront set is microlocal in the sense thatWF k,m b ( Au ) ⊂ WF k,m − s b ( u ) ∪ WF ′ b ( A )for each A ∈ Ψ s b ( X ); the proof is identical to that of [Vas2, Lemma 3.9]. From theconstruction of microlocal parametrices, one also obtains the following quantitativeversion: Lemma 3.12.
Let A be a bounded family in Ψ s b ( X ) and G ∈ Ψ s b ( X ) be such that WF ′ b ( A ) ⊂ ell b ( G ) . Suppose further that A and G have compact support in U ⊂ X . Let m ∈ R . Thenthere exists χ ∈ C ∞ c ( U ) and C > such that k Au k H ( X ) ≤ C ( k Gu k H ( X ) + k χu k H ,m ( X ) ) for every u ∈ H ,m loc ( X ) with WF ,s b ( u ) ∩ WF ′ b ( G ) = ∅ and every A ∈ A . We also make the following useful observation:
Lemma 3.13.
Let A be a bounded family in Ψ s b ( X ) and G ∈ Ψ s − ( X ) be such that WF ′ b ( A ) ⊂ ell b ( G ) . Suppose further that A and G have compact support in U ⊂ X . Let m ∈ R . Thenthere exists χ ∈ C ∞ c ( U ) such that k Au k L ( X ) ≤ C ( k Gu k H ( X ) + k χu k H ,m ( X ) ) for every u ∈ H ,m loc ( X ) with WF ,s − ( u ) ∩ WF ′ b ( G ) = ∅ and every A ∈ A .Proof. By a microlocal partition of unity, we can assume that U is a coordinatepatch with coordinates ( x, y , . . . , y n − ), and that at least one of the vector fields { xD x , D y , . . . , D y n − } is elliptic on WF ′ b ( A ). Then we can write A = n − X i =0 Q i G i + R where G i ∈ Ψ s − ( X ) has WF ′ b ( G i ) ⊂ ell b ( G ), and R ∈ Ψ −∞ b ( X ). Since Q i ∈ Diff ν ( X ),the proof is complete. (cid:3) The point of this lemma is that G has one order lower than the family A .3.5. Logarithmic twisting functions.
We will also need to consider more generaltwisting functions with logarithmic corrections. It should be stressed, however, thatthe material in this section is not needed in the any of the following three situations:(1) ν ∈ (0 , \ { / } ,(2) ν = 1 / ∂X is satisfied,(3) ν > F ∈ x ν − C ∞ ( X ) + ( x ν + log x ) C ∞ ( X )is a logarithmic twisting function if x − ν − F >
0. This is in contrast to the notion ofa smooth twisting function from the previous sections, where the logarithmic termswere absent. Throughout this section we assume ν ∈ (0 , R n + with standard coordinates ( z , . . . , z n − ), where x = z ∈ R + and y α = z α for α =1 , . . . , n − Lemma 3.14. If g ∈ L ∞ ( R n + ) satisfies D αy g ∈ L ∞ ( R n + ) for | α | ≤ and x ν − k D x g ( x, · ) k L ∞ ( R n − ) ∈ x ( n − / L ( R + ) , then multiplication by g defines a bounded map H ( R n + ) → H ( R n + ) . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 19
Proof.
It is clear that D αy g : H ( R n + ) → L ( R n + ) is continuous for | α | ≤
1. Next, let u ∈ H ( R n + ) and let u n ∈ x ν − C ∞ c ( R n + ) converge to u in H ( R n + ). We can use the Sobolevembedding from [Gri, Proposition 1.1’] to conclude thatsup x x − ν − k u n ( x, · ) k L ( R n − ) < C k u n k H ( R n + ) , where C > n . Now write k x ν − D x ( x − ν − gu n ) k L ( R n + ) ≤ C k u n k H ( R n + ) + k D x ( g ) u n k L ( R n + ) . The last term on the right hand side is estimate by k D x ( g ) u n k L ( R n + ) ≤ Z R (cid:16) x − ν − k u n ( x, · ) k L ( R n − ) (cid:17) (cid:16) x ν − k D x g ( x, · ) k L ∞ ( R n − ) (cid:17) x − n dx. By hypothesis, we conclude that k x ν − D x ( x − ν − gu n ) k L ( R n + ) ≤ C k u n k H ( R n + ) . In particular, the left-hand side is uniformly bounded n , since u n → u in H ( R n + ).Passing to a subsequence shows that x ν − D x x − ν − u ∈ L ( R n + ), and continuity of x ν − D x x ν − g : H ( R n + ) → L ( R n + )then follows from the closed graph theorem. (cid:3) Next, consider a logarithmic twisting function F on R n + , and assume that x − ν − D αy F is bounded for | α | ≤
1. We show that the function x ν − F − satisfies the hypotheses ofLemma 3.14. Since ( x − ν − F ) − is bounded it suffices to show that x ν − k D x ( x − ν − F ) k L ∞ ( R n + ) ∈ x ( n − / L ( R ) . We can write x − ν − F = F + ( x ν + − ν − log x ) F with F , F ∈ C ∞ ( X ), and the prooffollows since ν ∈ (0 , x − ν − F is a multiplier on H ( R n + ). Thefollowing result is an immediate corollary Lemma 3.15. If F is a logarithmic twisting function and B ∈ Diff ( X ) , then F BF − : H ( X ) → L ( X ) is bounded. Next we consider what happens when we conjugate A ∈ Ψ m b ( X ) by a logarithmictwisting function F . This necessitates the use of b-pseudodifferential operators with conormal coefficients . A function a ∈ L ∞ loc ( b T ∗ X ) is said to be a symbol with conormalcoefficients if in local coordinates ( x, y, σ, η ) it satisfies | ( x∂ x ) k ∂ αy ∂ ℓσ ∂ βη a ( x, y, σ, η ) | ≤ C kℓαβ h ( σ, η ) i m −| β |− ℓ . Roughly speaking, the conormal calculus consists of the quantizations of conormalsymbols; we denote by Ψ m b , c ( X ) ⊃ Ψ m b ( X ) the corresponding operators of order m . These operators form a filtered ∗ -algebra just as in the case of smooth coefficients.Furthermore, each A ∈ Ψ , c ( X ) defines a bounded operator on L ( X ).The only subtlety arises in the definition of the normal operator or the indicial familyof A ∈ Ψ m b , c ( X ). This will not pose a problem here, since we only consider operators A ∈ Ψ m b ( X ) + x δ Ψ m b , c ( X ) . for some δ >
0. If A = B + C with B ∈ Ψ m b ( X ) and C ∈ x δ Ψ m b , c ( X ), then we set b N ( A ) := b N ( B ). With this definition (3.11) still holds. Lemma 3.16. If A ∈ Ψ m b ( X ) and F is a logarithmic twisting function, then F AF − ∈ Ψ m b ( X ) + x ν − ε Ψ m b , c ( X ) for each ε > . Using Lemma 3.14, it is easy to see that A ∈ Ψ ( X ) + x ν − ε Ψ , c ( X ) maps H ( X )to itself. This is because given such an A , we can still find A ′ ∈ Ψ ( X ) + x ν − ε Ψ , c ( X )such that D x A = A ′ D x + A ′′ , the only difference being that now A ′′ ∈ Ψ ( X ) + x ν − ε − Ψ , c ( X ). This is because thecommutator of xD x with an element of x ν − ε Ψ , c ( X ) does not gain an extra order ofvanishing. On the other hand, x ν − − ε maps H ( X ) to L ( X ) by arguing as in theproof of Lemma 3.14 since ν ∈ (0 ,
1) and ε > m b ( X ) + x ν − ε Ψ m b , c ( X ) over m ∈ R forms a filtered ∗ -algebra, which is moreover closed under taking parametrices of invertible elements.Thus if A as above is elliptic on WF b ( B ) for some B ∈ Ψ k b ( X ), then the microlocalparametrix one obtains dividing B by A can also be taken in Ψ k − m b ( X )+ x ν − ε Ψ k − m b , c ( X ),and similarly for the residual terms. This makes it possible to test for WF k,m b usingthese operators for k = 0 , ± A = F BF − with B ∈ Ψ m b ( X ).This means that when carrying out commutator arguments with a chosen commutant A , it should actually be applied with F AF − (which has the same principal symbol).4. Asymptotically anti-de Sitter spacetimes
Basic definitions.
Let X be an n -dimensional manifold with boundary ∂X .Suppose that X ◦ is equipped with a smooth Lorentzian metric g of signature (1 , n − ∂X :(A) If x ∈ C ∞ ( X ) is a bdf, then ˆ g = x g extends smoothly to a Lorentzian metricon X . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 21 (B) The pullback ˆ g | ∂X of ˆ g to the boundary has Lorentzian signature.(C) ˆ g − ( dx, dx ) = − ∂X .These properties are independent of the choice of bdf. When all three are satisfiedwe say that ( X, g ) is an asymptotically anti-de Sitter (aAdS) spacetime. The pullbackˆ g | ∂X is only determined by g up to a conformal multiple (corresponding to a changeof bdf), hence ∂X is referred to as a conformal boundary .4.2. Special bdfs.
At various points it will be convenient to use a special bdf x ∈C ∞ ( X ) such that ˆ g ( dx, dx ) = − ∂X . In fact, if k is a representative of the conformal class ofboundary Lorentzian metrics, then x is uniquely determined near ∂X by specifying that k = ˆ g | ∂X . The existence of such an x is well known. For a proof in the Riemannian(conformally compact) setting, see [GL, Lemma 5.2]; the proof applies verbatim in theLorentzian case.Given U ⊂ ∂X with compact closure, we can always choose a collar neighborhoodof U diffeomorphic to [0 , ε ) × U in which the special bdf x ∈ C ∞ ( X ) is identified withprojection onto the first factor. Since ∂X will typically be non-compact, we make noclaim about the uniformity of ε = ε ( U ) > U varies. With this identification near U , g = − dx + kx , where x k ( x ) is a family of Lorentzian metrics on ∂X depending smoothly on x ∈ [0 , ε ) such that k (0) = k . In particular, one can choose local coordinates( x, y , . . . , y n − ) such that x is a special bdf, andˆ g − ( dx, dy α ) = 0 near ∂X, where α = 1 , . . . , n −
1. We call coordinates of this form special coordinates .We say that g is even modulo O ( x k +1 ) (in the sense of Guillarmou [Gui]) if theTaylor expansion of k at x = 0 contains only even terms modulo O ( x k +1 ). This con-dition is independent of the choice of special bdf (cf. [Gui, Lemma 2.1]). Furthermore,as shown in [Gui, Section 2], any two special coordinate systems ( x, y ) and (˜ x, ˜ y ) onoverlapping coordinates patches are related by˜ x = x k +1 X j =0 a j ( y ) x j + x k +4 C ∞ ( X ) , ˜ y = k +1 X j =0 b j ( y ) x j + x k +3 C ∞ ( X ) . (4.2)Evenness of g modulo O ( x ) is implicit in the works of [Hol, War1, HW], and isverified for solutions to the Einstein equations. In particular, it implies that ∂X is totally geodesic with respect to the conformal metric ˆ g . In this paper we do not insiston evenness assumptions for the metric.4.3. The Klein–Gordon operator.
Let (
X, g ) be an asymptotically AdS spacetime,and consider the Klein–Gordon operator P = (cid:3) g − λ, λ < ( n − . It is convenient to parametrize λ = ( n − − ν , with ν >
0. As in Section 3.1, set ν ± = n − ± ν . Fixing an arbitrary representative k in the conformal class of boundaryLorentzian metrics, we use the special bdf x as in the previous section.Since det g = x − n det ˆ g , it follows that in special coordinates on a coordinate patch[0 , ε ) × U , (cid:3) g = − ( x∂ x ) + ( n − x∂ x + ( xE ) x∂ x + x ˜ P, where ˜ P = (cid:3) k is a family of second order differential operators on ∂X dependingsmoothly on x ∈ [0 , ε ), and E ∈ C ∞ ( X ) is given by E = − ∂ x (log | det ˆ g | ) . (4.3)Observe that up to a scalar multiple, E | ∂X is the mean curvature of ∂X with respect toˆ g . Using the product structure near the boundary, we can identify the normal operator N ( P ) of P with an operator on X . Thus N ( P ) = ( xD x ) + i ( n − xD x − λ. In this case the indicial family b N ( P )( s ) = s + i ( n − s − λ can be identified with ascalar multiplication operator.In general, the difference between P and N ( P ) in an operator in x Diff ( X ). How-ever, P − N ( P ) = i ( xE ) xD x + x Diff ( X ) , (4.4)so if ∂X has zero mean curvature with respect to ˆ g , then actually P − N ( P ) ∈ x Diff ( X ).There are two important L spaces on X . The first is L ( X, dg ) = x n/ L ( X, d ˆ g ),and the second is L ( X, x dg ). It is this second space that is compatible with thenormalization in Section 3.1, and unless specified explicitly, we use L ( X ) = L ( X, x dg ) = x − L ( X, dg ) . The inner product of u, v ∈ L ( X ) will be denoted by h u, v i , whereas the inducedinner product of f, g ∈ L ( ∂X ) coming from the volume density dk will be denotedby h f, g i ∂X .Recall the notion of a twisting function, either smooth or logarithmic, as in Section3.4. For the next definition we fix such a function F . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 23
Definition 4.1.
Given u ∈ ˙ C ∞ ( X ), define the twisted differential d F u ∈ ˙ C ∞ ( X ; T ∗ X )by d F u = F d ( F − u ) = du + uF − · dF. The one-form F − · dF ∈ x − C ∞ ( X ; T ∗ X ) can be thought of as a singular magneticpotential. Note that d F u continues to make sense as a T ∗ X -valued distribution on X when u ∈ C −∞ ( X ). In local coordinates ( z , . . . , z n − ), set Q i = F D z i F − , so that − i · d F u = ( Q i u ) dz i . We define the twisted Dirichlet form for the metric bysetting E ( u, v ) = − Z G ( d F u, d F ¯ v ) dg = − Z ˆ G ( d F u, d F ¯ v ) x dg, where G is the induced inner product on the fibers of T ∗ X . Thus in coordinates, wehave G ( d F u, d F ¯ v ) = − g ij Q i u · Q j ¯ v. If u, v ∈ H ( X ), at least one of which has compact support, then E ( u, v ) is finite.We now come to the main reason for introducing the twisting function: as observed inthis context by [War1] and [HW], P = − ( d F ) † d F + F − P ( F ) , where ( d F ) † : ˙ C ∞ ( X ; T ∗ X ) → ˙ C ∞ ( X ) is the dg adjoint of d F . This is useful provided F is chosen so that multiplication by the function S F = F − P ( F ) ∈ C ∞ ( X ◦ )is bounded L ( X ) → x L ( X ). For instance, S F = O ( x ) is sufficient. We thus makethe following definition: Definition 4.2.
We say that a twisting function, either smooth or logarithmic, F is admissible if S F ∈ x L ∞ loc ( X ).We now discuss conditions under which it is possible to choose an admissible twistingfunction, which plays an important role when considering Neumann or Robin boundaryconditions; for Dirichlet boundary conditions any smooth twisting function suffices,essentially due to Lemma 3.5 (for more details, see Section 5.1). In particular, we onlyneed to consider the case ν ∈ (0 , ν = 1 /
2, so the difference between the indicial roots ν + − ν − = 2 ν of P is not an integer. Then there always exists an admissible smooth twisting function,which can actually be chosen to satisfy S F ∈ ˙ C ∞ ( X ) . (Cf. [Vas4, Lemma 4.13]). The terms in the polyhomogeneous expansion of F areuniquely determined by specifying x − ν − F | ∂X (which should be non-vanishing) for somechoice of bdf x . More explicitly, let x be a special bdf. Then in special coordinates,we can take F ( x, y ) = x ν − (1 + xf ( y )) , f ( y ) = ν − E (0 , y ) b N ( P )( − i (1 + ν − )) . We can simply multiply this F by a function of y to specify the restriction x − ν − F | ∂X .On the other hand, if ν = 1 /
2, then ν + − ν − = 1 differ by an integer. We can stillfind a logarithmic twisting function, but not necessarily a smooth one. Following thearguments in [Vas4, Lemma 4.12] and referring to (4.4), the existence of an admissiblesmooth twisting function when ν = 1 / ∂X with respect to ˆ g . This condition is independent of the choice ofspecial bdf.For simplicity, we henceforth focus on the case where there exists an admissiblesmooth twisting function. This is just to avoid some of the technicalities associatedwith operators with conormal coefficients as discussed Section 3.5. Hypothesis 4.3. If ν = 1 /
2, then the mean curvature of ∂X with respect to ˆ g vanishes. In other words, there exists a function F ∈ x ν − C ∞ ( X ) satisfying x − ν − F > S F ∈ x C ∞ ( X ) . It will also be convenient to define a sesquilinear pairing on one-forms relative to afixed C ∞ positive-definite inner product H . We use this to measure the inner productof differentials twisted by a fixed twisting function F (to be chosen). Introduce theDirichlet form associated to H by Q ( u, v ) = Z H ( d F u, d F ¯ v ) x dg. Thus the H norm-squared of u ∈ H ( X ) can be taken to be Q ( u, u ) + k u k L ( X ) .4.4. Asymptotic expansions.
Next, we discuss asymptotic expansions of solutionsto the Klein–Gordon and related equations. Given an aAdS spacetime (
X, g ) and k ∈ R ∪ {±∞} , define the spaces X k = { u ∈ H ,k loc ( X ) : P u ∈ x H ,k loc ( X ) } . We also abbreviate X = X . For finite k we equip these spaces with the seminorms u
7→ k φu k H ,k ( X ) + k x − φP u k H ,k ( X ) , and when k = ∞ we use the collection of all seminorms for all finite k . The maintechnical difficulty is that X k is not closed under applications of A ∈ Ψ ( X ). In fact, ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 25 it is not closed under multiplication by arbitrary C ∞ c ( X ) functions either. However,it turns out that closedness does hold if we work with b-pseudodifferential operatorssatisfying certain evenness properties with respect to a special bdf.Given an aAdS spacetime that is even modulo O ( x k +1 ), we introduce in AppendixB a class of b-pseudodifferential operators Ψ m b , even ( X ) ⊂ Ψ m b ( X ) whose coefficients areeven modulo O ( x k +3 ). Note that every aAdS spacetimes is even modulo O ( x ), sothis construction is always non-trivial. The definition mirrors a similar constructionby Albin within the 0-calculus [Alb]. The key property of Ψ b , even ( X ) is the following: Lemma 4.4.
Let B ∈ Ψ m b , even ( X ) have compact support in a boundary coordinate patchwith special coordinates ( x, y , . . . , y n − ) . There exists B ′ ∈ Ψ m b ( X ) and B ′′ ∈ x Ψ m b ( X ) such that Q F BF − = B ′ Q + B ′′ Proof.
Recall that Q = F D x F − . It suffices to show that (2.4) holds with A ′′ ∈ x Ψ m b ( X ), since the result follows by conjugating (2.4) by F . Tracing through theproof of (2.4) in [Vas2, Section 2], one can take A ′′ = x − [ xD x , B ] . The key here is that [ xD x , B ] has a vanishing indicial family, and hence lies in x Ψ m b ( X )in general. However, since xD x ∈ Ψ , even ( X ) and we are assuming that B ∈ Ψ m b , even ( X ),it follows that [ xD x , B ] ∈ x Ψ m b ( X )according to Lemma B.6. This shows that A ′′ ∈ x Ψ m b ( X ). (cid:3) The additional order of vanishing when A ∈ Ψ m b , even ( X ) is crucial when consideringthe action of A on X k . Lemma 4.5. If B ∈ Ψ m b , even ( X ) , then A = F BF − maps X k → X k − m for each k ∈ R .Proof. Since A maps H ,k loc ( X ) → H ,k − m loc ( X ), the only additional point to verify is that P u ∈ x H ,k loc ( X ) = ⇒ P Au ∈ x H ,k − m loc ( X ) . Now
P Au = AP u + [
P, A ] u , and since AP u ∈ x H ,k − m loc ( X ), it suffices to consider thecommutator. Also, x − [ P, A ] = [ x − P, A ] − [ x − , A ] P and [ x − , A ] ∈ x − Ψ m − ( X ), so [ x − , A ] P u ∈ H ,k − m loc ( X ). Recall that the distribu-tional action of x − P is given by x − P = Q ∗ Q + K, where K = Q ∗ α ˆ g αβ Q β + x − S F ∈ Ψ ( X ). Thus [ K, A ] ∈ Ψ m +1b ( X ), so [ K, A ] u ∈H ,k − m loc ( X ). As for the final term,[ Q ∗ Q , A ] = Q ∗ [ Q , A ] + [ Q ∗ , A ] Q = Q ∗ [ Q , A ] − [ Q , A ∗ ] ∗ Q . Now use Lemma 4.4 to write [ Q , A ] = A Q + A , where A ∈ x Ψ m b ( X ) and A ∈ Ψ m − ( X ). As a consequence, we can write Q ∗ [ Q , A ] = Q ∗ ( A Q + A ) = ( A ′ Q ∗ + A ′′ ) Q + Q ∗ A , where now A ′ , A ′′ ∈ Ψ m − ( X ). For the term Q ∗ A , we use the fact that Q ∗ x = Q x + Ψ ( X ) to see that Q ∗ A u ∈ H ,k − m loc ( X ). We can also write( A ′ Q ∗ + A ′′ ) Q = A ′ x − P − A ′ K + A ′′ Q , so when applied to u this is also in H ,k − m loc ( X ). In conclusion, Q ∗ [ Q , A ] u ∈ H ,k − m loc ( X ).The term [ Q , A ∗ ] ∗ Q is handled analogously. (cid:3) One application of Lemma 4.5 is when A is multiplication by a cutoff function χ ∈ C ∞ c , even ( X ) (see Appendix B for notation). We can always find a partition of unity { χ i } subordinate to a covering of X by either interior or special boundary coordinatespatches, such that χ i ∈ C ∞ c , even ( X ) in the latter case. This allows us to reduce the studyof X k to a local one.First we work locally, assuming that X = R n + . Assume that g is an aAdS metricgiven in standard coordinates ( x, y ) by g = − dx + k αβ ( x, y ) dy α dy β x . (4.5)We show that any u ∈ H ( R n + ) with compact support and satisfying P u ∈ x L ( R n + )admits a partial asymptotic expansion. Fix an admissible twisting function on R n + satisfying x − ν − F = 1 when x = 0. Lemma 4.6.
Let g be an asymptotically AdS metric on R n + of the form (4.5) , and set r = ( n − / . If u ∈ H ,k c ( R n + ) , P u ∈ x H ,k c ( R n + ) for k ≥ , then the restriction of u to any half-plane { x < ε } admits an asymptoticexpansion u = F u − + x ν + u + + x r +2 H k +2b ([0 , ε ); H k − ( R n − )) , (4.6) where u − ∈ H ν + k ( R n − ) and u + ∈ H − − ν + k ( R n − ) . Furthermore u − = γ − u .Proof. Assume that k = 0; the general case is proved analogously. We have that P isa differential operator of the form P = ( xD x ) + i ( n − xD x + i ( xE ) xD x + x ˜ P, ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 27 with E ∈ C ∞ ( R n + ) and ˜ P ∈ Diff ( R n + ). Since P − b N ( P ) ∈ x Diff ( X ), it follows that b N ( P ) u ∈ x r +1 L ( R + ; H − ( R n − )) . Applying the Mellin transform (which is uniformly square integrable on horizontal lineswith Im s > r + ) and deforming the contour to any horizontal line with Im s > r − ,we obtain u = x ν − u − + u (4.7)when x < ε , where u ∈ x r +1 H ([0 , ε ); H − ( R n − )) and u − ∈ H − ( R n − ). As inLemma 3.3, by interpolation u − ∈ H ν − ( R n − ). On the other hand, u − = γ − u , where γ − is the trace from Lemma 3.3, since u − in a partial expansion of the form (4.7) isunique. This shows that in fact u − ∈ H ν ( R n − ), which is a stronger statement since ν ∈ (0 , φ = φ ( x ) ∈ C ∞ c ( R + ) be identically one on supp u . Replacing u with φu , wecan write u = x ν − φu − + u on R + , with u of compact support. Then b N ( P ) u = P u − i ( xE ) xD x u − x ˜ P u , and the latter two terms lie in x r +2 L ( R + ; H − ( R n − )) by the properties of u . On theother hand, P u = P u − P ( x ν − φu − ) = − P ( x ν − φu − ) + x L ( R n + ) . Thus the only obstruction to having b N ( P ) u ∈ x r +2 L ( R + ; H − ( R n − )) is the term P ( x ν − φu − ), which a priori is merely in x r +1 L ( R + ; H − ( R n − )).This is remedied by replacing u − with F u − , which corresponds to replacing u with u + ( x ν − − F ) φu − . Thus u = F φu − + u , where we still have u ∈ x r +1 H ( R + ; H − ( R n − )). Now we repeat the same argumentto obtain an asymptotic expansion for u . When x < ε , this yields u = F u − + x ν + u + + u , where u ∈ x r +2 H ([0 , ε ); H − ( R n − )) and u + ∈ H − ( R n − ). The a priori regularity of u + can be improved by interpolation to give u + ∈ H − − ν ( R n − ). (cid:3) Suppose that in Lemma 4.6 we can take k = ∞ . Then the residual term on theright-hand side of (4.6), denoted in the proof by u , is in x H , ∞ loc ([0 , ε ) × R n − ). BySobolev embedding, this implies that Lu ∈ x ( n +1) / L ∞ loc ([0 , ε ) × R n − ) (4.8)for each b-differential operator L . Next, we show that X ∞ is dense in X k for each k ∈ R . Lemma 4.7.
Let ( X, g ) be an aAdS spacetime. If ν ∈ (0 , and k ∈ R , then X ∞ isdense in X k . Proof.
The proof is standard, making sure to use regularizers in Ψ −∞ b , even ( X ). Let { A r ∈ Ψ −∞ b ( X ) : r ∈ (0 , } be a compactly supported bounded family in Ψ ( X ) such that A r → δ b ( X ) foreach δ >
0. We may furthermore arrange that A r = F B r F − , B r ∈ Ψ −∞ b , even ( X ) . According to Lemma 4.5, A r u ∈ X ∞ . Tracing through the proof of Lemma 4.5 alsoshows that [ P, A r ] converges to zero strongly on X k , and so A r u → u in the graphnorm of X k . (cid:3) For simplicity, to define γ − on H ( X ) (which recall depends on the choice of bdf)let us fix once and for all a reference special bdf x . Given this choice of x , we also fixan admissible twisting function F normalized by x − ν − F | ∂X = 1. We then define thesecond trace γ + on X ∞ by γ + u = x − ν ∂ x ( F − u ) | ∂X . This is well defined in light of Lemma 4.6 which shows that the restriction of u to aspecial coordinate patch can be written in the form u = x ν − F u − + x ν + u + + u , u ∈ x H , ∞ loc ([0 , ε ) × R n − ) . Appealing to (4.8) shows that x − ν ∂ x ( F − u ) ∈ x − ν L ∞ loc ([0 , ε ) × R n − ) , which therefore vanishes when x = 0. Thus, in these coordinates, γ + u = 2 νu + . In thenext section we extend γ + to X k .4.5. Green’s formula.
Let ν ∈ (0 , u ∈ X ∞ and v ∈ x ν − C ∞ c ( X ).If either γ + u = 0 or γ − v = 0, then Z P u · ¯ v dg = E ( u, v ) + Z S F u · ¯ v dg. (4.9)Because X ∞ is dense in the graph space X , if v ∈ ˙ H ( X ) is fixed, then (4.9) is alsovalid for arbitrary u ∈ H ( X ) satisfying P u ∈ x L ( X ). More generally, however,there are boundary terms, and the correct Green’s formula is Z P u · ¯ v dg = E ( u, v ) + Z S F u · ¯ v dg + Z γ + u · γ − ¯ v dk (4.10)for u ∈ X ∞ and v ∈ H ( X ). This formula can be extended to more general u asfollows. Lemma 4.8.
Given k ∈ R , the map γ + extends to a bounded map X k → H k − ν loc ( X ) .Moreover, if u ∈ X k , then Green’s formula (4.10) holds for every v ∈ H , − k c ( X ) . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 29
Proof.
We focus on the case k = 0. Notice that the proof of Green’s formula wouldbe trivial if we a priori knew that γ + maps continuously into H − ν loc ( X ). As it stands,however, we must proceed differently. Given u ∈ H ( X ) with P u ∈ x L ( X ),consider the linear functional ℓ defined on v − ∈ C ∞ c ( ∂X ) given by ℓ ( v − ) = Z P u · ¯ v dg − E ( u, v ) − Z S F u · ¯ v dg, where v ∈ x ν − C ∞ c ( X ) is any element such that v − = γ − v . This is well defined in viewof (4.9). In particular, it is possible to take v = ρv − , where ρ : H ν c ( ∂X ) → H ( X )is a continuous extension that maps C ∞ c ( ∂X ) onto x ν − C ∞ c ( X ) (as in Lemma 3.4). If φ ∈ C ∞ c ( X ) is such that φ = 1 on supp ρv − , then | ℓ ( v − ) | ≤ C (cid:0) k x − φP u k L ( X ) + k φu k H ( X ) (cid:1) k ρv − k H ( X ) ≤ C k v − k H ν ( ∂X ) . By Hahn–Banach, there exists an element of H − ν loc ( X ), which we denote by ˜ u + , suchthat ℓ ( v − ) = h ˜ u + , v − i ∂X , and furthermore k φ ˜ u + k H − ν ( ∂X ) ≤ C (cid:0) k x − φP u k L ( X ) + k φu k H ( X ) (cid:1) . This shows that u ˜ u + is a continuous map from the graph space to H − ν loc ( X ), andthat (4.10) holds for arbitrary u in the graph space, provided γ + u is replaced with ˜ u + .It therefore remains to show that γ + u = ˜ u + . This is true if u ∈ X ∞ by (4.10), andthus also holds for u ∈ X by the density of X ∞ . (cid:3) The boundary value problem
The Dirichlet form.
The Dirichlet problem is given a weak formulation in theusual way. For a fixed smooth twisting function F , define E D ( u, v ) = E ( u, v ) + Z S F u · ¯ v dg. For an arbitrary F , in general one only has S F ∈ xC ∞ ( X ). By Hardy’s inequalityin one dimension, this is enough to guarantee that multiplication by S F is bounded˙ H ( X ) → x L ( X ). We then define the map P D : ˙ H ( X ) → H − ( X ) by h P D u, v i = E D ( u, v ) . This agrees with the distributional action of P on ˙ H ( X ). As in the discussionfollowing 3.10, we can also extend P D to a map ˙ H ,m loc ( X ) → H − ,m loc ( X ) for each m ∈ R .The only additional ingredient needed is an extension of Lemma 3.5 to spaces withgeneral conormal regularity. This latter statement is deduced from Lemma 3.5 bywriting xA = A ′ x , where A ′ = xAx − ∈ Ψ m b ( X ) whenever A ∈ Ψ m b ( X ). Next we consider the Robin problem under the assumption that an admissible twist-ing function exists. Formally, we consider u ∈ H ( X ) satisfying the boundary condi-tions γ + u − βγ − u = 0 , (5.1)where β ∈ C ∞ ( ∂X ). Eventually we will restrict to the case where β is real-valued.These boundary conditions are well-defined in the strong sense provided P u ∈ x L ( X ).For the weak formulation, set E R ( u, v ) = E ( u, v ) + Z S F u · ¯ v dg + Z βγ − u · γ − ¯ v dk . Note the importance of the requirement that S F ∈ x L ∞ loc ( X ): the weaker condition S F ∈ x C ∞ ( X ) does not guarantee that S F is bounded ˙ H ( X ) → x L ( X ), unlike inthe Dirichlet case. We then define P R : H ( X ) → ˙ H − ( X ) by h P R u, v i = E R ( u, v ) . If f ∈ L ( X ) ⊂ ˙ H − ( X ), then P R u = f implies that x − P u = f in distributions andthat the Robin boundary conditions (5.1) is satisfied in the strong sense. Remark 5.1.
We like to think of the distributional action of P as extending to amap X → x L ( X ). Note, however, that X ⊂ L ( X ), and in this sense P acts on L -based spaces with different weights. On the other hand, when defining the Robin(or Dirichlet) realization of P via a sesquilinear form, we have that P R maps H ( X )to a dual space relative to the x dg inner product. Since L ( X ) embeds in ˙ H − ( X ), itmakes sense to solve equations of the form P R u = f with f ∈ L ( X ). This inevitablyleads to statements like P R u = f = ⇒ x − P u = f in C −∞ ( X )above with weights that at first glance appear contradictory.Just as for the Dirichlet problem, we can also extend P R : H ,m loc ( X ) → ˙ H − ,m loc ( X ).In this case we must also show that the boundary pairing makes sense; this followsfrom the regularity γ − u ∈ H ν + m loc ( X ), valid with any m ∈ R for which u ∈ H ,m loc ( X ).5.2. Microlocal estimates.
We give some microlocal estimates for the Dirichlet form.We always work with b-pseudodifferential operators having compact support in a fixedcoordinate patch. Near the boundary, it is convenient to use special coordinates( x, y , . . . , y n − ), where x is our fixed special bdf. Lemma 5.2.
Let U ⊂ X be a boundary coordinate patch and m ≤ . Let A = { A r : r ∈ (0 , } be a bounded subset of Ψ s b ( X ) with compact support in U , such that A r ∈ Ψ m b ( X ) for each r ∈ (0 , . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 31
Let G ∈ Ψ s − / ( X ) be elliptic on WF ′ b ( A ) , with compact support in U . Then thereexists C > and χ ∈ C ∞ c ( U ) such that E ( A r u, A r u ) ≤ E ( u, A ∗ r A r u ) + C ( k G u k H ( X ) + k χu k H ,m ( X ) ) for every r ∈ (0 , and every u ∈ H ,m loc ( X ) , provided WF ,s − / ( u ) ∩ WF ′ b ( G ) = ∅ .Proof. Note that A ∗ r A r u ∈ H , − m c ( X ) for r ∈ (0 , E ( u, A ∗ r A r u ) iswell-defined (the precise order of each individual A r does not play an important role,and is simply chosen to justify the pairings; the order of the family A is the crucialpoint). Commuting twice, E ( u, A ∗ r A r u ) = h A r ˆ g jk Q j u, Q k A r u i + h ˆ g jk Q j u, [ Q k , A ∗ r ] A r u i = E ( A r u, A r u ) + h [ A r , ˆ g jk Q j ] u, Q k A r u i + h ˆ g jk Q j u, [ Q k , A ∗ r ] A r u i . The two commutator terms are of lower order; since both can be treated in the sameway we focus on the first of them. Here it is convenient to distinguish the commutatorswith Q j in the cases j = 0 or j = 0. Thus we write h [ A r , ˆ g jk Q j ] u, Q k A r u i = h [ A r , Q ] u, Q A r u i + h [ A r , ˆ g αβ Q α ] u, Q β A r u i (5.2)where α, β range only over 1 , . . . , n −
1. The second commutator term in (5.2) can behandled entirely within the b-calculus, so we consider only the first term.Let Λ / ∈ Ψ / ( X ) be everywhere elliptic, and let Λ − / ∈ Ψ − / ( X ) be a para-metrix, so that 1 = Λ / Λ − / + R where R ∈ Ψ −∞ b ( X ); in that case, h [ A r , Q ] u, Q A r u i = h Λ ∗ / [ A r , Q ] u, Λ − / Q A r u i + hh [ A r , Q ] u, RQ A r u i . (5.3)Now for any A ∈ Ψ s b ( X ) with compact support in U , repeated applications of Lemma3.7 show that the first term on the hand side of (5.3) can be written as h Λ ∗ / [ A, Q ] u, Λ − / Q Au i = h ( Q A ′ + A ′′ ) u, ( Q B ′ + B ′′ ) u i , for some A ′ , B ′ ∈ Ψ s − ( X ) and A ′′ , B ′′ ∈ Ψ s − ( X ), where the maps A ( A ′ , A ′′ , B ′ , B ′′ )are microlocal. Applying this in particular to A = A r ∈ A , it follows that the corre-sponding families A ′ = { A ′ r : r ∈ (0 , } , A ′′ = { A ′′ r : r ∈ (0 , } are bounded in Ψ s − ( X ) and Ψ s b ( X ) respectively, andWF ′ b ( A ′ ) ∪ WF ′ b ( A ′′ ) ⊂ WF ′ b ( A ) ⊂ ell b ( G ) . Since the same is true of the families formed by B ′ r and B ′′ r , we can bound |h Λ ∗ / [ A r , Q ] u, Λ − / Q A r u i| ≤ C ( k G u k H ( X ) + k χu k H ,m ( X ) )with C > r , for a suitable cutoff χ ∈ C ∞ c ( U ). The second term in(5.3) involving R can be bounded by k χu k H ,m ( X ) itself. (cid:3) Now we consider the Robin problem, always assuming the existence of an admissibletwisting function F with S F ∈ x C ∞ ( X ). Lemma 5.3.
Let U ⊂ X be a boundary coordinate patch and m ≤ . Let A = { A r : r ∈ (0 , } be a bounded subset of Ψ s b ( X ) with compact support in U , such that A r ∈ Ψ m b ( X ) for each r ∈ (0 , . Let G ∈ Ψ s b ( X ) and G ∈ Ψ s − / ( X ) both be elliptic on WF ′ b ( A ) , with compact supportin U . Then there exists C ε > and χ ∈ C ∞ c ( U ) such that E ( A r u, A r u ) − ε Q ( A r u, A r u ) ≤ C ε ( k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) + k G u k H ( X ) + k G P R u k H − ( X ) ) for every r ∈ (0 , and every u ∈ H ,m loc ( X ) , provided WF ,s − / ( u ) ∩ WF ′ b ( G ) = ∅ , WF − ,s b ( P R u ) ∩ WF ′ b ( G ) = ∅ . Proof.
Let f = P R u . By definition E R ( u, A ∗ r A r u ) = h A r f, A r u i , and according toLemma 3.13, |h A r f, A r u i| ≤ k A r f k ˙ H − ( X ) k A r u k H ( X ) ≤ ε k A r u k H ( X ) + C ε ( k G f k H − ( X ) + k χu k H ,m ( X ) ) ≤ ε Q ( A r u, A r u ) + C ε ( k G f k H − ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) ) . It remains to bound the difference between E R ( u, A ∗ r A r u ) and E ( A r u, A r u ). Firstconsider E R ( u, A ∗ r A r u ) − E ( u, A ∗ r A r u ) = h x − S F u, A ∗ r A r u i + h βγ − u, γ − ( A ∗ r A r u ) i ∂X . By Lemma 3.13, we have |h x − S F u, A ∗ r A r u i| ≤ C ( k χu k H ,m ( X ) + k G u k H ( X ) ) . This estimate would be true even with G ∈ Ψ s − ( X ). Next, we must handle theboundary terms. Recall that for any B ∈ Ψ m b ( X ), γ − ( Bu ) = ( x − ν − Bu ) | ∂X = b N ( B )( − iν − )( γ − u ) . As in Section 2.2, we have b N ( A ∗ r A r )( − iν − ) = b N ( ˜ A r )( − iν ) ∗ b N ( A r )( − iν ) , where ˜ A r = x ν − A r x − ν − , and the adjoint on the right is with respect to the induceddensity dk . Extend β arbitrarily to a C ∞ function on X , so multiplication by β on ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 33 the boundary can be written as b N ( β )( − iν − ). Then h βγ − u , γ − ( A ∗ r A r u ) i ∂X = h b N ( ˜ A r β )( − iν − ) γ − u, b N ( A r )( − iν − ) γ − u i ∂X = h γ − ( ˜ A r βu ) , γ − ( A r u ) i ∂X = h γ − ( β ˜ A r u + [ ˜ A r , β ] u ) , γ − ( A r u ) i ∂X . Note that ˜ A r has the same principal symbol as A r , so by (3.6), |h βγ − u , γ − ( A ∗ r A r u ) i| ≤ ε Q ( A r u, A r u ) + C ε ( k G u k H ( X ) + k χu k H ,m ( X ) )for every ε >
0, Again, this would even be true with G ∈ Ψ s − ( X ). Combining thesefacts with Lemma 5.2 finishes the proof. (cid:3) Lemma 5.3 also holds for the Dirichlet problem, this time taking u ∈ ˙ H ,m loc ( X ) with m ≤
0. Here it suffices to work with an arbitrary twisting function. There are twodifferences: firstly, there is no boundary integral to estimate, and secondly, because S F ∈ x C ∞ ( X ), |h A r x − S F , A r u i| = |h ( xA r x − ) x − S F u, x − A r u i|≤ C ( k G u k H ( X ) + k χu k H ,m ( X ) ) k A r u k H ( X ) by Hardy’s inequality in one dimension. Therefore |h A r x − S F , A r u i| ≤ ε Q ( A r u, A r u ) + C ε ( k G u k H ( X ) + k χu k H ( X ) )by Cauchy–Schwarz and Lemma 3.13.Next, we consider properties of Im E ( u, A ∗ Au ). This will be used in the positivecommutator arguments. If u ∈ H ( X ) and A ∈ Ψ ( X ) has compact support, then E ( u, Au ) − E ( Au, u ) = h ˆ g ij Q j u, Q i Au i − h ˆ g ij Q j Au, Q i u i = h ˆ g ij Q j u, [ Q i , A ] u i − h [ˆ g ij Q j , A ] u, Q i u i + h ( A ∗ − A )ˆ g ij Q j u, Q i u i . (5.4)If A is replaced with A ∗ A , then the third term vanishes. Therefore we have2 i Im E ( u, A ∗ Au ) = h ˆ g ij Q j u, [ Q i , A ∗ A ] u i − h [ˆ g ij Q j , A ∗ A ] u, Q i u i = h Q u, [ Q , A ∗ A ] u i − h [ Q , A ∗ A ] , Q u i + h [ Q α ˆ g αβ Q β , A ∗ A ] u, u i . With a = σ b , ( A ), h Q u, [ Q , A ∗ A ] u i − h [ Q , A ∗ A ] , Q u i = h Q u, Q A u i − h Q A u, Q u i + h Q u, A u i − h A u, Q u i , where σ b , − ( A ) = (1 /i ) ∂ σ ( a ) and σ b , ( A ) = (1 /i ) ∂ x ( a ). Propagation of singularities
The characteristic variety and bicharacteristics.
The principal symbol of x − P , as a function on T ∗ X ◦ \
0, is given by ˆ p = − ˆ g ij ζ i ζ j , where we have writtencovectors in coordinates ( z , . . . , z n − ) as ζ i dz i . Note that the mass parameter λ playsno role in this expression. Furthermore, ˆ p extends smoothly to a function on T ∗ X \ N = { ˆ p = 0 } ⊂ T ∗ X \ p . The compressed characteristic set ˙ N is the imageof N in b ˙ T ∗ X \ π : T ∗ X → b ˙ T ∗ X . We equip ˙ N with thesubspace topology inherited from b T ∗ X . This is the same as the quotient topology;cf. [Vas2, Lemma 5.1].There is a natural decomposition of b ˙ T ∗ X \ E = { q ∈ b ˙ T ∗ X \ π − ( q ) ∩ N = ∅} , G = { q ∈ b ˙ T ∗ X \ | π − ( q ) ∩ N | = 1 } , H = { q ∈ b ˙ T ∗ X \ | π − ( q ) ∩ N | = 2 } . Let U be a boundary coordinate patch with coordinates ( x, y , . . . , y n − ), where x isa special bdf and dx is orthogonal to each dy i . If ( x, y, σ, η ) are the correspondingcanonical coordinates on b T ∗ U X , then a point q = ( x , y , σ , η ) ∈ b ˙ T ∗ U X is in H precisely if q ∈ T ∗ ∂X (namely x = σ = 0) andˆ g αβ (0 , y )( η ) α ( η ) β > . Similarly, a point q ∈ T ∗ ∂X is in G when ˆ g αβ (0 , y )( η ) α ( η ) β = 0. There are severalequivalent definitions of GBBs in this setting, but we choose the following: Definition 6.1. If I ⊂ R is an interval, we say that a continuous map γ : I → ˙ N isa GBB if the following two conditions are satisfied for each s ∈ I :(1) If q = γ ( s ) ∈ G , then for every f ∈ C ∞ ( b T ∗ X ), dds ( f ◦ γ )( s ) = { ˆ p, π ∗ f } ( η ) , where η ∈ N is the unique point for which π ( η ) = q .(2) If q = γ ( s ) ∈ H , then there exists ε > < | s − s | < ε impliesthat x ( γ ( s )) = 0. ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 35
Continuity of γ implies that tangential momentum is conserved upon interactionwith the boundary. The second condition, at hyperbolic points, says that GBBs reflectinstantaneously.Since ˆ p is a smooth function on T ∗ X , various properties of GBBs that are true inthe setting of smooth boundary value problems are also valid here. In particular, theentire discussion in [Vas2, Section 5] applies verbatim.6.2. Elliptic estimates.
It is convenient to introduce a normal coordinate systemas follows: given a point p ∈ ∂X , fix a spacelike surface in ∂X (with respect to k )passing through p . We can then choose coordinates ( y , . . . , y n − ) such that k − is ofthe form k − = ∂ y n − − X h ab ∂ y a ∂ y b , where a, b range over 1 , . . . , n − h ab is positive definite. Transporting thesecoordinates to a collar neighborhood of X using the product structure, we see that g − = x ( − ∂ x − h ab ∂ y a ∂ y b + ∂ y n − + O ( x )) . This is useful for the following reason: if v has support in {| x | < δ } , then E ( v, v ) = k Q v k L ( X ) + h ˆ g αβ Q α v, Q β v i≥ k Q v k L ( X ) − (1 + Cδ ) k Q n − v k L ( X ) + (1 − Cδ ) h h αβ Q α v, Q β v i , (6.1)where C > δ >
0. Furthermore, the principal symbol ˆ p restrictedto T ∗ Y X is just ˆ p | T ∗ Y X = ξ + h ab η a η b − η n − . Let q ∈ b T ∗ Y X \ q / ∈ ˙ N . Thus there are two possibilities: either q isnot in the compressed b-cotangent bundle, or if it is, then h ab η a η b > η n − at q . Thisobservation is rephrased as follows: Lemma 6.2. If q ∈ b T ∗ Y X \ , then there is a conic neighborhood V of q in whichone of the following is true:(1) There is ε > such that σ < ε ( η n − + h ab η a η b ) and h ab η a η b > (1 + ε ) η n − .(2) There is C > such that | η n − | < C | σ | . Using this observation, it is simple to prove the following elliptic regularity foreither the Dirichlet or Robin (Neumann) problem. Since the proofs are identical uponsubstituting the appropriate function spaces, we focus on the Robin case.
Theorem 3.
Let u ∈ H ,m loc ( X ) for some m ≤ , and q ∈ b T ∗ Y X \ . If s ∈ R ∪ { + ∞} and q ∈ WF ,s b ( u ) \ WF − ,s b ( P R u ) , then q ∈ ˙ N . Proof.
Assume that q / ∈ ˙ N . We show that if q / ∈ WF − ,s b ( P R u ) and q / ∈ WF ,s − / ( u ),then q / ∈ WF ,s b ( u ) . The proof is then finished by induction; the inductive hypothesis is satisfied for any s ≤ / m by the assumption that u ∈ H ,m loc ( X ). Let A = { A r : r ∈ (0 , } be abounded subset of Ψ s b ( X ) such that A r ∈ Ψ m − ( X ) for each r ∈ (0 , ′ b ( A ) is contained in a sufficiently small neighborhood of V so thatWF ,s − / ( u ) ∩ WF ′ b ( A ) = ∅ . Fixing a boundary coordinate patch U containing q , assume in addition that A hascompact support in this patch. We consider two cases corresponding to those of Lemma6.2, letting V be a conic neighborhood of q as in the lemma.(1) If C > δ > − Cδ ) h ab η a η b − (1 + Cδ ) η n − = (1 − Cδ )( h ab η a η b − η n − ) − Cδη n − > ( ε (1 − Cδ ) − Cδ ) η n − > ( ε/ η n − on V . This implies that (1 − Cδ ) h ab η a η b − (1 + Cδ ) η n − is elliptic near V . If the family A has support in {| x | < δ } , we can choose B ∈ Ψ ( X ) with WF ′ b ( A ) ⊂ ell b ( B ) suchthat WF ′ b ((1 − Cδ ) h ab Q ∗ a Q b − (1 + Cδ ) Q ∗ n − Q n − − B ∗ B + T ) ∩ U = ∅ , where T ∈ Ψ ( X ). Therefore, for each r ∈ (0 , E ( A r u, A r u ) ≥ k Q A r u k L ( X ) + k BA r u k L ( X ) − h T A r u, A r u i≥ C − Q ( A r u, A r u ) − h T A r u, A r u i . Note that this pairing makes sense since A r ∈ Ψ m b ( X ) and T ∈ Ψ ( X ), using that u ∈ H ,m loc ( X ).(2) The second case is similar, noting that for v with support in {| x | < δ } , k Q v k L ( X ) ≥ δ − k xQ v k L ( X ) . Consider the operator (2 δ ) − ( xQ ) ∗ ( xQ ) − (1 + Cδ ) Q ∗ n − Q n − , which is in Ψ ( X )with principal symbol σ δ − (1 + Cδ ) η n − > cη n − on V . In particular, (2 δ ) − ( xQ ) ∗ ( xQ ) − (1 + Cδ ) Q ∗ n − Q n − is elliptic on V , so againwe can find B ∈ Ψ ( X ) with V ⊂ ell b ( B ) such thatWF b ((2 δ ) − ( xQ ) ∗ ( xQ ) − (1 + Cδ ) Q ∗ n − Q n − − B ∗ B + F ) ∩ U = ∅ , ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 37 where T ∈ Ψ ( X ). Again we find that for each r ∈ (0 , E ( A r u, A r u ) ≥ k Q Au k L ( X ) + (1 − Cδ ) h h αβ Q α v, Q β v i + k BA r u k L ( X ) − h T A r u, A r u i≥ C − Q ( A r u, A r u ) − h T A r u, A r u i . Let Λ / ∈ Ψ / ( X ) be elliptic with Λ − / ∈ Ψ − / ( X ) a parametrix. Then in bothcases we can write h T A r u, A r u i = h Λ ∗− / T A r u, Λ / A r u i + h T A r u, RA r u i with R ∈ Ψ −∞ b ( X ), which shows that |h T A r u, A r u i| is uniformly bounded in r ∈ (0 , u .Finally, we choose the family A . Let A ∈ Ψ s b ( X ) be elliptic at q , with compactsupport in U ∩ { x < δ } . Let { J r : r ∈ (0 , } be a bounded family in Ψ ( X ) such that J r ∈ Ψ m − s − ( X ) for each r ∈ (0 , ( X ) as r →
0. Wethen let A r = J r A, so that in particular A r u → Au in C −∞ c ( X ). Taking ε > A r u is uniformly bounded in H ( K ) for asuitable compact set K ⊂ X . Extracting a weakly convergent subsequence in H ( K )shows that Au ∈ H ( K ). (cid:3) Combined with standard elliptic regularity away from ∂X , we have shown the fol-lowing: Corollary 6.3. If u ∈ H ,m loc ( X ) for some m ≤ and s ∈ R ∪ { + ∞} , then WF ,s b ( u ) \ WF − ,s b ( P R u ) ⊂ ˙ N The same result holds for the Dirichlet problem, now taking u ∈ ˙ H ,m loc ( X ).6.3. The hyperbolic region.
Now we focus on the hyperbolic region. Fix a boundarycoordinate patch U with coordinates ( x, y , . . . , y n − ), and let q ∈ H ∩ b T ∗ U X . Incoordinates ( x, y, σ, η ), this means that q = (0 , y , , ζ ) , ˆ g αβ (0 , y )( η ) α ( η ) β > . Let u ∈ H ,m loc ( X ) for some m ≤
0, and suppose that q / ∈ WF − ,s +1b ( P R u ). In order toprove propagation of singularities through hyperbolic points, it suffices to show that q ∈ WF ,s b ( u ) implies q is an accumulation point of WF ,s b ( u ) ∩ { σ < } . Proposition 6.4.
Let u ∈ H ,m loc ( X ) for some m ≤ , and suppose that q / ∈ WF − ,s +1b ( P R u ) . If there exists a conic neighborhood W ⊂ T ∗ X \ of q such that W ∩ { σ < } ∩ WF ,s b ( u ) = ∅ , then q / ∈ WF ,s b ( u ) . It suffices to prove the proposition with any conic subset W ⊂ W containing q . If W is sufficiently small, we can assume W ∩ WF − ,s +1b ( P R u ) = ∅ . In particular, W ∩ WF ,s b ( u ) ⊂ ˙ N by elliptic regularity, so x = 0 on W ∩ { σ < } ∩ WF ,s b ( u ). Thus, if q ∈ WF ,s b ( u ),then q is the limit of points in the wavefront set intersected with the interior.The rest of this section is dedicated to the proof of Proposition 6.4. The proofproceeds iteratively, increasing the regularity by 1 / q / ∈ WF ,s − / ( u ). We thenshow that q / ∈ WF ,s b ( u ). Note that the inductive hypothesis is always satisfied for s ≤ / m .As in Section 6.2, we can further choose coordinates ( y , . . . , y n − ) such thatˆ g αβ (0 , y ) η α η β = η n − − h ab ( y ) η a η b . In particular, η n − is non-vanishing at q . If κ : b T ∗ X \ → b S ∗ X is the canonicalprojection , then in a sufficiently small neighborhood of κ ( q ) in b S ∗ X we may useprojective coordinates x, y, ˆ σ = ρ − σ, ˆ η a = ρ − η a , where we define ρ = | η n − | . Closely following [Vas2, Section 6], define the functions ω = | x | + | y − y | + X | ˆ η a − (ˆ η ) a | , φ = ˆ σ + 1 β δ ω. The parameters δ, β will be chosen later; β > δ > q . Choose cutofffunctions χ , χ with the following properties: • χ is supported in [0 , ∞ ), with χ ( s ) = exp( − /s ) for s > • χ is supported in [0 , ∞ ), with χ ( s ) = 1 for s ≥
1, and χ ′ ≥ κ ( q ) define the functions a = χ (2 − φ/δ ) χ (2 + ˆ σ/δ ) . (6.2)For each fixed β >
0, the support of a is controlled by the parameter δ > a to a globally defined symbol in S ( b T ∗ X ). Lemma 6.5.
Given a neighborhood V ⊂ b S ∗ X of κ ( q ) and β > , there exists δ > such that supp a ⊂ V for each δ ∈ (0 , δ ) . Recall that b S ∗ X is the b -cosphere bundle , obtained by quotienting b T ∗ X \ R + action of dilations. ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 39
Proof.
Necessary conditions to lie in the support of a are φ ≤ δ and − δ ≤ ˆ σ . Fromthe definition of φ , | ˆ σ | ≤ δ, ≤ ω ≤ β δ (2 δ − ˆ σ ) ≤ β δ on supp a , i.e., supp a ⊂ {| ˆ σ | ≤ δ, ω / ≤ βδ } . (6.3)Finally, observe that any neighborhood of V of κ ( q ) contains a set of the form {| ˆ σ | ≤ δ, ω / ≤ βδ } provided δ is sufficiently small. (cid:3) Fix a conic neighborhood V of q having compact closure in which ˆ g αβ η α η β >
0. If ψ ∈ C ∞ c ( b S ∗ X ) is identically one V with support in a sufficiently small neighborhoodof V , then η n − = 0 on supp ψ . To facilitate the regularization argument, fix a bounded family of operators { J r : r ∈ (0 , } in Ψ s +1 / ( X ) such that J r ∈ Ψ m b ( X ) for r ∈ (0 , J r so that itsprincipal symbol j r = σ b , ( J r ) is given by j r = ψ · ρ s +1 / h rρ i − s − / m . In particular, J r is elliptic near V . We then define A r = AJ r , which is bounded inΨ s +1 / ( X ).We also need some additional auxiliary operators. Let B ∈ Ψ − / ( X ) have principalsymbol b = ρ − / δ − / ( χ ′ χ ) / χ . Here the arguments of χ , χ are as in (6.2). Let B r ∈ Ψ s +1b ( X ) have principal symbol σ b ,s +1 ( B r ) = ρj r b , and let ˜ B r ∈ Ψ s b ( X ) have principal symbol σ b ,s ( ˜ B r ) = j r b . Finally,let C ∈ Ψ ( X ) have principal symbol σ b , ( C ) = ρ − (ˆ g αβ η α η β ) / ψ . (6.4)This makes sense, since ˆ g αβ η α η β > ψ . In order to handle terms boundedby the inductive hypothesis, fix G ∈ Ψ s − / ( X ) and G ∈ Ψ s +1b ( X ) such thatWF ′ b ( G ) ∩ WF ,s − / ( u ) = ∅ , WF ′ b ( G ) ∩ WF − ,s +1b ( P R u ) = ∅ and q ∈ ell b ( G ) ∩ ell b ( G ).Throughout the rest of this section there appear various operators with wavefrontsets contained in { ˆ σ < } . Given β, δ , we use the notation E r to denote a genericoperator such that E = { E r : r ∈ (0 , } forms a bounded family (in a space ofoperators of fixed order, to be specified) andWF ′ b ( E ) ⊂ {− δ ≤ ˆ σ ≤ − δ, ω / ≤ βδ } . (6.5) Similarly, we denote by T r a generic operator such thatWF ′ b ( T ) ⊂ {| ˆ σ | ≤ δ, ω / ≤ βδ } , (6.6)where we set T r = { T r : r ∈ (0 , } . Terms of the form T r will arise as lower order errorsbounded by the inductive hypothesis. For notational flexibility we allow the operators E r , T r to change from line to line, but their orders will always be made explicit. Lemma 6.6.
Let g ∈ S k ( b T ∗ X ) . Given β > and c > , there exists δ , C > suchthat for each δ ∈ (0 , δ ) , | ∂ x φ | + ρ − k |{ φ, g }| ≤ C ( δ + βδ + β − ) whenever | ˆ σ | < cδ and ω / ≤ cβδ .Proof. Note that we must take δ > φ , sinceit is only locally defined near q . First consider the Poisson bracket { φ, g } , consistingof three terms { φ, g } = { ρ − , g } σ + { σ, g } ρ − + ( β δ ) − { ω, g } . The term ρ − k |{ ρ − , g } σ | is bounded by a constant times δ . Furthermore, because ρ − k |{ ω, g }| + ρ − k | x∂ x g | ≤ Cω / locally uniformly, the desired bound holds. Similarly, | ∂ x ω | ≤ Cω / , which completesthe proof. (cid:3) Lemma 6.6 can be applied to compute the commutator [ A ∗ r A r , G ] for G ∈ Ψ k b ( X ).Recall the convention regarding generic operators E r , T r discussed before Lemma 6.6. Lemma 6.7.
Let G ∈ Ψ k b ( X ) . Given β > , there exists δ > such that for each δ ∈ (0 , δ ) , i [ A ∗ r A r , G ] = B ∗ r D r B r + E r + T r , where E r ∈ Ψ s + k b ( X ) , T r ∈ Ψ s + k − ( X ) , and D r ∈ Ψ k − ( X ) . Moreover, there exists C > such that for every r ∈ (0 , , ρ − k | σ b ,k − ( D r ) | ≤ C ( βδ + δ + β − ) . Proof.
Let g = σ b ,k ( G ), so the principal symbol of i [ A ∗ r A r , G ] ∈ Ψ s + k b ( X ) is 2 a r { a r , g } .The claim is that we can write { a r , g } = d r b r + e r with d r , e r representing the principal symbols of the operators D r , E r . The lower orderterm T r then arises since we have arranged equality at the level of principal symbols.Recall that a r = χ (2 − φ/δ ) χ (2 + σ/δ ) j r , ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 41 so the Poisson bracket with g is a sum of terms with derivatives landing on either χ , χ , or j r .Fix a cutoff ψ ∈ C ∞ c ( b S ∗ X ) such that ψ = 1 on {| ˆ σ | ≤ δ, ω / ≤ βδ } andsupp ψ ⊂ {| ˆ σ | < δ, ω / < βδ } .(1) When χ is differentiated we obtain a term − ρ − { φ, g } b r . Now set d ,r = − ψρ − { φ, g } , to which Lemma 6.6 applies. The term d ,r will partly comprise d r ; note that d ,r is infact independent of r .(2) The second constituent of d r , in addition to d ,r above, arises when j r is differ-entiated. Thus we have a term of the form a { j r , g } . By construction χ ( s ) = s χ ′ ( s )for s >
0, so ρ / a = ρ / (2 − φ/δ )( χ ′ χ ) / χ = δ / (2 − φ/δ ) ρb. Also note that | − φ/δ | < b . Using that j r is elliptic on V , given δ > a { j r , g } = d ,r b r , where ρ − k | d ,r | ≤ C δ . Finally, we let d r = d ,r + d ,r .(3) The term e r arises when χ is differentiated, hence has the desired supportproperties. (cid:3) Now we expand 2 i Im E ( u, A ∗ r A r u ), recalling from the end of Section 5.2 that2 i Im E ( u, A ∗ r A r u ) = h Q u, Q A ,r u i − h Q A ,r u, Q u i + h Q u, A ,r u i − h A ,r u, Q u i + h [ Q α ˆ g αβ Q β , A ∗ r A r ] u, u i , (6.7)where σ b , − ( A ,r ) = (1 /i ) ∂ σ ( a r ) and σ b , ( A ,r ) = (1 /i ) ∂ x ( a r ). Lemma 6.8.
There exist C , c, β, δ > , a cutoff χ ∈ C ∞ c ( X ) , and an operator G ∈ Ψ s b ( X ) with WF ′ b ( G ) ⊂ W ∩ { σ < } , such that c k ˜ B r u k H ( X ) ≤ − E ( u, A ∗ r A r u ) + C k G u k H ( X ) + C ( k G P R u k H − ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) ) for every δ ∈ (0 , δ ) .Proof. Note that WF ′ b ( G ) ∩ WF ,s b ( u ) = ∅ by the hypotheses on u . According to (6.1), Q ( ˜ B r u, ˜ B r u ) ≤ C ( k B r u k L ( X ) + E ( ˜ B r u, ˜ B r u ) + k G u k H ( X ) ) provided δ > B r and D η n − ˜ B r havethe same principal symbol (up to a sign which is irrelevant when taking norms). Onthe other hand, Lemma 5.3 applies to the family { ˜ B r : r ∈ (0 , } , and thus Q ( ˜ B r u, ˜ B r u ) ≤ C k B r u k L ( X ) + C k G P R u k H − ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) ) . Therefore it suffices to bound k B r u k L ( X ) using (6.7).Arguing as in Lemma 6.7, we can expand iA ,r = ˜ B ∗ r ˜ B r + E r + T r , where E r ∈ Ψ s b ( X )and T r ∈ Ψ s − ( X ). Therefore h Q u, iQ A ,r u i = h Q u, Q ( ˜ B ∗ r ˜ B r + E r + T r ) u i . The pairing involving ˜ B ∗ r ˜ B r can be re-expressed in terms of the Dirichlet form itself: h Q , iQ A ,r u i = E ( u, ˜ B ∗ r ˜ B r u ) − h ˆ g αβ Q α u, Q β ˜ B ∗ r ˜ B r u i + h Q u, Q ( E r + T r ) u i , (6.8)recalling that E r , T r are allowed to change from line to line. The last term on theright hand side of (6.8) is bounded by a constant times k G u k H ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) . As for E ( u, ˜ B ∗ r ˜ B r u ), it is bounded by acceptable terms by arguing as inLemmas 5.2 and 5.3. Finally, we can write h ˆ g αβ Q α u, Q β ˜ B ∗ r ˜ B r u i = k CB r u k L ( X ) . modulo acceptable errors, where C has principal symbol (6.4).It remains to consider the second and third lines of (6.7). For the third term, weapply Lemma 6.7 to write h i [ Q α ˆ g αβ Q β , A ∗ r A r ] u, u i = B ∗ r D r B r + E r + T r , where D r ∈ Ψ ( X ) satisfies | σ b , ( D r ) | ≤ C ( βδ + δ + β − ) . Given ε >
0, we can first fix β > δ > |h [ Q α ˆ g αβ Q β , A ∗ r A r ] u, u i| ≤ ε k B r u k L ( X ) + C ( k G u k H ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) ) . As for the term h Q , iA ,r u i , note that we can write iA ,r = B ∗ r D ′ r B r + T r , where T r ∈ Ψ s b ( X ) and D ′ r ∈ Ψ ( X ) satisfies | σ b , ( D ′ r ) | ≤ C βδ. Thus we can similarly bound |h Q , iA ,r u i| ≤ ε k B r u k L ( X ) + C ( k G u k H ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) ) . We can now take ε sufficiently small, noting that C is elliptic on WF ′ b ( B ), where B = { B r : r ∈ (0 , } . (cid:3) ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 43
The final step in the proof of Proposition 6.4 is to bound Im E ( u, A ∗ r A r u ). Lemma 6.9.
Given ε > , there exists β > and δ > such that Im E ( u, A ∗ r A r u ) ≤ ε k ˜ B r u k H ( X ) + C k G u k H ( X ) + C ( k G P R u k H − ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) ) for every δ ∈ (0 , δ ) .Proof. Let Λ − / ∈ Ψ / ( X ) be elliptic. First, we bound E R ( u, A ∗ r A r u ) similar to thebeginning of proof of Lemma 5.3: |E R ( u, A ∗ r A r u ) | ≤ ε k Λ − / A r u k H ( X ) + C ε ( k G f k H − ( X ) + k G u k H ,m ( X ) + k χu k H ,m ( X ) ) . On the other hand, as in the proof of Lemma 6.7 we can write Λ − / A r = LB r + T r with L ∈ Ψ ( X ) and T r ∈ Ψ s − ( X ); thus we can bound k Λ − / A r u k H ( X ) ≤ C ( k ˜ B r u k H ( X ) + k G u k H ( X ) ) . It thus remains to bound the difference between Im E R ( u, A ∗ r A r u ) and Im E ( u, A ∗ r A r u ).There are two terms to consider. Since S F is real-valued, the first is h A r x − S F u, A r u i − h A r u, A r x − S F u i = h A ∗ r [ A r , x − S F ] u, u i − h u, A ∗ r [ A r , x − S F ] u i . Note that A ∗ r [ A r , x − S F ] is uniformly bounded in Ψ s b ( X ), so if G ∈ Ψ s − / ( X ) iselliptic on WF ′ b ( A ), then | Im h x − S F u, A ∗ r A r u i| ≤ C ( k G u k H ( X ) + k χu k H ,m ( X ) ) . Of course this actually true if the order of G is merely s − k G u k H ( X ) + k χu k H ,m ( X ) ; see Remark 3.6.Unfortunately, it is not obvious how to do this in general. Instead, write h βγ − u, γ − ( A ∗ r A r u ) i ∂X − h γ − ( A ∗ r A r u ) , βγ − u i ∂X = h γ − ([ A ∗ r A r , β ] u ) , γ − u i ∂X + h βγ − (( ˜ A ∗ r ˜ A r − A ∗ r A r ) u ) , γ − u i ∂X , where ˜ A r has the same principal symbol as A r . Here we have extended β arbitrarily toa function on X ; it is particularly convenient to choose this extension to be independentof x . Now consider [ A ∗ r A r , β ] ∈ Ψ s b ( X ). Lemma 6.7 applies to this commutator with m = 0. Because we are assuming that β is independent of x , we can take E r = 0.Thus we can write i [ A ∗ r A r , β ] = ˜ B r ˜ D r ˜ B r + T r , (6.9)where T r ∈ Ψ s − ( X ), and ˜ D r ∈ Ψ ( X ). Then h γ − ([ A ∗ r A r , β ] u ) , γ − u i ∂X = h γ − ( ˜ D r ˜ B r u ) , γ − ( ˜ B r u ) i ∂X . modulo terms bounded by k G u k H ( X ) + k χu k H ,m ( X ) . Now write h γ − ( ˜ D r ˜ B r u ) , γ − ( ˜ B r u ) i ∂X = h b N ( − iν − )( ˜ D r ) b N ( − iν − )( ˜ B r ) γ − u, b N ( − iν − )( ˜ B r ) γ − u i ∂X . In coordinates ( x, y, σ, η ), the total symbol of b N ( − iν − )( ˜ D r ) ∈ Ψ ( ∂X ) is just σ b , ( ˜ D r )(0 , y, − iν − , η ) ∈ S ( T ∗ ∂X ) , hence b N ( − iν − )( ˜ D r ) forms a bounded family. Using (3.5), we can bound |h γ − ( ˜ D r ˜ B r u ) , γ − ( ˜ B r u ) i ∂X | ≤ ε k ˜ B r u k H ( X ) + C ( k G u k H ( X ) + k χu k H ,m ( X ) ) . Notice that the supremum of | σ b , ( ˜ D r ) | can in fact be made small by choosing β, δ appropriately, but that is not needed here.Next, consider h βγ − (( ˜ A ∗ r ˜ A r − A ∗ r A r ) u ) , γ − u i ∂X . Note that ˜ A ∗ r ˜ A r = x ν − A ∗ r A r x − ν − ,and hence ˜ A ∗ r ˜ A r − A ∗ r A r = x ν − [ A ∗ r A r , x − ν − ] . Thus the principal symbol of ˜ A ∗ r ˜ A r − A ∗ r A r ∈ Ψ s b ( X ) is just 2 iν − ∂ σ ( a r ), and we canwrite ˜ A ∗ r ˜ A r − A ∗ r A r = (2 iν − ) ˜ B ∗ r ˜ B r + E r + T r (6.10)as in Lemma 6.8. Again using (3.5), the corresponding boundary term can be estimatedby ε k ˜ B r u k H ( X ) + C ( k G u k H ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) ). (cid:3) Combining Lemmas 6.8 and 6.9 shows that ˜ B r u is uniformly bounded in H ( K ) foran appropriate compact set K ⊂ X . Since the limiting operator B ∈ Ψ s b ( X ) as r → q , it follows that q / ∈ WF ,s b ( u ) by the same argument as in the proofof Theorem 3. Arguing inductively, this establishes Proposition 6.4 for finite s . Forinfinite s one merely needs to be more careful in choosing the operators G , G , G ateach step of the induction, and is done exactly as in the proof of [Vas2, Proposition6.2]. (cid:3) The glancing region.
Next we consider the glancing set G . Fix a boundarycoordinate patch as in Section 6.3. If q ∈ G ∩ T ∗ ∂X ∩ b T ∗ U X , then in coordinates( x, y, σ, η ), q = (0 , y , , η ) , ˆ g αβ (0 , y )( η ) α ( η ) β = 0 . Since η n − = 0 at q , we continue to use projective coordinates ( x, y, ˆ σ, ˆ η a ) on b S ∗ X near κ ( q ). We need the following result for the Dirichlet form: Lemma 6.10.
Let U ⊂ X be a boundary coordinate patch, and m ≤ . Let A = { A r : r ∈ (0 , } be a bounded subset of Ψ s b ( X ) with compact support in U such that A r ∈ Ψ m b ( X ) for each r ∈ (0 , . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 45
Let V δ = { q ∈ b T ∗ U X \ g αβ ζ α ζ β ≤ δ | ζ n − | } , and assume that WF ′ b ( A ) ⊂ V δ . Let G ∈ Ψ s b ( X ) and G ∈ Ψ s − / ( X ) both be elliptic on WF ′ b ( A ) , with compact supportin U . Then there exists C ε > and χ ∈ C ∞ c ( U ) such that k Q A r u k L ( X ) − ε Q ( A r u, A r u ) ≤ δ k Q n − A r u k L ( X ) + C ε ( k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) + k G u k H ( X ) + k G P R u k H − ( X ) ) for every r ∈ (0 , and every u ∈ H ,m loc ( X ) , provided WF ,s − / ( u ) ∩ WF ′ b ( G ) = ∅ , WF − ,s b ( P R u ) ∩ WF ′ b ( G ) = ∅ . Proof.
We have k Q A r u k = h ˆ g αβ Q α A r u, Q β A r u i + E ( A r u, A r u ). According to Lemma5.3, k Q A r u k − ε Q ( A r u, A r u ) ≤ h Q β ˆ g αβ Q α A r u, A r u i + C ε ( k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) + k G u k H ( X ) + k G f k H − ( X ) ) . Choose F ∈ Ψ ( X ) such that WF ′ b ( Q n − F Q n − + Q β ˆ g αβ Q α ) ∩ WF ′ b ( A ) = ∅ andWF ′ b ( F ) ⊂ V δ . In particular sup | σ b , ( F ) | ≤ δ, which implies that |h Q β ˆ g αβ Q α A r u, A r u i| ≤ δ k Q n − A r u k L ( X ) + C k χu k H ,m ( X ) as de-sired. (cid:3) To formulate the relevant estimates in the glancing region, we follow [Vas2, Section7] and introduce in local coordinates the map ˜ π : T ∗ U X → T ∗ ( U ∩ ∂X ) given by˜ π ( x, y, ξ, η ) = ( y, η ) . Let W denote the “gliding vector field” on T ∗ ( U ∩ ∂X ) given in coordinates by W ( y, η ) = n − X i =1 ( ∂ η i ˆ p )(0 , y, , η ) ∂ y i − ( ∂ y i ˆ p )(0 , y, , η ) ∂ η i . Note that W is just the b-Hamilton vector field of ˆ g αβ η α η β ∈ S ( b T ∗ X ) (which is hencetangent to T ∗ ∂X ) restricted to T ∗ ∂X .We then prove the following analogue of [Vas2, Proposition 7.3]: Proposition 6.11.
Let u ∈ H ,m loc ( X ) with m ≤ . If K ⊂ b S ∗ U X is compact and K ⊂ ( G ∩ T ∗ ∂X ) \ WF − ,s +1b ( P R u ) , then there exist C , δ > such that for each q ∈ K and δ ∈ (0 , δ ) the followingholds. Let α ∈ N be such that π ( α ) = q . If the conditions α ∈ N , | ˜ π ( α ) − exp( − δW )(˜ π ( α )) | ≤ C δ , | x ( α ) | ≤ C δ (6.11) imply π ( α ) / ∈ WF ,s b ( u ) , then q / ∈ WF ,s b ( u ) . The proof Proposition 6.11 goes through nearly exactly as in [Vas2, Section 7],working directly with the Dirichlet form, just as we did in Section 6.3. For this reasonwe only briefly sketch a proof of the proposition, referring to [Vas2, Section 7] for moredetails.Fix q = (0 , y , , η ) ∈ K . Note that ρ − W descends to a vector field on S ∗ ∂X ,at least near ( y , η ), recalling that ρ = | η n − | . Note that ρ − W does not vanish near( y , η ), since ρ − W y n − = 2 sgn( η n − )near q . Thus in a neighborhood of ( y , (ˆ η ) a ) ∈ S ∗ ∂X we straighten the ρ − W flow,finding homogeneous degree zero functions ρ , . . . , ρ n − on T ∗ ∂X with linearly inde-pendent differentials such that ρ − W ρ = 1 , ρ − W ρ i = 0 for i = 2 , . . . , n − . In fact, since W annihilates ˆ p (0 , y, , η ), and since this function has a non-vanishingdifferential, we can always take ρ ( y, η ) = ˆ p (0 , y, , η )The functions ρ , . . . , ρ n − are then extended to be independent of ( x, σ ), so that( x, ˆ σ, ρ , . . . ρ n − ) form a valid coordinate system near κ ( q ), say in some neighborhood V . Now for q ∈ K ∩ V , introduce the function ω = n − X i =1 ( ρ i − ρ i ( q )) , ω = x + ω . We then define φ = ρ + ( β δ ) − ω and φ = ρ + ( β δ ) − ω . With χ , χ denoting thesame cutoff functions as in Section 6.3, set a = χ (2 − φ/δ ) χ (1 + ( ρ + δ ) / ( βδ )) . Note that when a is differentiated, any derivatives falling onto χ yield a term sup-ported on {− δβ ≤ ρ ≤ − δ, ω / ≤ βδ } . (6.12)Suppose we take β = c δ for some fixed c >
0. If α ∈ N and π ( α ) is contained in theset (6.12), then α satisfies (6.11) for C > c .Recall that the restriction of the Poisson bracket ρ − { ˆ g αβ η α η β , φ } to b T ∗ ∂X X is ρ − W φ = 1. Since | x | ≤ ω / , | ρ − { ˆ g αβ η α η β , φ } − | ≤ C (1 + β − δ − ω / ) ω / . Observe that C > V for q fixed, but it is also uniform on V as q ranges over V ∩ K , where V ⊂ V is a neighborhood of κ ( q ). Since K iscompact, it suffices to prove Proposition 6.11 with V ∩ K replacing K , and as we shallsee, the uniformity of C implies the uniformity of c in the preceding paragraph. ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 47
Let A have principal symbol a . With J r denoting the same operator as in Section6.3, let A r = AJ r . Also define B r and ˜ B r as in Section 6.3. We let E r , T r denoteoperators as in (6.5), (6.6), except that everywhere ˆ σ should be replaced with ρ , andthe right hand side of (6.5) should be replaced with (6.12). Finally, let G , G be thesame as in Section 6.3. We then have the analogue of Lemma 6.8: Lemma 6.12.
There exist C , c, β, δ > , a cutoff χ ∈ C ∞ c ( X ) , and an operator G ∈ Ψ s b ( X ) with WF ′ b ( G ) ⊂ W ∩ {− δβ < ρ < − δ/ , ω / < βδ } such that c k ˜ B r u k H ( X ) ≤ − E ( u, A ∗ r A r u ) + C k G u k H ( X ) + C ( k G P R u k H − ( X ) + k G u k H ( X ) + k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) ) for every δ ∈ (0 , δ ) and c δ < β < .Proof. As in the proof of Lemma 6.8, consider the expansion (6.7). In this case, since a is independent of σ , we can write A ,r = T r with T r ∈ Ψ s − ( X ). As for A ,r , wehave that σ b , s +1 ( A ,r ) = (1 /i ) ∂ x a r , and since the only dependence on x is through φ ,we can write iA ,r = ˜ B ∗ r D r Q n − ˜ B r + T r , where T r ∈ Ψ s b ( X ) and D r ∈ Ψ ( X ). Furthermore, since | x | ≤ ω / ,sup | σ b , ( D r ) | ≤ Cβ − Finally, consider the term i [ Q α ˆ g αβ Q β , A ∗ r A r ] ∈ Ψ s +2b ( X ), which in analogy withLemma 6.7 we can write as i [ Q α ˆ g αβ Q β , A ∗ r A r ] = B ∗ r (1 + D ′ r ) B ∗ r + E r + T r . As in Lemma 6.7, D ′ r has principal symbol d ,r + d ,r , corresponding to when χ or j r is differentiated. When χ is differentiated we are left with a term − ρ − { ˆ g αβ η α η β , φ } b r , and hence sup | d ,r | ≤ C ( βδ + δ ); in this case d ,r can be taken independent of r .To handle the terms where j r is differentiated we argue as in Lemma 6.8 to see thatsup | d ,r | ≤ Cδ .As observed in Lemma 6.8, it suffices to control k B r u k L ( X ) . Write h i [ Q α ˆ g αβ Q β , A ∗ r A r ] u, u i = k B r u k L ( X ) + h D ′ r B r u, B r u i + h E r u, u, i + h T r u, u i . Modulo error terms bounded by the a priori hypotheses, we can write |h Q u, iA ,r u i| = h ˜ B r Q u, D r Q n − ˜ B r u i = h Q ˜ B r u, D r Q n − ˜ B r u i and hence |h Q u, iA ,r u i| ≤ k Q ˜ B r u k L ( X ) k ˜ D r Q n − ˜ B r u k L ( X ) + C ( k G u k H ( X ) + k χu k H ,m ( X ) )Now WF ′ b ( ˜ B r ) ⊂ { ω / ≤ βδ } , and since | ρ ( y, η ) | = | ˆ p (0 , y, , η ) | ≤ ω / and | x | ≤ ω / , it follows that | ˆ g αβ η α η β | ≤ βδ on WF ′ b ( ˜ B r ). If we are given ε > c δ < β < c > |h Q u, iA ,r u i| ≤ ε Q ( ˜ B r u, ˜ B r u ) + C ε k G u k H ( X ) + C ε ( k χu k H ,m ( X ) + k χP R u k H − ,m ( X ) + k G u k H ( X ) + k G P R u k H − ( X ) ) . Since we have controlled k B r u k L ( X ) − ε Q ( ˜ B r u, ˜ B r u ), the proof is complete. (cid:3) The proof of Proposition 6.11 now follows by combining Lemma 6.12 and the ana-logue of Lemma 6.9. Note that estimating the boundary terms as in the latter lemmais done slightly differently. For the analogue of (6.9) there appears an extra term E r ∈ Ψ s b ( X ) (which is of course harmless). The analogue of (6.10) is simpler: since a r is independent of σ , ˜ A ∗ r ˜ A r − A ∗ r A r = T r with T r ∈ Ψ s − ( X ).We now proceed inductively, the difference being that at each step of the iterationthe commutant must be modified slightly. This is done exactly as in [Vas2, Section 7].6.5. Propagation of singularities.
Combining Theorem 3 and Propositions 6.4, 6.11allows us to prove Theorem 1.Given the ingredients discussed above, the details of proof are identical to those of[Vas2, Theorem 8.1], hence are omitted.7.
Propagators and their singularities
Retarded and advanced propagators.
We begin by discussing well-posednessfor the Klein–Gordon equation on an aAdS spacetime. In the case of Dirichlet bound-ary conditions, well-posedness for the forward problem was first studied by Vasy [Vas4](see also [Hol] for a related study of the Cauchy problem). In the setting of Robinboundary conditions, well-posedness of the Cauchy problem was studied by Warnick[War1], including certain results on higher-order conormal regularity. It should alsobe noted that [War1] considers the situation where the metric is even modulo O ( x ),whereas [Vas4] does not.We will need a more refined study of the forward Klein–Gordon problem, akin to theresults of [Vas4], in the case of Robin boundary conditions. In order to give a globalformulation we make the following assumptions as in [Vas4, Wro] (cf. [H¨or, Section24.1]): ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 49
Hypothesis 7.1.
We assume (
X, g ) is an aAdS spacetime with the following proper-ties.(TF) There exists t ∈ C ∞ ( X ; R ) such that the level sets of t are spacelike with respectto ˆ g .(PT) The map t : X → R is proper.Given a choice of t , we orient N by declaring dt to be future-oriented. Let N ± denote the corresponding future/past light-cones. This yields a decomposition˙ N = ˙ N + ∪ ˙ N − of the compressed characteristic set. The main well-posedness result we need is thefollowing: Theorem 4.
Let s, t ∈ R . If f ∈ ˙ H − ,s +1loc ( X ) is supported in { t ≥ t } , then thereexists a unique u ∈ H ,s loc ( X ) supported in { t ≥ t } such that P R u = f. Furthermore, for each K ⊂ X compact, there exists K ′ ⊂ X compact and C > suchthat k u k H ,s ( K ) ≤ C k f k ˙ H − ,s +1 ( K ) . Observe that Theorem 4 exhibits the loss of one conormal derivative from the sourceterm to the solution. Now H ,s c ( X ) ⊂ H ,s +1c ( X ) for each s ∈ R , so by transposition H ,s loc ( X ) ⊂ ˙ H − ,s +1loc ( X ) . In particular, given f ∈ L ( X ), we obtain a solution u ∈ H ( X ) of x − P u = f satisfying Robin boundary conditions in the strong sense. Replacing t with − t yieldsa result for the backward problem as well.We sketch a proof of this result adapting the approach of [Vas4], which along the waygives a different perspective on the twisted stress-energy tensor introduced in [HS3].As usual, the key step is an appropriate energy estimate.We make a preliminary reduction: we will need that the level sets of t meet ∂X orthogonally with respect to ˆ g . Note that the level sets of t automatically meet ∂X transversally, but since t is provided by the problem there is no reason to assume theintersection is orthogonal. On the other hand, to prove Theorem 4, given δ > t ′ with an analogous function t ′ satisfying the required orthogonalityproperty, such that (cid:12)(cid:12) t − t ′ (cid:12)(cid:12) < δ on any given compact subset of X . This is arranged in[Vas4, Lemma 4.9], and we henceforth assume dt and dx are orthogonal along ∂X .Let u ∈ H , ( X ). Given an appropriate real b-vector field V ∈ V b ( X ) with compactsupport, set V ′ = F V F − ∈ Diff ( X ) and compute 2 Re h P R u, V ′ u i ; note that this pairing makes sense by our assumption on the conormal regularity of u . For illustrativepurposes, let us assume that the Robin function β and S F both vanish identically; theseterms can easily be handled in general. In particular h P R u, v i = E ( u, v ). Setting A = iV ′ and applying (5.4),2 Re E ( u, V ′ u ) = h ˆ g ij Q j u, [ Q i , V ′ ] u i + h ˆ g ij [ Q j , V ′ ] u, Q i u i + h [ˆ g ij , V ′ ] Q j u, Q i u i + h ( V ′ + ( V ′ ) ∗ )ˆ g ij Q j u, Q i u i . The commutator [ Q i , V ′ ] can be computed exactly using (3.8): if V = V j ∂ z j , then[ Q i , V ′ ] = F [ D z i , F − V ′ F ] F − = F [ D z i , V ] F − = ∂ z i ( V k ) Q k . It is interesting to observe that if one instead uses V rather than its twisted version V ′ (despite the fact that both are b-operators), there arise error terms that cannot beestimated appropriately.We also have [ˆ g ij , V ′ ] = [ˆ g ij , V ]. Finally, ( V ′ ) ∗ = F − V ∗ F , and since the adjoint istaken with respect to x dg ,( V ′ ) ∗ = − F − V F − div ˆ g V + ( n − x − V ( x )= − V + F V ( F − ) − div ˆ g V + ( n − x − V ( x ) . In particular, this shows that2 Re E ( u, V ′ u ) = h B ij Q i u, Q j u i (7.1)where B = (div ˆ g V + 2 F V ( F − ) + ( n − x − V ( x )) · ˆ g − − L V ˆ g − is symmetric. Now we take V = f W for a suitable function f and a real b-vector field W ; expanding out the tensor B yields B = ( W f + f · div ˆ g W + 2 F f V ( F − ) + ( n − f x − W ( x ))ˆ g − − f L W ˆ g − − ∇ ˆ g f ) ⊗ s W, where ⊗ s denotes the symmetric tensor product. Observe that all the terms in B where f is differentiated can be written in the form − T ˆ g ( W, ∇ ˆ g f ), where T ˆ g ( W, ∇ ˆ g f ) = ( ∇ ˆ g f ) ⊗ s W − ˆ g ( ∇ ˆ g f, W ) · ˆ g − is the stress-energy tensor (with respect to ˆ g ) of a scalar field associated to W and ∇ ˆ g f . ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 51
We make our choices of W and f as follows. First, set W = ∇ ˆ g t , which is indeed in V b ( X ) by the assumption that the level sets of t meet ∂X orthogonally. Fix t < t ,and let δ > u ∈ H , ( X ) satisfiessupp u ⊂ { t + δ ≤ t ≤ t } . Let χ ∈ C ∞ c ( R ) satisfy the following properties: • supp χ ⊂ [ t , t + δ ], • χ ≥ χ ( s ) > s ∈ [ t + δ, t ], • χ ′ ( s ) ≤ s ∈ [ t + δ, t + δ ].We then set f = e − γt ( χ ◦ t ). Notice that ∇ ˆ g f = ( χ ′ ◦ t − γχ ◦ t ) e − γt ∇ ˆ g t . Since ∇ ˆ g t is strictly timelike, the stress-energy tensor term controls γ Z X e − γt H ( d F u, d F ¯ u ) x dg. We can also control the L ( X ) norm of u by writing2 Re h u, V u i = −h div ˆ g ( V ) u + (2 − n ) x − V ( x ) u, u i . Again using that − div ˆ g V = − W f − f · div ˆ g W , we can bound the first term on theright-hand side − W f = − ˆ g ( ∇ ˆ g t, ∇ ˆ g f ) ≥ c γe − γt ( χ ◦ t ) . This allows us to control γ k e − γt u k L ( X ) , and hence γ k e − γt u k H ( X ) as well. All of theremaining terms can be absorbed into the latter positive term for large γ > γ k e − γt u k H ( X ) ≤ Cγ − k e − γt P R u k ˙ H − , ( X ) . Equipped with this energy estimate, standard functional-analytic arguments as in[Vas4, Section 4 & Theorem 8.12] and Theorem 1 on propagation of singularities allowone to deduce Theorem 4.Let us denote by H ± ( X ) the space of future/past supported elements of H ( X ),i.e. H ± ( X ) = (cid:8) u ∈ H ( X ) : supp u ⊂ {± t ≥ ± t } for some t ∈ R (cid:9) , (7.2)and let us define ˙ H − ± ( X ) analogously. By hypothesis (PT), the elements of the inter-section H ( X ) ∩ H − ( X ) are compactly supported in X . Corollary 7.2.
There exist unique retarded/advanced propagators P − R, ± , i.e. contin-uous operators P − R, ± : ˙ H − ,s +1 ± ( X ) → H ,s ± ( X ) (7.3) such that P R P − R, ± = 1 on ˙ H − ± ( X ) and P − R, ± P R = 1 on H ± ( X ) . Furthermore, P − R, ± maps continuously P − R, ± : ˙ H − , ∞ c ( X ) → H , ∞ loc ( X ) . By uniqueness, the formal adjoint of P − R, ± equals P − R, ∓ .7.2. Symplectic space of solutions.
We continue to focus on Robin (or Neumann)boundary conditions. The difference of the two propagators, G R := P − R, + − P − R, − : ˙ H − ,s +1c ( X ) → H ,s loc ( X ) (7.4)will be called the causal propagator (associated to the choice of boundary conditions).A natural space of solutions is given by (cid:8) u ∈ H , ∞ loc ( X ) : P R u = 0 (cid:9) . Exactly as in [Wro] we can show that this space is the range of G R acting on a quotientspace which is suitable for field quantization. A similar statement is true for Dirichletboundary conditions. Proposition 7.3.
The causal propagator (7.4) induces a bijection [ G R ] : ˙ H − , ∞ c ( X ) P R H , ∞ c ( X ) −→ (cid:8) u ∈ H , ∞ loc ( X ) : P R u = 0 (cid:9) . (7.5) Moreover, i ( ·| G R · ) L induces a non-degenerate Hermitian form on the quotient space ˙ H − , ∞ c ( X ) /P R H , ∞ c ( X ) . We remark that a similar statement was obtained in [DF2, Section 4] in the case ofthe Poincar´e patch of AdS; cf. [DDF, Proposition 34] for static spacetimes with smoothtimelike boundary.
Remark 7.4.
We can use the present framework and Proposition 7.3 to extend the re-cent results in [DW] on quantum holography to the case of Neumann and Robin bound-ary conditions. This can be done by replacing the space H − , ∞ , b , c ( X ), resp. H , ∞ , b , c ( X )considered therein by ˙ H − , ∞ c ( X ), resp. H , ∞ c ( X ), and by replacing the map ∂ + thereinby the pair of trace maps ( γ − , γ + ). In this way one obtains a direct analogue of [DW,Theorem 3.7], the statement of which is non-trivial if one has the following unique con-tinuation property for some open set O ⊂ ∂X : for any u ∈ H , −∞ loc ( X ) solving P R u = 0,if γ − u = 0 and γ + u = 0 then u = 0 on some non-empty V ( O ) ⊂ X . Results of thistype were obtained by Holzegel and Shao [HS1, HS2] in the case of high regularitysolutions. ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 53
Microlocal regularity of traces.
We will need more precise mapping state-ments for γ ± in the case of high conormal regularity. We show that as in the case ofconventional traces, the wavefront set of γ ± u is controlled in term of the b-wavefrontset of u ; cf. [Wro, Proposition 3.9] for a similar result, proved therein only in theDirichlet case and assuming P u = 0. Here we are assuming ν ∈ (0 , γ + case, we need the following observation: if u ∈ X ∞ and B ∈ Ψ , even ( X ), then F BF − u ∈ X ∞ as well by Lemma 4.5. In particular, γ + ( F BF − u )can be computed by the formula γ + ( F BF − u ) = x − ν ∂ x ( BF − u ) | ∂X . Now write x − ν x∂ x BF − = x − ν ([ B, x∂ x ] F − + Bx∂ x F − )= x − ν B ′ F − u + B ′′ x − ν ∂ x ( F − u ) , where B ′ = x − [ B, x∂ x ] ∈ Ψ ( X ) since B ∈ Ψ , even ( X ), and B ′′ = x − ν Bx ν ∈ Ψ ( X ).Since u ∈ X ∞ and ν ∈ (0 , x − ν B ′ F − u | ∂X = 0by (4.6) and (4.8). Furthermore, B ′′ x − ν ∂ x ( F − u ) | ∂X = b N ( B )( − iν )( γ + u ). Since X ∞ is dense in X k , we conclude that if B ∈ Ψ , even ( X ) and u ∈ X k , then γ + ( F BF − u ) = b N ( B )( − iν )( γ + u ) . (7.6)The other observation we need is the following: Lemma 7.5.
Let u ∈ X −∞ , and suppose that q / ∈ WF , ∞ b ( u ) ∪ WF , ∞ b ( x − P u ) . If B ∈ Ψ , even ( X ) and WF ′ b ( B ) is a sufficiently small conic neighborhood of q , then x − P Au ∈ H , ∞ loc ( X ) if we set A = F BF − .Proof. By taking WF ′ b ( B ) to be a sufficiently small conic neighborhood of q , we mayassume that Au ∈ H , ∞ loc ( X ). Tracing through the proof of Lemma 4.5 and furthershrinking the microsupport of B as necessary, we see that x − P Au ∈ H , ∞ loc ( X ) as well;this is a consequence of the microlocality the operations involved in Lemma 4.5. (cid:3) Lemma 7.6. If u ∈ H ( X ) , then WF( γ − u ) ⊂ WF , ∞ b ( u ) ∩ T ∗ ∂X. (7.7) Moreover, if
P u ∈ x L ( X ) , then WF( γ + u ) ⊂ (WF , ∞ b ( u ) ∪ WF , ∞ b ( x − P u )) ∩ T ∗ ∂X. Proof.
The proof for γ − is standard: let q ∈ T ∗ ∂X , and suppose that q / ∈ WF , ∞ b ( u ),meaning that there exists B ∈ Ψ ( X ) elliptic at q such that Bu ∈ H , ∞ loc ( X ). Then b N ( B )( − iν − ) γ − u = γ − Bu ∈ C ∞ ( X ) , which finishes the argument since b N ( B )( − iν − ) is elliptic at q .Now consider the γ + case. Let q ∈ T ∗ ∂X , and suppose that q / ∈ WF , ∞ b ( u ) ∪ WF , ∞ b ( x − P u ). By Lemma 7.5 we may find B ∈ Ψ , even ( X ) elliptic at q such that F BF − u ∈ X ∞ . In particular, γ + ( F BF − u ) ∈ C ∞ ( ∂X ) by Lemma 4.8. Finally, according to (7.6), b N ( B )( − iν )( γ + u ) = γ + ( F BF − u ) ∈ C ∞ ( ∂X ) , which shows that q / ∈ WF( γ + u ). (cid:3) Holographic Hadamard condition.
Let ν ∈ (0 , W −∞ b ( X )the set of bounded operators from ˙ H − , −∞ c ( X ) to H , ∞ loc ( X ). Following [Wro] (thoughusing different spaces of distributions) we introduce an operatorial b-wave front setwhich is a subset of b S ∗ X × b S ∗ X . Definition 7.7.
Suppose Λ : ˙ H − , −∞ c ( X ) → H , −∞ loc ( X ) is continuous. We say that( q , q ) ∈ b S ∗ X × b S ∗ X is not in WF Opb (Λ) if there exist B i ∈ Ψ ( X ), elliptic at q i ( i = 1 , B Λ B ∗ ∈ W −∞ b ( X ).The notion of holographic Hadamard two-point functions introduced in [Wro] hasthe following straightforward adaptation to the present case. Definition 7.8.
We say that two continuous operators Λ ± R : ˙ H − , −∞ c ( X ) → H , −∞ loc ( X )are (Robin) two-point functions if: i ) P R Λ ± R = Λ ± R P R = 0 ,ii ) Λ + R − Λ − R = iG R and Λ ± R ≥ H − , ∞ c ( X ) . (7.8)We say that Λ ± R are holographic Hadamard two-point functions if in addition theysatisfy WF Opb (Λ ± R ) ⊂ ˙ N ± × ˙ N ± . (7.9)The first definition is a direct adaptation of the standard definition of two-pointfunctions to the setup provided by Proposition 7.3. We refer to e.g. [GOW, Section7.1] for a brief introduction to two-point functions, field quantization and for remarkson the relation with the more commonly used real formalism. We point out that oncesome two-point functions Λ ± are given, the general formalism of quasi-free states and ofthe GNS representation applies, and as an outcome one obtains quantum fields (whichare not discussed here in any detail). ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 55
The second definition provides a replacement for the celebrated
Hadamard condition ,which is widely used on globally hyperbolic spacetimes, and which is formulated interms of the (usual, smooth) wave front set since the work of Radzikowski [Rad]. Themain interest for Hadamard two-point functions comes from Radzikowski’s theorem,which asserts their uniqueness modulo smoothing operators. Thus, their singularitieshave a universal form that comes from the local geometry and which can be subtractedby a renormalization procedure. In our setup, Radzikowski’s theorem is replaced bythe following result.
Theorem 5.
Holographic Hadamard two-point functions exist and are unique modulo W −∞ b ( X ) .Proof. The proof is largely analogous to that of [Wro, Theorem 5.11] and [Wro, Propo-sition 5.13], so we only sketch it.As in [Wro, Theorem 5.9], one can reduce the existence problem to a neighborhoodof a time slice. By a spacetime deformation argument, this allows one to reduce theproblem further to the case of a standard static spacetime. By the same argument onecan assume without loss of generality that S F ≥ λx for some λ > β = 0.Then P equals f ( ∂ t + L ) f , for some smooth multiplication operators f , f ∈C ∞ ( X ), where the spatial part L is a differential operator on S associated with a qua-dratic form like E , except that the signature is Riemannian. We can associate to L a positive self-adjoint operator consistent with Neumann boundary conditions, as dis-cussed in Appendix A. From this point on, the construction of Λ ± can be done exactlyas in [Wro, Lemma 4.5] and the proof of the holographic condition from [Wro, Theo-rem 5.9] can be repeated verbatim. The only subtle point is the mapping propertiesof L − / on H , ∞ ( S ), which is discussed in Lemma A.1 (cid:3) We now turn our attention to singularities of parametrices for P . If q , q ∈ b S ∗ X ,then we write q ˙ ∼ q if q , q ∈ ˙ N and q , q can be connected by a GBB. We can alsodefine the backward flow-out of a point q ∈ ˙ N to be F q = { q ′ ∈ ˙ N : q ′ ˙ ∼ q, t ( q ′ ) ≤ t ( q ) } . Since GBBs exhibit possible branching behavior at glancing points, it is not completelytrivial that F q is closed; this instead follows from the compactness of the set of GBBswith values in a fixed compact subset of b S ∗ X (as discussed in [Vas2, Proposition 5.5]).With our new propagation of singularities result at hand, it is straightforward torepeat the proof of [Wro, Theorem 5.12] to obtain the following result. In this setting,by parametrices we mean bounded operators from ˙ H − , −∞ c ( X ) to H , −∞ loc ( X ) that areinverses of P modulo errors in W −∞ b ( X ). Theorem 6. If ν ∈ (0 , , then: WF Opb ( P − R, ± ) \ diag ∗ ⊂ { ( q , q ) : q ˙ ∼ q , ± t ( q ) > ± t ( q ) } , (7.10) where diag ∗ = { ( q , q ) ∈ b S ∗ X × b S ∗ X } . Furthermore, suppose that Λ ± R are holo-graphic Hadamard two-point functions. Then WF Opb (Λ ± R ) ⊂ { ( q , q ) ∈ ˙ N ± × ˙ N ± : q ˙ ∼ q } . (7.11) Moreover, setting P − R, F := i − Λ + R + P − R, − and P − R, F := − i − Λ − R + P − R, − , we have WF Opb ( P − R, F ) \ diag ∗ ⊂ { ( q , q ) : q ˙ ∼ q , and ± t ( q ) < ± t ( q ) if q ∈ ˙ N ± } , WF Opb ( P − R, F ) \ diag ∗ ⊂ { ( q , q ) : q ˙ ∼ q , and ∓ t ( q ) < ∓ t ( q ) if q ∈ ˙ N ± } . (7.12) Furthermore, the respective condition in (7.10) or (7.12) characterizes P − R, + , P − R, − , P − R, F and P − R, F uniquely modulo terms in W −∞ b ( X ) among parametrices of P R . The analogue of this theorem also holds for the Dirichlet realization P D (and any ν > Induced two-point functions at the boundary.
We continue to assume ν ∈ (0 , H − , −∞ c ( X ) → H , −∞ loc ( X ) induces an operator onthe boundary: γ − Λ γ ∗− : E ′ ( X ) → D ′ ( ∂X ) . Defining γ + Λ γ ∗ + is more delicate and requires additional hypotheses; sufficient condi-tions are given in the next lemma. Observe that X ∞ is dense in H , ∞ loc ( X ), so we canview ˙ H − , −∞ c ( X ) as a dense subspace of the dual ( X ∞ ) ′ . Lemma 7.9.
Let ν ∈ (0 , , and suppose that P Λ : ˙ H − , −∞ c ( X ) → x H , −∞ loc ( X ) , P Λ ∗ : ˙ H − , ∞ c ( X ) → x H , ∞ loc ( X ) ,P ( P Λ) ∗ : x − H , ∞ c ( X ) → x H , ∞ loc ( X ) . Then Λ extends to a continuous map ( X ∞ ) ′ → X −∞ .Proof. First observe that Λ ∗ : ˙ H − , ∞ c ( X ) → H , ∞ loc ( X ), so by the mapping properties of P Λ ∗ we conclude that Λ ∗ : ˙ H − , ∞ c ( X ) → X ∞ . This shows that Λ admits an extensionΛ : ( X ∞ ) ′ → H , −∞ loc ( X ) . Similarly, the mapping properties of P Λ and P ( P Λ) ∗ imply that in fact Λ : ( X ∞ ) ′ →X −∞ . (cid:3) ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 57
Since γ + : X ∞ → C ∞ ( ∂X ), we can view γ ∗ + : E ′ ( ∂X ) → ( X ∞ ) ′ , which allows us todefine γ + Λ γ ∗ + : E ′ ( ∂X ) → D ′ ( ∂X ) . We show that the wave front sets of γ ± Λ γ ∗± can be estimated in terms of WF Opb (Λ).
Lemma 7.10.
Let ν ∈ (0 , and Λ : ˙ H − , −∞ c ( X ) → H , −∞ loc ( X ) be a continuousoperator. Then WF ′ ( γ − Λ γ ∗− ) ∩ ( S ∗ ∂X × S ∗ ∂X ) ⊂ WF Opb (Λ) ∩ ( S ∗ ∂X × S ∗ ∂X ) . (7.13) Furthermore, if Λ restricts to a continuous map ˙ H − , ∞ c ( X ) → H , ∞ loc ( X ) and satisfies P Λ : ˙ H − , −∞ c ( X ) → x H , ∞ loc ( X ) , P Λ ∗ : ˙ H − , −∞ c ( X ) → x H , ∞ loc ( X ) ,P ( P Λ) ∗ : x − H , −∞ c ( X ) → x H , ∞ loc ( X ) , then the γ + analogue of (7.13) is true.Proof. We focus on the more delicate γ + case, which is essentially the microlocalizationof the proof of Lemma 7.9. Notice that we are assuming stronger mapping propertiesas compared to Lemma 7.9. Now suppose ( q , q ) / ∈ WF Opb (Λ) ∩ ( S ∗ ∂X × S ∗ ∂X ), sothat there exists B i ∈ Ψ , even ( X ) elliptic at q i such that A Λ A ∗ ∈ W −∞ b ( X ) , where we have set A i = F B i F − . Note that Λ : ( X ∞ ) ′ → X −∞ , hence the same is trueof A Λ A ∗ . The claim is that A Λ A ∗ extends to a map ( X ∞ ) ′ → X ∞ . First, we showthat the range of A Λ A ∗ in this extended sense is contained in H , ∞ loc ( X ). To see this,note that ( A Λ A ∗ ) ∗ = A Λ ∗ A ∗ : ˙ H − , −∞ c ( X ) → H , ∞ loc ( X )since A Λ A ∗ ∈ W −∞ b , and then Lemma 7.5 shows that ( A Λ A ∗ ) has its range containedin X ∞ . Thus A Λ A ∗ maps ( X ∞ ) ′ → H , ∞ loc ( X ). It then suffices to show that P A Λ A ∗ : ( X ∞ ) ′ → x H , ∞ loc ( X ) . Again applying Lemma 7.5, we see that
P A Λ A ∗ maps ˙ H − , −∞ c ( X ) → x H , ∞ loc ( X ). Itthen suffices to show that P ( P A Λ A ∗ ) ∗ : x − H , −∞ c ( X ) → x H , ∞ loc ( X ) . (7.14)Using our previously established mapping properties, we can write this composition as P A ( P A Λ) ∗ = P A ( P Λ) ∗ A ∗ + P A ([ P, A ]Λ) ∗ . According to Lemma 7.5, we see that
P A ( P Λ) ∗ A ∗ has the requisite mapping proper-ties. Now following the proof of Lemma 4.5, we can write [ P, A ] = B ′ P + QB ′′ + B ′′′ ,where B ′ ∈ Ψ − ( X ) , B ′′ ∈ x Ψ ( X ) , B ′′′ ∈ x Ψ ( X ) , Q ∈ Diff ν ( X ) . We then write([
P, A ]Λ) ∗ = (( B ′ P + QB ′′ + B ′′′ )Λ) ∗ = ( P Λ) ∗ B ′ + Λ ∗ ( B ′′ Q ) ∗ + Λ ∗ ( B ′′′ ) ∗ . (7.15)Write B ′′′ = B ′′′ C + x R , where C ∈ Ψ ( X ) satisfies WF ′ b (1 − C ) ∩ WF ′ b ( B ′′′ ) = ∅ and R ∈ Ψ −∞ b ( X ). Consider the termΛ ∗ ( B ′′′ ) ∗ = Λ ∗ C ∗ ( B ′′′ ) ∗ + Λ ∗ R ∗ x . Note that ( B ′′′ ) ∗ maps x − H , −∞ c ( X ) → ˙ H − , −∞ c ( X ), and that A Λ ∗ C ∗ = ( C Λ A ) ∗ : ˙ H − , −∞ c ( X ) → H , ∞ loc ( X )if C has sufficiently small wavefront set near q , which can be arranged by choosing A appropriately. Similarly, A Λ ∗ R ∗ = ( R Λ A ) ∗ : ˙ H − , −∞ c ( X ) → H , ∞ loc ( X ) . This shows that A Λ ∗ ( B ′′′ ) ∗ maps x − H , −∞ c ( X ) → H , ∞ loc ( X ), and since P Λ ∗ ( B ′′′ ) ∗ maps x − H , −∞ c ( X ) → x H , ∞ loc ( X ), a final application of Lemma 7.5 shows that P A Λ ∗ ( B ′′′ ) ∗ : x − H , −∞ c ( X ) → x H , ∞ loc ( X ) . The desired mapping properties of the other terms arising from (7.15) are obtainedsimilarly, which establishes (7.14).We have shown that A Λ A ∗ extends to a map ( X ∞ ) ′ → X ∞ . In particular, γ + ( A Λ A ∗ ) γ ∗ + : E ′ ( ∂X ) → C ∞ ( ∂X ) . Finally, notice that we can use (7.6) to write γ + A i = b N ( B i )( − iν ) γ + on X ∞ , where˜ B i = b N ( B i )( − iν ) ∈ Ψ ( ∂X ) is elliptic at q i . Thus˜ B γ + Λ γ ∗ + ˜ B ∗ ∈ Ψ −∞ ( ∂X ) , which finishes the proof. (cid:3) We can now conclude as in [Wro, Theorem 5.16] the following result, which appliesto holographic Hadamard two-point functions of the form Λ ± R . Theorem 7.
Suppose ( X, g ) is an asymptotically AdS spacetime and ν ∈ (0 , . If Λ ± R is a pair of holographic Hadamard two-point functions then WF ′ ( γ + Λ ± R γ ∗ + ) ∩ ( S ∗ ∂X × S ∗ ∂X ) ⊂ WF Opb (Λ ± R ) ∩ ( S ∗ ∂X × S ∗ ∂X ) ⊂ ( ˙ N ± × ˙ N ± ) ∩ ( S ∗ ∂X × S ∗ ∂X ) , and the same is true for γ − Λ ± R γ ∗− . Furthermore, if ˜Λ ± R is another pair of holographicHadamard two-point functions then γ + ( ˜Λ ± R − Λ ± R ) γ ∗ + and γ − ( ˜Λ ± R − Λ ± R ) γ ∗− have smoothSchwartz kernel. ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 59
The operators γ − Λ ± R γ ∗− and γ + Λ ± R γ ∗ + are interpreted as two-point functions of aninduced theory at the boundary, in the formalism of generalized free fields , see [San] (theword “generalized” refers to the fact that they are not solutions of a natural differentialequation). Theorem 7 asserts that these two-point functions satisfy a generalizedversion of the Hadamard condition which is nevertheless sufficient in applications (seealso [San]). Appendix A. The elliptic setting
A.1.
Mapping properties.
In this appendix we consider the analogue of the operator P R in the Euclidean signature. The manifold with boundary will be denoted by S and will be assumed compact. With x ∈ C ∞ ( S ) a boundary defining function, fix F ∈ x ν − C ∞ ( S ). Given a smooth Riemannian metric h on S , we then consider thequadratic form L ( u, v ) = Z H ( d F u, d F v ) + αu ¯ v x dh, where H is the sesquilinear pairing on one-forms induced by the metric, and α > L coercive on H ( S ). We can associate to L an operator L : H ( S ) → ˙ H − ( S ) , which extends to a positive self-adjoint operator L : D ( L ) → L ( S ) with form do-main D ( L / ) = H ( S ). The key mapping property we need is that L − / preservesconormality: Lemma A.1.
The operator L − / maps H , ∞ ( S ) → H , ∞ ( S ) continuously.Proof. First, note that for u ∈ H , ∞ ( S ) we have a trivial estimate k u k H ( S ) ≤ k ( L + µ ) u k ˙ H − ( S ) for µ >
0. Arguing as in Lemma 5.3, induction then shows that for each integer s ≥ k u k H ,s ( S ) ≤ k ( L + µ ) u k ˙ H − ,s ( S ) . (A.1)We also have the estimate k ( L + µ ) − k H ( S ) →H ( S ) ≤ Cµ − , (A.2)since L ± / commutes with the resolvent and the estimate is clearly true for L ( S )replacing H ( S ). The claim is that for each s ≥ k ( L + µ ) − k H ,s ( S ) →H ,s ( S ) ≤ Cµ − , (A.3)for some C > s . Notice that the analogue of Theorem 3 (in Riemanniansignature) applied to L shows that L + µ is invertible ˙ H − ,s ( S ) → H ,s ( S ) for each s ∈ R , so (A.3) is well-defined. Let A s ∈ Ψ s b ( S ) be elliptic and A − s ∈ Ψ − s b ( X ) be a parametrix, so A s A − s + R = 1 and A − s A s + R ′ = 1 for some R, R ′ ∈ Ψ −∞ b ( S ). Given u ∈ H , ∞ ( S ), k ( L + µ ) − u k H ,s ( S ) ≤ C k A s ( L + µ ) − u k H ( S ) + C k ( L + µ ) − u k H ( S ) . The second term on the right-hand side we bound by µ − k u k H ( S ) ≤ µ − k u k H ,s ( S ) ,using (A.2). Now write u = ( A − s A s + R ′ ) u . We then bound k A s ( L + µ ) − A − s A s u k H ( S ) ≤ k A s ( L + µ ) − A − s ( L + µ ) k H ( S ) →H ( S ) k ( L + µ ) − A s u k H . Now write A s ( L + µ ) − A − s ( L + µ ) = A s ( L + µ ) − A − s L + A s A − s − A s ( L + µ ) − LA − s , and apply (A.1) to see that this operator mapping H ( S ) → H ( S ) is uniformlybounded in µ . On the other hand, k ( L + µ ) − A s u k H ≤ Cµ − k u k H ,s ( S ) , which showsthat k A s ( L + µ ) − A − s A s u k H ( S ) ≤ Cµ − k u k H ,s ( S ) . The term k A s ( L + µ ) R ′ u k H ( S ) is bounded similarly.Since we can write L − / = 1 π Z ∞ µ − / ( L + µ ) − / dµ, the estimate (A.3) shows that L − / indeed maps H , ∞ ( S ) → H , ∞ ( S ) (cid:3) Appendix B. Even b-calculus
B.1.
Even b-pseudodifferential operators.
Let (
X, g ) be an aAdS spacetime whichis even modulo O ( x k +1 ). We sketch a construction of the b-pseudodifferential opera-tors of order m on X that are even modulo O ( x k +3 ), which we denote by Ψ m b , even ( X );the construction closely mirrors that of an even subcalculus by Albin in the 0-calculus[Alb]. The operators we consider are described in terms of their Schwartz kernels, andfor this reason we assume familiarity with the construction of the usual b-calculus interms of conormal distributions on the b-stretched product [Mel, Chapter 4, 5]. Definition B.1.
We say that f ∈ C ∞ ( X ) is even modulo O ( x k +3 ), written f ∈C ∞ even ( X ), if in any special coordinate system ( x, y ) the Taylor of expansion of f at ∂X contains only even terms modulo O ( x k +3 )This space is well defined (i.e., independent of the choice of special coordinates) inview of (4.2). Similarly, we can define the space C ∞ odd ( X ) of odd functions modulo O ( x k +4 ).Next we consider b-pseudodifferential operators. We write X for the b-stretchedproduct, which recall is obtained by blowing up the corner ∂X × ∂X in X × X . Forsimplicity we will neglect various b-half-density factors. Let ( x, y ) and ( x ′ , y ′ ) be special ROPAGATION OF SINGULARITIES ON ADS SPACETIMES 61 local coordinates on X , such that x = x ′ . Thus ( x, y, x ′ , y ′ ) are valid local coordinateson X near ∂X × ∂X . Near the front face in X , local coordinates are given by τ = x − x ′ x + x ′ , r = x + x ′ , y, y ′ , where r is a bdf for the front face. Definition B.2.
We say that a smooth function f ∈ C ∞ ( X ) is even modulo O ( r k +3 ),written f ∈ C ∞ even ( X ), if the Taylor expansion of f at the front face { r = 0 } incoordinates ( τ, r, y, y ′ ) contains only even terms modulo O ( r k +3 ).In other words, f ∈ C ∞ ( X ) is even modulo O ( r k +3 ) if we can write f ( τ, r, y, y ′ ) = f ( τ, r , y, y ′ ) + r k +3 f ′ ( τ, r, y, y ′ )for smooth functions f , f ′ . Again by (4.2), this definition is independent of thechoice of special coordinates in either factor. We can now define the space of evenb-pseudodifferential operators, recalling that that elements of Ψ m b ( X ) have Schwartzkernels on X that are conormal to the lifted diagonal. In local coordinates ( τ, r, y, y ′ ),we can view these as distributions conormal to { τ = 0 , y = y ′ } with smooth parametricdependence on r . Definition B.3. If A ∈ Ψ m b ( X ), then we say that A ∈ Ψ −∞ b , even ( X ) if its Schwartz kernel K A in local coordinates ( τ, r, y, y ′ ) as above has a Taylor expansion at the front face { r = 0 } containing only even terms modulo O ( r k +3 ).The Schwartz kernel K A also has the usual infinite order of vanishing at the sidefaces. Roughly speaking, this definition means that the total symbol of K A in localcoordinates ( r, y, σ, η ) is even in r modulo O ( r k +3 ).Next, we discuss composition of even b-pseudodifferential operators. First, we notethat C ∞ even ( X ) · C ∞ even ( X ) ⊂ C ∞ even ( X ) , and that the lifts of C ∞ even ( X ) functions to X from either the left or right factors landin C ∞ even ( X ). Using partitions of unity χ i ∈ C ∞ even ( X ), one can reduce to composition ofeven operators with localized Schwartz kernels, just as considered in [Mel, Section 5.9].It is then straightforward to see that even operators are closed under composition: Proposition B.4. A ∈ Ψ m b , even ( X ) and B ∈ Ψ m ′ b , even ( X ) , then AB ∈ Ψ m + m ′ b , even ( X ) . On R n + consider the quantization procedure given byOp b ( a ) u ( x, y ) = Z e i (( x/x ′ − σ + h y − y ′ ,η i ) φ ( x/x ′ ) a ( x, y, σ, η ) dx ′ x ′ dy ′ dσdη, where φ ∈ C ∞ c ( R ) satisfies supp φ ⊂ (1 / ,
2) and φ ( s ) = 1 near s = 1. If a ∈ S m ( R n + )is of the form a ( x, y, σ, η ) = a ( x , y, σ, η ) + x k +3 a ′ ( x, y, σ, η ) , then Op b ( a ) ∈ Ψ m b , even ( R n + ) irrespective of the choice of aAdS metric on R n + of the form − dx + k αβ ( x, y ) dy αβ x . By using an even partition of unity to patch these local quantization procedures to-gether, we can quantize even symbols on an arbitrary aAdS spacetime (
X, g ). Inparticular, we can always construct elliptic operators A ∈ Ψ m b , even ( X ).B.2. The indicial family.
Apart from composition and the existence of elliptic el-ements, the other key property of even b-pseudodifferential operators is an improvedstatement about the kernel of the indicial family map. Given A ∈ Ψ m b ( X ), recall that b N ( A )( s ) is defined invariantly as follows: first, one restricts the kernel K A to the frontface in X . The front face is then identified with the inward pointing spherical normalbundle to ∂X × ∂X , which is identified with ∂X × ∂X × R + . Finally, the indicial familyis the Mellin transform of the resulting function in the R + factor. The vanishing of b N ( A ) is thus equivalent to the statement that A ∈ x Ψ m b ( X ).Now suppose that ( X, g ) is an aAdS spacetime; in particular, it is trivially even mod-ulo O ( x ). The even calculus on X is thus defined modulo cubic terms. In particular,we have the following: Lemma B.5. If A ∈ Ψ m b , even ( X ) and b N ( A ) = 0 , then A ∈ x Ψ m b ( X ) . This implies the following corollary:
Lemma B.6.
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E-mail address : [email protected] Department of Mathematics, Lunt Hall, Northwestern University, Evanston, IL60208, USA
E-mail address : [email protected]@univ-grenoble-alpes.fr