Propagation of singularities for the wave equation on manifolds with corners

Transcription

1 Annals of Mathematics, 168 (2008), Propagation of singularities for the wave equation on manifolds with corners By András Vasy* Astract In this paper we descrie the propagation of C and Soolev singularities for the wave equation on C manifolds with corners M equipped with a Riemannian metric g. That is, for = M R t, P = D 2 t M, and u H 1 loc () solving P u = 0 with homogeneous Dirichlet or Neumann oundary conditions, we show that WF (u) is a union of maximally extended generalized roken icharacteristics. This result is a C counterpart of Leeau s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified oundary, [11]. Our methods rely on -microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperolic points even if M has a smooth oundary (and no corners). 1. Introduction In this paper we descrie the propagation of C and Soolev singularities for the wave equation on a manifold with corners M equipped with a smooth Riemannian metric g. We first recall the asic definitions from [12], and refer to [20, 2] as a more accessile reference. Thus, a tied (or t-) manifold with corners of dimension n is a paracompact Hausdorff topological space with a C structure with corners. The latter simply means that the local coordinate charts map into [0, ) k R n k rather than into R n. Here k varies with the coordinate chart. We write l for the set of points p such that in any local coordinates φ = (φ 1,..., φ k, φ k+1,..., φ n ) near p, with k as aove, precisely l of the first k coordinate functions vanish at φ(p). We usually write such local coordinates as (x 1,..., x k, y 1,..., y n k ). A oundary face of codimension l is the closure of a connected component of l. A oundary face of codimension 1 is called a oundary hypersurface. A manifold with corners is a tied manifold with corners such that all oundary hypersurfaces are emedded sumanifolds. This implies the existence of gloal defining functions ρ H for *This work is partially supported y NSF grant #DMS , a fellowship from the Alfred P. Sloan Foundation and a Clay Research Fellowship.

2 750 ANDRÁS VASY each oundary hypersurface H (so that ρ H C (), ρ H 0, ρ H vanishes exactly on H and dρ H 0 on H); in each local coordinate chart intersecting H we may take one of the x j s (j = 1,..., k) to e ρ H. While our results are local, and hence hold for t-manifolds with corners, it is convenient to use the emeddedness occasionally to avoid overurdening the notation. Moreover, in a given coordinate system, we often write H j for the oundary hypersurface whose restriction to the given coordinate patch is given y x j = 0, so that the notation H j depends on a particular coordinate system having een chosen (ut we usually ignore this point). If is a manifold with corners, denotes its interior, which is thus a C manifold (without oundary). Returning to the wave equation, let M e a manifold with corners equipped with a smooth Riemannian metric g. Let = g e the positive Laplacian of g, let = M R t, P = D 2 t, and consider the Dirichlet oundary condition for P : P u = 0, u = 0, with the oundary condition meaning more precisely that u H0,loc 1 (). Here H0 1() is the completion of C c () (the vector space of C functions of compact support on, vanishing with all derivatives at ) with respect to u 2 H 1 () = du L 2 () + u L2 (), L 2 () = L 2 (, dg dt), and H0,loc 1 () is its localized version; i.e., u H0 1() if for all φ C c (), φu H0 1 (). At the end of the introduction we also consider Neumann oundary conditions. The statement of the propagation of singularities of solutions has two additional ingredients: locating singularities of a distriution, as captured y the wave front set, and descriing the curves along which they propagate, namely the icharacteristics. Both of these are closely related to an appropropriate notion of phase space, in which oth the wave front set and the icharacteristics are located. On manifolds without oundary, this phase space is the standard cotangent undle. In the presence of oundaries the phase space is the -cotangent undle, T, ( stands for oundary), which we now riefly descrie following [19], which mostly deals with the C oundary case, and especially [20]. Thus, V () is, y definition, the Lie algera of C vector fields on tangent to every oundary face of. In local coordinates as aove, such vector fields have the form aj (x, y)x j xj + j (x, y) yj with a j, j smooth. Correspondingly, V () is the set of all C sections of a vector undle T over : locally x j xj and yj generate V () (over C ()), and thus (x, y, a, ) are local coordinates on T. j

3 PROPAGATION OF SINGULARITIES 751 The dual undle of T is T ; this is the phase space in our setting. Sections of these have the form (1.1) σj (x, y) dx j + ζ j (x, y) dy j, x j j and correspondingly (x, y, σ, ζ) are local coordinates on it. Let o denote the zero section of T (as well as other related vector undles elow). Then T \ o is equipped with an R + -action (fierwise multiplication) which has no fixed points. It is often natural to take the quotient with the R + -action, and work on the -cosphere undle, S. The differential operator algera generated y V () is denoted y Diff (), and its microlocalization is Ψ (), the algera of -, or totally characteristic, pseudodifferential operators. For A Ψ m (), σ,m(a) is a homogeneous degree m function on T \ o. Since is not compact, even if M is, we always understand that Ψ m () stands for properly supported ps.d.o s, so its elements define continuous maps C () C () as well as C () C (). Here C () denotes the suspace of C () consisting of functions vanishing at with all derivatives, C c () the suspace of C () consisting of functions of compact support. Moreover, C () is the dual space of C c (); we may call its elements tempered or extendile distriutions. Thus, Cc ( ) C () and C () C ( ). We are now ready to define the wave front set WF (u) for u Hloc 1 (). This measures whether u has additional regularity, locally in T, relative to H 1. For u Hloc 1 (), q T \ o, m 0, we say that q / WF 1,m (u) if there is A Ψ m () such that σ,m(a)(q) 0 and Au H 1 (). Since compactly supported elements of Ψ 0 () preserve H1 loc (), it follows that for u Hloc 1 (), WF1,0 (u) =. For any m, WF1,m (u) is a conic suset of T \o; hence it is natural to identify it with a suset of S. Its intersection with T \ o, which can e naturally identified with T \ o, is WF m+1 (u). Thus, in the interior of, WF 1,m (u) measures whether u is microlocally in H m+1. The main result of this paper, stated at the end of this section, is that for u H0 1 () with P u = 0, WF1,m (u) is a union of maximally extended generalized roken icharacteristics, which are defined elow. In fact, the requirement u H0 1 () can e relaxed and m can e allowed to e negative, see Definitions We also remark that for such u, the H 1 ()-ased -wave front set, WF 1,m (u), could e replaced y an L 2 ()-ased -wave front set; see Lemma 6.1. In addition, our methods apply, a fortiori, for elliptic prolems such as g on (M, g), e.g. showing that u H0,loc 1 (M) and ( g λ)u = 0 imply u H 1,,loc (M), so that u is conormal; see the end of Section 4. This propagation result is the C (and Soolev space) analogue of Leeau s result [11] for analytic singularities of u when M and g are real analytic. Thus, the geometry is similar in the two settings, ut the analytic techniques are

4 752 ANDRÁS VASY rather different: Leeau uses complex scaling and the analytic wave front set of the extension of u as 0 to a neighorhood of (in an extension of the manifold ), while we use positive commutator estimates and -microlocalization relative to the form domain of the Laplacian. It should e kept in mind though that positive commutator estimates can often e thought of as infinitesimal versions of complex scaling (if complex scaling is availale at all), although this is more of a moral than a technical statement, for the techniques involved in working infinitesimally are quite different from what one can do if one has room to deform contours of integration! In fact, our microlocalization techniques, especially the positive commutator constructions, are very closely related to the methods used in N-ody scattering, [24], to prove the propagation of singularities (meaning microlocal lack of decay at infinity) there. Although Leeau allows more general singularities than corners for, provided that sits in a real analytic manifold with g extending to, we expect to generalize our results to settings where no analogous C extension is availale; see the remarks at the end of the introduction. We now descrie the setup in more detail so that our main theorem can e stated in a precise fashion. Let F i, i I, e the closed oundary faces of M (including M), F i = F i R, F i,reg the interior ( regular part ) of F i. Note that for each p, there is a unique i such that p F i,reg. Although we work on oth M and, and it is usually clear which one we mean even in the local coordinate discussions, to make matters clear we write local coordinates on M, as in the introduction, as (x, y) (with x = (x 1,..., x k ), y = (y 1,..., y dim M k )), with x j 0 (j = 1,..., k) on M, and then local coordinates on, induced y the product M R t, as (x, ȳ), ȳ = (y, t) (so that is given y x j 0, j = 1,..., k). Let p, and let F i e the closed face of with the smallest dimension that contains p, so that p F i,reg. Then we may choose local coordinates (x, y, t) = (x, ȳ) near p in which F i is defined y x 1 =... = x k = 0, and the other oundary faces through p are given y the vanishing of a suset of the collection x 1,..., x k of functions; in particular, the k oundary hypersurfaces H j through p are locally given y x j = 0 for j = 1,..., k. (This may require shrinking a given coordinate chart (x, ȳ ) that contains p so that the x j that do not vanish identically on F i do not vanish at all on the smaller chart, and can e relaelled as one of the coordinates y l.) Now, there is a natural non-injective inclusion π : T T induced y identifying T with T (and hence also their dual undles) with each other in the interior of, where the condition on tangency to oundary faces is vacuous. In view of (1.1), in the canonical local coordinates (x, ȳ, ξ, ζ) on T (so one-forms are ξ j dx j + ζj dȳ j ), and canonical local coordinates (x, ȳ, σ, ζ) on T, π takes the form π(x, ȳ, ξ, ζ) = (x, ȳ, xξ, ζ), with xξ = (x 1 ξ 1,..., x k ξ k ).

5 PROPAGATION OF SINGULARITIES 753 Thus, π is a C map, ut at the oundary of, it is not a local diffeomorphism. Moreover, the range of π over the interior of a face F i lies in T F i (which is welldefined as a suspace of T ) while its kernel is N F i, the conormal undle of F i in. In local coordinates as aove, in which F i is given y x = 0, the range T F i over F i is given y x = 0, σ = 0 (i.e. y x 1 =... = x k = 0, σ 1 =... = σ k = 0), while the kernel N F i is given y x = 0, ζ = 0. Then we define the compressed -cotangent undle T to e the range of π: T = π(t ) = i I T F i,reg T. We write o for the zero section of T as well, so that and then π restricts to a map T \ o = i I T F i,reg \ o, T \ i N F i T \ o. Now, the characteristic set Char(P ) T \o of P is defined y p 1 ({0}), where p C (T \ o) is the principal symol of P, which is homogeneous degree 2 on T \o. Notice that Char(P ) N F i = for all i, i.e. the oundary faces are all non-characteristic for P. Thus, π(char(p )) T \o. We define the elliptic, glancing and hyperolic sets y E = {q T \ o : π 1 (q) Char(P ) = }, G = {q T \ o : Card(π 1 (q) Char(P )) = 1}, H = {q T \ o : Card(π 1 (q) Char(P )) 2}, with Card denoting the cardinality of a set; each of these is a conic suset of T \ o. Note that in T, π is the identity map, so that every point q T is either in E or G depending on whether q / Char(P ) or q Char(P ). Local coordinates on the ase induce local coordinates on the cotangent undle, namely (x, y, t, ξ, ζ, τ) on T near π 1 (q), q T F i,reg, and corresponding coordinates (y, t, ζ, τ) on a neighorhood U of q in T F i,reg. The metric function on T M has the form g(x, y, ξ, ζ) = i,j A ij (x, y)ξ i ξ j + i,j 2C ij (x, y)ξ i ζ j + i,j B ij (x, y)ζ i ζ j with A, B, C smooth. Moreover, these coordinates can e chosen (i.e. the y j can e adjusted) so that C(0, y) = 0. Thus, p x=0 = τ 2 ξ A(y)ξ ζ B(y)ζ, with A, B positive definite matrices depending smoothly on y, so that E U = {(y, t, ζ, τ) : τ 2 < ζ B(y)ζ, (ζ, τ) 0}, G U = {(y, t, ζ, τ) : τ 2 = ζ B(y)ζ, (ζ, τ) 0}, H U = {(y, t, ζ, τ) : τ 2 > ζ B(y)ζ, (ζ, τ) 0}.

6 754 ANDRÁS VASY and The compressed characteristic set is Σ = π(char(p )) = G H, ˆπ : Char(P ) Σ is the restriction of π to Char(P ). Then Σ has the suspace topology of T, and it can also e topologized y ˆπ, i.e. requiring that C Σ e closed (or open) if and only if ˆπ 1 (C) is closed (or open). These two topologies are equivalent, though the former is simpler in the present setting; e.g., it is immediate that Σ is metrizale. Leeau [11] (following Melrose s original approach in the C oundary setting, see [17]) uses the latter; in extensions of the present work, to allow e.g. iterated conic singularities, that approach will e needed. Again, an analogous situation arises in N-ody scattering, though that is in many respects more complicated if some susystems have ound states [24], [25]. We are now ready to define generalized roken icharacteristics, essentially following Leeau [11]. We say that a function f on T \ o is π-invariant if f(q) = f(q ) whenever π(q) = π(q ). In this case f induces a function f π on T which satisfies f = f π π. Moreover, if f is continuous, then so is f π. Notice that if f = π f 0, f 0 C ( T ), then f C (T ) is certainly π-invariant. Definition 1.1. A generalized roken icharacteristic of P is a continuous map γ : I Σ, where I R is an interval, satisfying the following requirements: (i) If q 0 = γ(t 0 ) G then for all π-invariant functions f C (T ), d (1.2) dt (f π γ)(t 0 ) = H p f( q 0 ), q 0 = ˆπ 1 (q 0 ). (ii) If q 0 = γ(t 0 ) H T F i,reg then there exists ε > 0 such that (1.3) t I, 0 < t t 0 < ε γ(t) / T F i,reg. (iii) If q 0 = γ(t 0 ) G T F i,reg, and F i is a oundary hypersurface (i.e. has codimension 1), then in a neighorhood of t 0, γ is a generalized roken icharacteristic in the sense of Melrose-Sjöstrand [13]; see also [4, Def ]. Remark 1.2. Note that for q 0 G, ˆπ 1 ({q 0 }) consists of a single point, and so (1.2) makes sense. Moreover, (iii) implies (i) if q 0 is in a oundary hypersurface, ut it is stronger at diffractive points; see [4, 24.3]. The propagation of analytic singularities, as in Leeau s case, does not distinguish etween gliding and diffractive points, hence (iii) can e dropped to define what we may

7 PROPAGATION OF SINGULARITIES 755 call analytic generalized roken icharacteristics. It is an interesting question whether in the C setting there are also analogous diffractive phenomena at higher codimension oundary faces, i.e. whether the following theorem can e strengthened at certain points. We remark also that there is an equivalent definition (presented in lecture notes aout the present work, see [26]), which is more directly motivated y microlocal analysis and which also works in other settings such as N-ody scattering in the presence of ound states. Our main result is: Theorem (See Corollary 8.4). Suppose that P u = 0, u H0,loc 1 (). Then WF 1, (u) Σ, and it is a union of maximally extended generalized roken icharacteristics of P in Σ. The analogue of this theorem was proved in the real analytic setting y Leeau [11], and in the C setting with C oundaries (and no corners) y Melrose, Sjöstrand and Taylor [13], [14], [22]. In addition, Ivriĭ [8] has otained propagation results for systems. Moreover, a special case with codimension 2 corners in R 2 had een considered y P. Gérard and Leeau [3] in the real analytic setting, and y Ivriĭ [5] in the smooth setting. It should e mentioned that due to its relevance, this prolem has a long history, and has een studied extensively y Keller in the 1940s and 1950s in various special settings; see e.g. [1], [10]. The present work (and ongoing projects continuing it, especially joint work with Melrose and Wunsch [15], see also [2], [16]), can e considered a justification of Keller s work in the general geometric setting (curved edges, variale coefficient metrics, etc.). A more precise version of this theorem, with microlocal assumptions on P u, is stated in Theorem 8.1. In particular, one can allow P u C (), which immediately implies that the theorem holds for solutions of the wave equation with inhomogeneous C Dirichlet oundary conditions that match across the oundary hyperfaces, see Remark 8.2. In addition, this theorem generalizes to the wave operator with Neumann oundary conditions, which need to e interpreted in terms of the quadratic form of P (i.e. the Dirichlet form). That is, if u Hloc 1 () satisfies d M u, d M v t u, t v = 0 for all v H 1 c (), then WF 1, (u) Σ, and it is a union of maximally extended generalized roken icharacteristics of P in Σ. In fact, the proof of the theorem for Dirichlet oundary conditions also utilizes the quadratic form of P. It is slightly simpler in presentation only to the extent that one has more flexiility to integrate y parts, etc., ut in the end the proof for Neumann oundary conditions simply requires a slightly less conceptual (in terms of the traditions of microlocal analysis) reorganization, e.g. not using commutators

8 756 ANDRÁS VASY [P, A] directly, ut commuting A through the exterior derivative d M and t directly. It is expected that these results will generalize to iterated edge-type structures (under suitale hypotheses), whose simplest example is given y (isolated) conic points, recently analyzed y Melrose and Wunsch [16], extending the product cone analysis of Cheeger and Taylor [2]. This is suject of an ongoing project with Richard Melrose and Jared Wunsch [15]. It is an interesting question whether this propagation theorem can e improved in the sense that, under certain non-focusing assumptions for a solution u of the wave equation, if a icharacteristic segment carrying a singularity of u hits a corner, then the reflected singularity is weaker along nongeometrically related generalized roken icharacteristics continuing the aforementioned segment than along geometrically related ones. Roughly, geometrically related continuations should e limits of icharacteristics just missing the corner. In the setting of (isolated) conic points, such a result was otained y Cheeger, Taylor, Melrose and Wunsch [2], [16]. While the analogous result (including its precise statement) for manifolds with corners is still some time away, significant progress has een made, since the original version of this manuscript was written, on analyzing edge-type metrics (on manifolds with oundaries) in the project [15]. The outline of these results, including a discussion of how it relates to the prolem under consideration here, is written up in the lecture notes of the author on the present paper [26]. To make clear what the main theorem states, we remark that the propagation statement means that if u solves P u = 0 (with, say, Dirichlet oundary condition), and q T \ o is such that u has no singularities on icharacteristics entering q (say, from the past), then we conclude that u has no singularities at q, in the sense that q / WF 1, (u); i.e., we only gain -derivatives (or totally characteristic derivatives) microlocally. In particular, even if WF 1, (u) is empty, we can only conclude that u is conormal to the oundary, in the precise sense that V 1... V k u Hloc 1 () for any V 1,..., V k V (), and not that u Hloc k () for all k. Indeed, the latter cannot e expected to hold, as can e seen y considering e.g. the wave equation (or even elliptic equations) in 2-dimensional conic sectors. This already illustrates that from a technical point of view a major challenge is to comine two differential (and pseudodifferential) algeras: Diff() and Diff () (or Ψ ()). The wave operator P lies in Diff(), ut microlocalization needs to take place in Ψ (): if Ψ( ) is the algera of usual pseudodifferential operators on an extension of, its elements do not even act on C (): see [4, 18.2] when has a smooth oundary (and no corners). In addition, one needs an algera whose elements A respect the oundary conditions, so that e.g. Au depends only on u. This is exactly the origin of the algera of totally characteristic pseudodifferential operators, denoted y

9 PROPAGATION OF SINGULARITIES 757 Ψ (), in the C oundary setting [18]. The interaction of these two algeras also explains why we prove even microlocal elliptic regularity via the quadratic form of P (the Dirichlet form), rather than y standard arguments, valid if one studies microlocal elliptic regularity for an element of an algera (such as Ψ ()) with respect to the same algera. The ideas of the positive commutator estimates, in particular the construction of the commutants, are very similar to those arising in the proof of the propagation of singularities in N-ody scattering in previous works of the author the wave equation corresponds to the relatively simple scenario there when no proper susystems have ound states [24]. Indeed, the author has indicated many times in lectures that there is a close connection etween these two prolems, and it is a pleasure to finally spell out in detail how the N-ody methods can e adapted to the present setting. The organization of the paper is as follows. In Section 2 we recall asic facts aout Ψ () and analyze its commutation properties with Diff(). In Section 3 we descrie the mapping properties of Ψ () on H 1 ()-ased spaces. We also define and discuss the -wave front set ased on H 1 () there. The following section is devoted to the elliptic estimates for the wave equation. These are otained from the microlocal positivity of the Dirichlet form, which implies in particular that in this region commutators are negligile for our purposes. In Section 5 we descrie asic properties of icharacteristics, mostly relying on Leeau s work [11]. In Sections 6 and 7, we prove propagation estimates at hyperolic, resp. glancing, points, y positive commutator arguments. Similar arguments were used y Melrose and Sjöstrand [13] for the analysis of propagation at glancing points for manifolds with smooth oundaries. In Section 8 these results are comined to prove our main theorems. The arguments presented there are very close to those of Melrose, Sjöstrand and Leeau. Here we point out that Ivriĭ [8], [6], [7], [9] also used microlocal energy estimates to otain propagation results of a different flavor for symmetric systems in the smooth oundary setting, including at hyperolic points. Roughly, Ivriĭ s results give conditions for hypersurfaces Σ through a point q 0 under which the following conclusion holds: the point q 0 is asent from the wave front set of a solution provided that, in a neighorhood of q 0, one side of Σ is asent from the wave front set with further restrictions on the hypersurface in the presence of smooth oundaries. In some circumstances, using other known results, Ivriĭ could strengthen the conclusion further. Since the changes for Neumann oundary conditions are minor, and the arguments for Dirichlet oundary conditions can e stated in a form closer to those found in classical microlocal analysis (essentially, in the Neumann case one has to pay a price for integrating y parts, so one needs to present the proofs in an appropriately rearranged, and less transparent, form) the proofs in

10 758 ANDRÁS VASY the ody of the paper are primarily written for Dirichlet oundary conditions, and the required changes are pointed out at the end of the various sections. In addition, the hypotheses of the propagation of singularities theorem can e relaxed to u H 1,m,0,loc (), m 0, defined in Definition Since this simply requires replacing the H 1 () norms y the H 1,m norms (which are only locally well defined), we suppress this point except in the statement of the final result, to avoid overurdening the notation. No changes are required in the argument to deal with this more general case. See Remark 8.3 for more details. To give the reader a guide as to what the real novelty is, Sections 2-3 should e considered as variations on a well-developed theme. While some of the features of microlocal analysis, especially wave front sets, are not discussed on manifolds with corners elsewhere, the modifications needed are essentially trivial (cf. [4, Ch. 18]). A slight novelty is using H 1 () as the point of reference for the -wave front sets (rather than simply weighted L 2 spaces), which is very useful later in the paper, ut again only demands minimal changes to standard arguments. The discussions of icharacteristics in Section 5 essentially quotes Leeau s paper [11, III]. Moreover, given the results of Sections 4, 6 and 7, the proof of propagation of singularities in Section 8 is standard, essentially due to Melrose and Sjöstrand [14, 3]. Indeed, as presented y Leeau [11, Prop. VII.1], asically no changes are necessary at all in this proof. The novelty is thus the use of the Dirichlet form (hence the H 1 -ased wave front set) for the proof of oth the elliptic and hyperolic/glancing estimates, and the systematic use of positive commutator estimates in the hyperolic/glancing regions, with the commutants arising from an intrinsic pseudodifferential operator algera, Ψ (). This approach is quite roust, hence significant extensions of the results can e expected, as was already indicated. Acknowledgments. I would like to thank Richard Melrose for his interest in this project, for reading, and therey improving, parts of the paper, and for numerous helpful and stimulating discussions, especially for the wave equation on forms. While this topic did not ecome a part of the paper, it did play a role in the presentation of the arguments here. I am also grateful to Jared Wunsch for helpful discussions and his willingness to read large parts of the manuscript at the early stages, when the ackground material was still mostly asent; his help significantly improved the presentation here. I would also like to thank Rafe Mazzeo for his continuing interest in this project and for his patience when I tried to explain him the main ideas in the early days of this project, and Victor Ivriĭ for his interest in, and his support for, this work. At last, ut not least, I am very grateful to the anonymous referee for a thorough reading of the manuscript and for many helpful suggestions.

11 PROPAGATION OF SINGULARITIES Interaction of Diff() with the -calculus One of the main technical issues in proving our main theorem is that unless =, the wave operator P is not a -differential operator: P / Diff 2 (). In this section we descrie the asic properties of how Diff k (), which includes P for k = 2, interacts with Ψ (). We first recall though that for p F i,reg, local coordinates in T over a neighorhood of p are given y (x, y, t, σ, ζ, τ) with σ j = x j ξ j. Thus, the map π in local coordinates is (x, y, t, ξ, ζ, τ) (x, y, t, xξ, ζ, τ), where y xξ we mean the vector (x 1 ξ 1,..., x k ξ k ). In fact, in this section y and t play a completely analogous role, hence there is no need to distinguish them. The difference will only arise when we start studying the wave operator P in Section 4. Thus, we let ȳ = (y, t) and ζ = (ζ, τ) here to simplify the notation. We riefly recall asic properties of the set of classical (one-step polyhomogeneous, in the sense that the full symols are such on the fiers of T ) pseudodifferential operators Ψ () = m Ψ m () and the set of standard (conormal) -pseudodifferential operators, Ψ c () = m Ψ m c (). The difference etween these two classes is in terms of the ehavior of their (full) symols at fier-infinity of T ; elements of Ψ c () have full symols that satisfy the usual symol estimates, while elements of Ψ () have in addition an asymptotic expansion in terms of homogeneous functions, so that Ψ m () Ψm c (). Conceptually, these are est defined via the Schwartz kernel of A Ψ m c () in terms of a certain low-up 2 of ; see [20]. The Schwartz kernel is conormal to the lift diag of the diagonal of 2 to 2 with infinite order vanishing on all oundary faces of 2 which are disjoint from diag. Modulo Ψ (), however, the explicit quantization map we give elow descries Ψ m c () and Ψm (). Here Ψ c () = Ψ () = m Ψ m c () = m Ψ m () is the ideal of smoothing operators. The topology of Ψ c () is given in terms of the conormal seminorms of the Schwartz kernel K of its elements; these seminorms can e stated in terms of the Besov space norms of L 1 L 2... L k K as k runs over non-negative integers, and the L j over first order differential operators tangential to diag ; see [4, Def ]. Recall in particular that these seminorms are (locally) equivalent to the C seminorms away from the lifted diagonal diag. There is a principal symol map σ,m : Ψ m c () Sm ( T )/S m 1 ( T ); here, for a vector undle E over, S k (E) denotes the set of symols of order k on E (i.e. these are symols in the fiers of E, smoothly varying over ). Its restriction to Ψ m () can e re-interpreted as a map σ,m : Ψ m () C ( T \ o) with values in homogeneous functions of degree m; the range can of course also e identified with C ( S ) if m = 0 (and with sections of

12 760 ANDRÁS VASY a line undle over S in general). There is a short exact sequence 0 Ψ m 1 c () Ψ m c () Sm ( T )/S m 1 ( T ) 0 as usual; the last non-trivial map is σ,m. There are also quantization maps (which depend on various choices) q = q m : S m ( T ) Ψ m c (), which restrict to q : Scl m( T ) Ψ m (), cl denoting classical symols, and σ,m q m is the quotient map S m S m /S m 1. For instance, over a local coordinate chart U as aove, with a supported in TK, K U compact, we may take, with n = dim, (2.1) q(a)u(x, ȳ) = (2π) n e i(x x ) ξ+(ȳ ȳ ) ζφ ( x x x ) a(x, y, xξ, ζ)u(x, ȳ ) dx dȳ dξ dζ, understood as an oscillatory integral, where φ Cc (( 1/2, 1/2) k ) is identically 1 near 0 and x x x = ( x1 x 1 x 1,..., xk x k x k ), and the integral in x is over [0, ) k. Here the role of φ is to ensure the infinite order vanishing at the oundary hypersurfaces of 2 disjoint from diag ; it is irrelevant as far as the ehavior of Schwartz kernels near the diagonal is concerned (it is identically 1 there). This can e extended to a gloal map via a partition of unity, as usual. Locally, for q(a), supp a TK as aove, the conormal seminorms of the Schwartz kernel of q(a) (i.e. the Besov space norms descried aove) can e ounded in terms of the symol seminorms of a; see the eginning of [4, 18.2], and conversely. Moreover, any A Ψ c () with properly supported Schwartz kernel defines continuous linear maps A : C () C (), A : C () C (). Remark 2.1. We often do not state it elow, ut in general most pseudodifferential operators have compact support in this paper. Sometimes we use properly supported ps.d.o s, in order not to have to state precise support conditions; these are always composed with compactly supported ps.d.o s or applied to compactly supported distriutions, so that, effectively, they can e treated as compactly supported. See also Remark 4.1. If g is any C Riemannian metric on, and K is compact, any A Ψ 0 c () with Schwartz kernel supported in K K defines a ounded operator on L 2 () = L 2 (, d g), with norm ounded y a seminorm of A in Ψ 0 c (). Indeed, this is true for A Ψ () with compact support, as follows from the Schwartz lemma and the explicit description of the Schwartz kernel of A on 2. The standard square root argument then shows the oundedness for A Ψ 0 c (), with norm ounded y a seminorm of A in Ψ0 c (); see [20, Eq. (2.16)]. In fact, we get more from the argument: letting a = σ,0 (A), there exists A Ψ 1 () such that for all v L2 (), Av 2 sup a v + A v.

13 PROPAGATION OF SINGULARITIES 761 (The factor 2 of course can e improved, as can the order of A.) This estimate will play an important role in our propagation estimates. It will make it unnecessary to construct a square root of the commutator, which would e difficult here as we will commute P with an element of Ψ (), so that the commutator will not lie in Ψ (). We remark here that it is more usual to take a -density in place of d g, i.e. a gloally non-vanishing section of Ω 1 = Ω, which thus takes the form (x 1... x k ) 1 d g locally near a codimension k corner, to define an L 2 -space, namely L 2 () = L2 d g (, x 1...x k ); then L 2 () = x 1/ x 1/2 k L 2 () appears as a weighted space. Elements of Ψ0 c () are ounded on oth L 2 spaces, in the manner stated aove. The two oundedness results are very closely related, for if A Ψ 0 c (), then so is xλ j Ax λ j, λ C. There is an operator wave front set associated to Ψ c () as well: for A Ψ m c (), WF (A) is a conic suset of T \ o, and has the interpretation that A is in Ψ c () outside WF (A). (We caution the reader that unlike the previous material, as well as the rest of the ackground in the next three paragraphs, WF is not discussed in [20]. This discussion, however, is standard; see e.g. [4, 18.1], especially after Definition , in the oundaryless case, and [4, 18.3] for the case of a C oundary, where one simply says that the operator is order on certain open cones; see e.g. the proof of Theorem there.) In particular, if WF (A) =, then A Ψ (). For instance, if A = q(a), a S m ( T ), q as in (2.1), WF (A) is defined y the requirement that if p / WF (A) then p has a conic neighorhood U in T \ o such that A = q(a), a is rapidly decreasing in U; i.e., a(x, ȳ, σ, ζ) C N (1 + σ + ζ ) N for all N. Thus, WF (A) is a closed conic suset of T \ o. Moreover, if K S is compact, and U is a neighorhood of K, there exists A Ψ 0 () such that A is the identity on K and vanishes outside U, i.e. WF (A) U, WF (Id A) K =. We can construct a to e homogeneous degree zero outside a neighorhood of o, such that this homogeneous function regarded as a function on S (and still denoted y a) satisfies a 1 near K, supp a U, and then let A = q(a). (This roughly says that Ψ () can e used to localize in S, i.e. to -microlocalize.) Since Ψ c () forms a filtered -algera, A j Ψ mj c (), j = 1, 2, implies A 1 A 2 Ψ m1+m2 c (), and A j Ψmj c () with σ,m1+m 2 (A 1 A 2 ) = σ,m1 (A 1 )σ,m2 (A 2 ), σ,mj (A j) = σ,mj (A). Here the formal adjoint is defined with respect to L 2 (), the L 2 -space of any C Riemannian metric on ; the same statements hold with respect to L 2 () as well, since conjugation y x 1... x k preserves Ψ m c () (as well as Ψm ()), as already remarked for m = 0. Moreover, [A 1, A 2 ] Ψ m1+m2 1 c () with σ,m1+m 2 1([A 1, A 2 ]) = 1 i {a 1, a 2 }, a j = σ,mj (A j );

14 762 ANDRÁS VASY {, } is the Poisson racket lifted from T via the identification of T with T. If A j Ψ mj (), then A 1A 2 Ψ m1+m2 (), A j Ψmj (), and [A 1, A 2 ] Ψ m1+m2 1 (). In addition, operator composition satisfies WF (A 1A 2 ) WF (A 1) WF (A 2). If A Ψ m c (A) is elliptic, i.e. σ,m(a) is invertile as a symol (with inverse in S m ( T \o)/s m 1 ( T \o)), then there is a parametrix G Ψ m c () for A, i.e. GA Id, AG Id Ψ c (). This construction microlocalizes, so if σ,m (A) is elliptic at q T \ o, i.e. σ,m (A) is invertile as a symol in an open cone around q, then there is a microlocal parametrix G Ψ m c () for A at q, so that q / WF (GA Id), q / WF (AG Id), so GA, AG are microlocally the identity operator near q. More generally, if K S is compact, and σ,m (A) is elliptic on K then there is G Ψ m c () such that K WF (GA Id) =, K WF (AG Id) =. For A Ψm (), σ,m(a) can e regarded as a homogeneous degree m function on T \ o, and ellipticity at q means that σ,m (A)(q) 0. For such A, one can take G Ψ m () in all the cases descried aove. The other important ingredient, which however rarely appears in the following discussion, although when it appears it is crucial, is the notion of the indicial operator. This captures the mapping properties of A Ψ () in terms of gaining any decay at. It plays a role here as P / Diff (); so even if we do not expect to gain any decay for solutions u of P u = 0 say, we need to understand the commutation properties of Diff() with Ψ (), which will in turn follow from properties of the indicial operator. There is an indicial operator map (which can also e considered as a non-commutative analogue of the principal symol), denoted y ˆN i, for each oundary face F i, i I, and ˆN i maps Ψ m c () to a family of -pseudodifferential operators on F i. For us, only the indicial operators associated to oundary hypersurfaces H j (given y x j = 0) will e important; in this case the family is parametrized y σ j, the -dual variale of x j. It is characterized y the property that if f C (H j ) and u C () is any extension of f, i.e. u Hj = f, then ˆN j (A)(σ j )f = (x iσj j Ax iσj j u) Hj, where x iσj j Ax iσj j Ψ m c (), hence x iσj j Ax iσj j u C (), and the right-hand side does not depend on the choice of u. (In this formulation, we need to fix x j, at least mod x 2 j C (), to fix ˆN j (A). Note that the radial vector field, x j D xj, is independent of this choice of x j, at least modulo x j V ().) If A Ψ m c () and ˆN i (A) = 0, then in fact A CF i () Ψ m c (), where C F i () is the ideal of C () consisting of functions that vanish at F i. In particular, for a oundary hypersurface H j defined y x j, if A Ψ m c () and ˆN j (A) = 0, then A = x j A with A Ψ m c (). The indicial operators satisfy ˆN i (AB) = ˆN i (A) ˆN i (B). The indicial family of x j D xj at H j is multiplication y σ j, while the indicial

15 PROPAGATION OF SINGULARITIES 763 family of x k D xk, k j, is x k D xk and that of Dȳk is Dȳk. In particular, ˆN j ([x j D xj, A]) = [ ˆN j (x j D xj ), ˆN j (A)] = 0, so (2.2) [x j D xj, A] x j Ψ m c (), which plays a role elow. All of the aove statements also hold with Ψ c () replaced y Ψ (). The key point in analyzing smooth vector fields on, and therey differential operators such as P, is that while D xj / V (), for any A Ψ m () there is an operator à Ψm () such that (2.3) D xj A ÃD x j Ψ m (), and analogously for Ψ m () replaced y Ψm c (). Indeed, D xj A = x 1 j By (2.2), applied for Ψ rather than Ψ c, (x j D xj )A = x 1 j [x j D xj, A] + x 1 j Ax j D xj. x 1 j [x j D xj, A] Ψ m (). Thus, we may take à = x 1 j Ax j, proving (2.3). We also have, more trivially, that (2.4) Dȳj A ÃD ȳ j Ψ m (), à Ψm (), σ,m(a) = σ,m (Ã). Since σ,m (A) = σ,m (x 1 j Ax j ), we deduce the following lemma. Lemma 2.2. Suppose V V(), A Ψ m (). Then [V, A] = A j V j +B with A j Ψ m 1 (), V j V(), B Ψ m (). Similarly, [V, A] = V j A j + B with A j Ψm 1 (), V j V(), B Ψ m (). Analogous results hold with Ψ () replaced y Ψ c (). Proof. It suffices to prove this for the coordinate vector fields, and indeed just for the D xj. Then with the notation of (2.3), D xj A AD xj = (à A)D x j + B, and σ,m (Ã) = σ,m(a), so that à A Ψm 1 (), proving the claim. More generally, we make the definition: Definition 2.3. Diff k Ψ s () is the vector space of operators of the form (2.5) P j A j, P j Diff k (), A j Ψ s (), j where the sum is locally finite in. Diff k () Ψ s c () is defined analogously.

16 764 ANDRÁS VASY Remark 2.4. Since any point q T \ o has a conic neighorhood U in T \ o on which some vector field V V () is elliptic, i.e. σ,1 (V ) 0 on U, we can always write A j Ψ s+k kj () with WF (A) U, k j k, as A j = Q j A j + R j with Q j Diff k kj (), A j Ψs (), R j Ψ (). Thus, any operator which is given y a locally finite sum of the form j P j A j, P j Diff kj (), A j Ψ s+k kj (), can in fact e written in the form (2.5). In particular, Diff k Ψ s c () Diff k Ψ s c () provided that k k and k + s k + s, and Diff k Ψ s () Diff k Ψ s () provided that k k, k + s k + s and s s is an integer. Lemma 2.5. Diff Ψ c () is a filtered algera with respect to operator composition, with B j Diff kj Ψ sj c (), j = 1, 2, implying Moreover, with B 1, B 2 as aove, B 1 B 2 Diff k1+k2 Ψ s1+s2 c (). [B 1, B 2 ] Diff k1+k2 Ψ s1+s2 1 c (). Proof. To prove that Diff Ψ c () is an algera, we only need to prove that if A Ψ s c (), P Diffk (), then AP Diff k () Ψ s c (). When P is a sum of products of vector fields in V(), the claim follows from Lemma 2.2. Writing B j = V j,1... V j,k1 A j, A j Ψ sj c (), V j,i V(), and expanding the commutator [B 1, B 2 ], one gets a finite sum, which is a product of the factors V j,1,... V j,k1, A j with two factors (one with j = 1 and one with j = 2) removed and replaced y a commutator. In view of the first part of the lemma, it suffices to note that [V 1,i, V 2,i ] V(), Diff k1+k2 1 Ψ s1+s2 c [A 1, A 2 ] Ψ s1+s2 1 c () [V j,i, A 3 j ] Diff 1 Ψ s3 j 1 c (), () Diff k1+k2 Ψ s1+s2 1 c (), where the last statement is a consequence of Lemma 2.2, when we take into account that Ψ m c () Diff1 Ψ m 1 c (). We can also define the principal symol on Diff k Ψ s (). Thus, using π : T T, we can pull ack σ,s (A), A Ψ s (), to T, and define: Definition 2.6. Suppose B = P j A j Diff k Ψ s (), P j Diff k (), A j Ψ s (). The principal symol of B is the C homogeneous degree k + s function on T \ o defined y (2.6) σ k+s (B) = σ k (P j )π σ,s (A j ).

17 PROPAGATION OF SINGULARITIES 765 Lemma 2.7. σ k+s (B) is independent of all choices. Proof. Away from, B is a pseudodifferential operator of order k + s, and σ k+s (B) is its invariantly defined symol. Since the right-hand side of (2.6) is continuous up to, and is independent of all choices in T, it is independent of all choices in T. We are now ready to compute the principal symol of the commutator of A Ψ m () with D x j. Lemma 2.8. Let xj, σj denote local coordinate vector fields on T in the coordinates (x, ȳ, σ, ζ). For A Ψ m () with Schwartz kernel supported in the coordinate patch, a = σ,m (A) C ( T \ o), we have [D xj, A] = A 1 D xj + A 0 Diff 1 Ψ m 1 () with A 0 Ψ m (), A 1 Ψ m 1 () and (2.7) σ,m 1 (A 1 ) = 1 i σ j a, σ,m (A 0 ) = 1 i x j a. This result also holds with Ψ () replaced y Ψ c () everywhere. Remark 2.9. Notice that σ m ([D xj, A]) = 1 i {ξ j, π a} = 1 i x j ξ π a, {.,.} denoting the Poisson racket on T and xj ξ denoting the appropriate coordinate vector field on T (where ξ is held fixed rather than σ during the partial differentiation), since oth sides are continuous functions on T \o which agree on T \ o. A simple calculation shows that the lemma is consistent with this result. The statement of the lemma would follow from this oservation if we showed that the kernel of σ m on Diff 1 Ψ m 1 () is Diff 1 Ψ m 2 (). The proof given elow avoids this point y reducing the calculation to Ψ (). Proof. The lemma follows from Indeed, when (2.8) A 0 = x 1 j D xj A AD xj = x 1 j [x j D xj, A] + x 1 j [A, x j ]D xj. [x j D xj, A] Ψ m (), A 1 = x 1 [A, x j ] Ψ m 1 (), the principal symols can e calculated in the -calculus. Since they are given y the standard Poisson racket in T, hence in T, y continuity the same calculation gives a valid result in T. As ξj = x j σj, xj ξ = xj σ + ξ j σj, we see that for = σ j or = x j, the Poisson racket {, a} is given y x j ( σj )( xj σ a + ξ j σj a) x j ( σj a)( xj σ + ξ j σj ) so that we get and (2.7) follows from (2.8). {σ j, a} = x j xj σ a, {x j, a} = x j σj a, j = x j ( σj ) xj σ a x j ( σj a) xj σ

18 766 ANDRÁS VASY 3. Function spaces and microlocalization We now turn to actions of Ψ () on function spaces related to differential operators in Diff(), and in particular to H 1 () which corresponds to first order differential operators, such as the exterior derivative d. We first recall that Cc () is the space of C functions of compact support on (which may thus e non-zero at ), while C c () is the suspace of Cc () consisting of functions which vanish to infinite order at. Although we will mostly consider local results, and any C Riemannian metric can e used to define L 2 loc (), L2 c() (as different choices give the same space), it is convenient to fix a gloal Riemmanian metric, g = g+dt 2, on, where g is the metric on M. With this choice, L 2 () is well-defined as a Hilert space. For u Cc (), we let u 2 H 1 () = du 2 L 2 () + u 2 L 2 (). We then let H 1 () e the completion of Cc () with respect to the H 1 () norm. Then we define H0 1() as the closure of C c () inside H 1 (). Remark 3.1. We recall alternative viewpoints of these Soolev spaces. Good references for the C oundary case (and no corners) include [4, App. B.2] and [23, 4.4]; only minor modifications are needed to deal with the corners for the special cases discussed elow. We can define H 1 ( ) as the suspace of L 2 () consisting of functions u such that du, defined as the distriutional derivative of u in, lies in L 2 (, Λ 1 ); we then equip it with the aove norm. This is locally equivalent to saying that V u L 2 loc () for all C vector fields V on, where V u refers to the distriutional derivative of u on. In fact, H 1 ( ) = H 1 (), since H 1 ( ) is complete with respect to the H 1 norm and Cc () is easily seen to e dense in it. For instance, locally, if is given y x j 0, j = 1,..., k, and u is supported in such a coordinate chart, one can take u s (x, ȳ) = u(x 1 + s,..., x k + s, ȳ) for s > 0, and see that u s u in Hc 1 ( ). Then a standard regularization argument on R n, n = dim, gives the claimed density of Cc () in Hc 1 ( ). Thus, H 1 ( ) = H 1 () indeed, which shows in particular that H 1 () L 2 (). (Note that u L2 () u H1 () only guarantees that there is a continuous inclusion H 1 () L 2 (), not that it is injective, although that can e proved easily y a direct argument; cf. the Friedrichs extension method for operators; see e.g. [21, Th..23].) If is a manifold without oundary, and is emedded into it, one can also extend elements of H 1 () to elements Hloc 1 ( ) exactly as in the C oundary case (or simply locally extending in x 1 first, then in x 2, etc., and using the C oundary result); see [23, 4.4]. Thus, with the notation of [4, App. B.2], Hloc 1 () = H loc 1 ( ). As is clear from the completion definition,

19 PROPAGATION OF SINGULARITIES 767 H 1 0,loc () can e identified with the suset of H1 loc ( ) consisting of functions supported in. Thus, H0,loc 1 () = Ḣ1 loc () with the notation of [4, App. B.2]. All of the discussion aove can e easily modified for H m in place of H 1, m 0 an integer. We are now ready to state the action on Soolev spaces. These results would e valid, with similar proofs, if we replaced H 1 () y H m (), m 0 an integer. We also refer to [4, Th ] for further extensions when has a C oundary (and no corners). Lemma 3.2. Any A Ψ 0 c () with compact support defines continuous linear maps A : H 1 () H 1 (), A : H0 1() H1 0 (), with norms ounded y a seminorm of A in Ψ 0 c (). Moreover, for any K compact, any A Ψ 0 c () with proper support defines a continuous map from the suspace of H 1 () (resp. H0 1 ()) consisting of distriutions supported in K to Hc 1 () (resp. H0,c 1 ()). Remark 3.3. Note that all smooth vector fields V of compact support define a continuous operator H 1 () L 2 (), so that, in particular, V V () do so. Now, any A Ψ 1 c () can e written as (D xj x j )A j + Dȳj A j + A with A j, A j, A Ψ 0 c () y writing σ,1(a) = σ j a j + ζj a j, and taking A j, A j with principal symol a j, a j. Therefore the lemma implies that any A Ψ 1 c () defines a continuous linear operator H1 () L 2 (), and in particular restricts to a map H0 1() L2 (). Proof. For A Ψ 0 c (), y (2.3) D x j Au = ÃD x j u + Bu, with Ã Ψ 0 c (), B Ψ0 c (), the seminorms of oth in Ψ0 c () ounded y seminorms of A in Ψ 0 c (). Thus, for u C c () D xj Au L2 () à B(L 2 (),L 2 ()) D xj u L2 () + B B(L2 (),L 2 ()) u L2 (). Since there is an analogous formula for D xj replaced y Dȳj, we deduce that for some C > 0, depending only on a seminorm of A in Ψ 0 c (), d Au L2 () C( d u L2 () + u L2 ()). Thus, A Ψ 0 c () extends to a continuous linear map from the completion of Cc () with respect to the H 1 () norm to itself, i.e. from H 1 () to itself as claimed. As it maps C c () C c (), it also maps the H 1 -closure of C () to itself, i.e. it defines a continuous linear map H0 1() H1 0 (), which finishes the proof of the first half of the lemma. For the second half, we only need to note that Au = Aφu if φ 1 near K and has compact support; now Aφ has compact support so that the first half of the lemma is applicale.

20 768 ANDRÁS VASY Note that H 1 () L 2 () C (), with C () denoting the dual space of C c (), i.e. the space of extendile distriutions. (Here we use d g = dg dt to trivialize Ω.) Since for any m, A Ψ m c () maps C () C (), we could view A already defined as a map H 1 () C (); then the aove lemma is a continuity result for m = 0. We let H 1 () e the dual of H0 1() and Ḣ 1 () e the dual of H 1 (), with respect to an extension of the sesquilinear form u, v = u v d g, i.e. the L 2 inner product. As H0 1() is a closed suspace of H1 (), H 1 () is the quotient of Ḣ 1 () y the annihilator of H0 1 (). In terms of the identification of the H 1 spaces in the penultimate paragraph of Remark 3.1, H 1 loc () = H 1 loc ( ) in the notation of [4, App. B.2], i.e. its elements are the restrictions to of elements of H 1 loc ( ). Analogously, Ḣ 1 loc () consists of those elements of H 1 loc ( ) which are supported in. Any V Diff 1 () of compact support defines a continuous map L 2 () H 1 () via V u, v = u, V v for u L 2 (), v H0 1 (); this is the same map as that induced y extending V to an element Ṽ of Diff1 ( ), extending u to, say as 0, and letting V u = Ṽ ũ. Thus, any P Diff2 () of compact support defines continuous maps H 1 () H 1 (), and in particular H0 1() H 1 (), since we can write P = V j W j with V j, W j Diff 1 (). Similarly, any P Diff 2 () defines continuous maps Hloc 1 () H 1 loc (), and in particular H0,loc 1 () H 1 loc (). Thus, for P = g + 1, u, v H1 () = u, P v if u H0 1() and v H1 (). Similarly, for P = Dt 2 g, D t u, D t v d M u, d M v = u, P v, if u H0 1() and v H1 (). We also note that as H 1 () and H0 1 () are Hilert spaces, their duals are naturally identified with themselves via the inner product. Thus, if f is a continuous linear functional on H0 1(), then there is a v H1 0 () such that f(u) = u, v + du, dv. Thus, regarding H0 1() as a suspace of H1 ( ), for an extension of, as in Remark 3.1, we deduce that f(u) = u, ( g + 1)v, and so the identification of H 1 () with H0 1 () (regarded as its own dual) is given y H0 1() v ( g + 1)v H 1 (). Since Ψ 0 c () is closed under taking adjoints, the following result is an immediate consequence of Lemma 3.2. Corollary 3.4. Any A Ψ 0 c () with compact support defines continuous linear maps A : H 1 () H 1 (), A : Ḣ 1 () Ḣ 1 (), with norm ounded y a seminorm of A in Ψ 0 c (). We now define suspaces of H 1 () which possess additional regularity with respect to Ψ (). Definition 3.5. For m 0, we define H 1,m,c () as the suspace of H1 () consisting of u H 1 () with supp u compact and Au H 1 () for some