NON-TRAPPING ESTIMATES NEAR NORMALLY HYPERBOLIC TRAPPING

Transcription

1 NON-TRAPPING ESTIMATES NEAR NORMALLY HYPERBOLIC TRAPPING PETER HINTZ AND ANDRAS VASY Astract. In this paper we prove semiclassical resolvent estimates for operators with normally hyperolic trapping which are lossless relative to nontrapping estimates ut take place in weaker function spaces. In particular, we otain non-trapping estimates in standard L 2 spaces for the resolvent sandwiched etween operators which localize away from the trapped set Γ in a rather weak sense, namely whose principal symols vanish on Γ. 1. Introduction The purpose of this paper is to otain semiclassical estimates for pseudodiffererential operators P h (z) with normally hyperolic trapping for z real which are lossless relative to non-trapping estimates, ut take place in weaker function spaces which are defined in a manner related to the Hamiltonian dynamics. Thus, the main result is an estimate of the form u Hh,Γ Ch 1 P h (z)u H h,γ, with certain function spaces H h,γ and Hh,Γ, descried elow; away from the trapped set these are just standard L 2 spaces. As the main application of such estimates is in so-called -spaces, e.g. Kerr-de Sitter spaces, for which the estimates follow from the semiclassical ones immediately in the presence of dilation invariance, we also prove their counterpart in the general, non-dilation-invariant, -setting. Extensions of special cases of these estimates play an important role in the recent gloal analysis of nonlinear wave equations on asymptotically Kerr-de Sitter spaces y the authors [7]. So at first we consider a family P h (z) of semiclassical pseudodifferential operators P h (z) Ψ h (X) on a closed manifold X, depending smoothly on the parameter z C, with normally hyperolic trapping at the trapped set Γ, and assume that P h (z) is formally self-adjoint near Γ for z R; moreover, we add complex asorption W in such a way that all forward and ackward icharacteristics outside Γ either enter the elliptic set of W in finite time or tend to Γ, and in at least one of the two directions they tend to the elliptic set of W. The icharacteristics tending to Γ in the forward/ackward directions are forward/ackward trapped; denote y Γ, resp. Γ + the forward, resp. ackward trapped set, 1 and assume that these are Date: May 12, The authors were supported in part y A.V. s National Science Foundation grants DMS and DMS and P. H. was supported in part y a Gerhard Casper Stanford Graduate Fellowship and the German National Academic Foundation. They are grateful to the referee for comments that significantly improved the exposition in the manuscript. 1 In the notation of Wunsch and Zworski [13], which we recall elow in Section 2.1, Γ± are the ackward/forward trapped sets for all (not necessarily null) icharacteristics near the, say, zero 1

2 2 PETER HINTZ AND ANDRAS VASY smooth codimension one sumanifolds of T X which intersect transversally in Γ, which we moreover assume to e symplectic. In this normally hyperolic setting, under additional hypotheses, Wunsch and Zworski [13] have shown polynomial semiclassical resolvent estimates u Ch N P h (z)u, 0 < h < h 0, (1.1) in small strips Im z ch, c > 0 sufficiently small, N > 1, and indeed for z real, the loss (as compared to non-trapping estimates, which hold in many cases where there is no trapping, and which lose a power of h 1 ) is merely logarithmic, i.e. one has u Ch 1 (log h 1 ) P h (z)u, 0 < h < h 0, (1.2) where is the L 2 -norm; Bony, Burq and Ramond [1] showed that (1.2) is indeed sharp. Dyatlov [6] improved these estimates in Im z < 0 y making c and N explicit; in a more general setting, Nonnenmacher and Zworski [10] otained the optimal value for c. We are concerned with improved estimates (for z almost real) if one localizes u and P h (z)u away from the trapping Γ in a rather weak sense, such as y applying pseudodifferential operators with symols vanishing at Γ. To place this into context, recall that Datchev and Vasy [3, 4] have shown that under our assumptions, with Im z = O(h ), if A, B Ψ h (X) with WF h(a) Γ = WF h(b) Γ =, B elliptic on WF h(a), then for all M there is N such that Au Ch 1 BP h (z)u + C h M u + C h N (Id B)P h (z)u. (1.3) Thus, if P h (z)u is O(h N 1 ) at Γ (corresponding to the Id B term in the estimate), then on the elliptic set of A, hence off Γ y appropriate choice of A, u satisfies nontrapping semiclassical estimates: AP h (z) 1 Av Ch 1 v, with A as aove (take B as aove with WF h(id B) WF h(a) = ). Here the O(h ) ound on Im z arises from the a priori estimate, (1.1), and if 1 < N < 3/2, e.g. as is on, and sufficiently near, the real axis, 2 then one can take Im z = O(h 1+2N ). The purpose of this paper is to improve this result y relaxing the conditions on WF h(a) and WF h(b) in (1.3). The main point of the theorem elow is thus that its estimate degenerates only at, as opposed to near, Γ. The proof given here is closely related to the proof of Wunsch and Zworski [13, 4] ut can take place in a significantly simpler, standard semiclassical pseudodifferential algera, at the cost of eing suoptimal in terms of the L 2 -estimate, even though it is optimal (i.e. non-trapping) when a pseudodifferential operator with vanishing principal symol at Γ is applied from oth sides. To set this up, let Q ± Ψ h (X) e self-adjoint and have symols which are defining functions of Γ ± near Γ, say on a neighorhood O of Γ. Let Q 0 Ψ 0 h (X) e a semiclassical operator with WF h(q 0 ) Γ = which is elliptic on O c (and thus on a neighorhood of O c ), with real principal symol for convenience. One considers normally isotropic spaces at Γ, denoted H h,γ, with squared norms given y u 2 H h,γ = Q 0 u 2 + Q + u 2 + Q u 2 + h u 2 ; level set of the semiclassical principal symol p h,z, and Γ λ ± are the corresponding sets within the λ-level set of p h,z. 2 In the latter case y the Phragmén-Lindelöf theorem.

3 NON-TRAPPING ESTIMATES 3 this is just the standard L 2 -space microlocally away from Γ as one of Q +, Q or Q 0 is elliptic there, and it does not depend on the choice of Q 0 as on O \ Γ one of Q + and Q is elliptic at every point. The dual space relative to L 2 is then 3 H h,γ = h 1/2 L 2 + Q + L 2 + Q L 2 + Q 0 L 2 (which is L 2 as a space, ut with this norm); P h (z)u will then e measured in H h,γ. Theorem 1.1. Let P = P h (z), Q ± e as aove, Im z = O(h 2 ). Then and thus y (1.2), Q + u + Q u Ch 1 P u H h,γ + C h 1/2 u, (1.4) u Hh,Γ Ch 1 P u H h,γ. (1.5) In fact, we also otain a direct proof of (1.5) without using (1.2) at the end of Section 2. Note that this theorem in particular implies the main result of [3] in this setting, in that the estimates are of the same kind, except that in [3] P u is assumed to e microlocalized away from Γ, and u is estimated microlocally away from Γ. The aforementioned -estimates will e proved in Section 3, see Theorem Semiclassical resolvent estimates on the real line 2.1. Notation and definitions. We will review some definitions of semiclassical analysis, partially in order to fix our notation. For a general reference, see Zworski [14]. Let X e a compact n-dimensional manifold without oundary, and fix a smooth density on X. For u L 2 (X), denote y u its L 2 (X) norm; moreover, denote y, the (sesquilinear) inner product on L 2 (X). A family of functions u = (u h ) h (0,1) on X is polynomially ounded if u Ch N for some N. If k R, we say that u O(h k ) if u C k h k, and u O(h ) if u C N h N for every N. For a = (a h ) h (0,1) C (T X), we say a h k S m (T X) if a satisfies α z β ζ a h(z, ζ) C αβ h k ζ m β for all multiindices α, β and all N N in any coordinate chart, where the z are coordinates in the ase and ζ coordinates in the fier. We define the semiclassical quantization Op h (a) of a y Op h (a)u(z) = (2πh) n e izζ/h a(z, ζ)û(ζ/h) dζ for u Cc (X) supported in a chart and for general u Cc (X) y using a partition of unity. We write Op h (a) h k Ψ m h (X). The quantization depends on the choice of partition of unity, ut the resulting class of operators does not, modulo operators that have Schwartz kernel in h C (X 2 ). We say that a is a symol of Op h (a). The equivalence class of a in h k S m (T X)/h k 1 S m 1 (T X) is invariantly defined and is called the principal symol of Op h (a). All operators elow except Q 0 Ψ 0 h (X) will in fact have compact microsupport in 3 One really has Q ± and Q 0 in this formula, ut the reality of the principal symols assures that one may replace them y Q ± and Q 0 modulo hl 2. See [9, Appendix A] for a general discussion of the underlying functional analysis; also see Footnote 11.

4 4 PETER HINTZ AND ANDRAS VASY the sense that they are quantizations of symols a h k S m (T X) satisfying in addition for all N α z β ζ a h(z, ζ) C N h N ζ N for all multiindices α, β for ζ outside of a compact suset of T X. We denote the class of such symols y h k S(T X) and the corresponding class of operators y h k Ψ h (X). If A, B Ψ h (X), then [A, B] hψ h (X), and its principal symol is h i H a, where we define the Hamilton vector field in a coordinate chart y H a = ( ζ a) z ( z a) ζ. By a icharacteristic of A we mean an integral curve of the Hamilton vector field of the principal symol of A. We denote the integral curve passing through the point ρ T X y γ ρ, i.e. γ ρ (0) = ρ and γ ρ(s) = H a (γ ρ (s)). We shall also write φ s (ρ) := γ ρ (s) for the icharacteristic flow. For a polynomially ounded family (u h ) h (0,1) and k R { }, we say that u = O(h k ) at a point ρ T X if there exists a S(T X) with a(ρ) 0 such that Op h (a)u = O(h k ). We define the semiclassical wave front set WF h (u) of u as the complement of the set of all ρ T X at which u = O(h ). The microsupport of A = Op h (a) h k Ψ h (X), denoted WF h(a), is the complement of the set of all ρ T X so that α a = O(h ) near ρ for every multiindex α, in any (and therefore in every) coordinate chart. For A h k Ψ h (X) with principal symol a h k S(T X), we say that A is elliptic at ρ T X if there is a constant C > 0 such that a(ρ ) Ch k for ρ near ρ and h sufficiently small. For a suset E T X, we say that A is elliptic on E if A is elliptic at each point of E. If A h k Ψ h (X) is elliptic on E T X and Au = f with u, f polynomially ounded and f is O(1) on E, then microlocal elliptic regularity states that u is O(h k ) on E. The semiclassical characteristic set of the semiclassical operator A Ψ h (X) with principal symol a is defined y Σ h = {ρ T X : a(ρ) = 0}. If A Ψ h (X) has a principal symol with non-positive imaginary part, u, f are polynomially ounded, Au = f, and u = O(h k ) at ρ, f = O(h k+1 ) on γ ρ ([0, T ]) for some T > 0, then the propagation of singularities states that u = O(h k ) at γ ρ (T ). Let P Ψ h (X) e a semiclassical operator. Let U X denote an open suset so that the cotangent undle over U contains what will e the trapped set, and place complex asoring potentials in a neighorhood of U c. 4 We recall the notion of normal hyperolicity from [13]: Define the ackward, resp. forward, trapped set Γ +, resp. Γ, y Γ ± = {ρ T X : γ ρ (s) / TU cx for all s 0}. Let Γ λ ± = Γ ± p 1 (λ) e the ackward/forward trapped set within the energy surface p 1 (λ), and define the trapped set Γ λ := Γ λ + Γ λ. We say that P is normally hyperolically trapping if: (1) There exists δ > 0 such that dp 0 on p 1 (λ) for λ < δ; (2) Γ ± p 1 ( δ, δ) are smooth codimension one sumanifolds intersecting transversally at Γ p 1 ( δ, δ), and Γ p 1 ( δ, δ) is symplectic; 4 See [13, 11] for details; the point here is that the relevant part of our analysis takes place microlocally near the trapped set, and the complex asoring potentials allow us to cut off the icharacteristic flow in a neighorhood of the trapped set.

5 with c ± > 0 near Γ, and with 5 {φ +, φ } > 0 NON-TRAPPING ESTIMATES 5 (3) the flow is hyperolic in the normal directions to Γ λ within the energy surface: There exist suundles E ± λ of T Γ λ (Γ λ ±) such that T Γλ Γ λ ± = T Γ λ E ± λ, where dφs : E ± λ E± λ, and there exists θ > 0 such that for all λ < δ dφ s (v) Ce θ t v for all v E λ, ±t Details on the setup and proof of the main result. Let p = p h,z e the semiclassical principal symol of P = P h (z). Recall from the work of Wunsch and Zworski [13, Lemma 4.1], with a corrected argument in [12], that for defining functions φ ± of Γ ± (near Γ, namely in a neighorhood O of Γ) one can take φ ± with H p φ ± = c 2 ±φ ± near Γ. This is the only relevant feature of normal hyperolicity for this paper; thus these identities and estimates could e taken as its definition for our purposes. By shrinking O if necessary we may assume that this Poisson racket as well as c ± have positive lower ounds on O. Then notice that H p φ 2 + = 2c 2 +φ 2 +, H p φ 2 = 2c 2 φ 2. As indicated in the introduction, we consider normally isotropic spaces at Γ, denoted H h,γ, with squared norms given y u 2 H h,γ = Q 0 u 2 + Q + u 2 + Q u 2 + h u 2 ; we can take Q ± with principal symol φ ±, while Q 0 is elliptic on O c with real principal symol. This is just the standard L 2 -space microlocally away from Γ as one of Q +, Q and Q 0 is elliptic there, and it does not depend on the choice of Q 0 as on O \ Γ one of Q + and Q is elliptic at every point. Notice that in fact (Q + iq ) (Q + iq ) = Q +Q + + Q Q i[q +, Q ] and if B Ψ h (X) with WF h(b) O then h Bv 2 C Re i[q +, Q ]Bv, Bv + Ch N v 2, C > 0, in view of {φ +, φ } > 0 on O, so Q +Q + + Q Q = 1 2 (Q +Q + + Q Q + (Q + iq ) (Q + iq ) + i[q +, Q ]) shows that, for h > 0 small, the norm on H h,γ is equivalent to just the norm u 2 H h,γ,2 = Q 0 u 2 + Q + u 2 + Q u 2. As mentioned in the introduction, the dual space relative to L 2 is then H h,γ = h 1/2 L 2 + Q + L 2 + Q L 2 + Q 0 L 2. Then Ψ h (X) acts on H h,γ, and thus on H h,γ, for B Ψ h(x) preserves h 1/2 L 2 and gives Q + Bu BQ + u + [Q +, B]u C Q + u + h u L 2, 5 These defining functions exist gloally when Γ± is orientale; ut even if Γ ± is not such, the square is gloally defined. There is only a minor change required elow if φ ± are not well defined; see Footnote 6.

6 6 PETER HINTZ AND ANDRAS VASY with a similar result for Q and Q 0. We remark that the notation H h,γ is justified as the space depends only on Γ, not on the particular defining functions φ ± as any other defining functions would change Q ± y an elliptic factor modulo an element of hψ h (X), whose contriution to the squared norm can e asored into Ch 2 u 2 L 2, and thus dropped altogether (for h small) in view of the equivalence of the two norms discussed aove. We are now ready to prove Theorem 1.1. We remark that the microlocal version of the two estimates of the theorem is that given any neighorhood O of Γ with closure in O, there exist B 0 Ψ h (X) elliptic at Γ, B 1, B 2 Ψ h (X) with WF h(b 2 ) Γ + =, WF h(b j ) O for j = 0, 1, 2 such that B 0 Q + u + B 0 Q u h 1 B 1 P u H h,γ + B 2 u L 2 + C h 1/2 u L 2, (2.1) respectively B 0 u Hh,Γ h 1 B 1 P u H h,γ + B 2 u L 2 + C h u L 2; (2.2) see (2.9). The theorem is then proved y controlling the B 2 u term using the ackward non-trapped nature of Γ \ Γ. Proof of Theorem 1.1. We first prove (1.4), which proves (1.5) y (1.2). In fact, one can also give a direct proof of (1.5) without using (1.2); see the discussion following this proof. Let χ 0 (t) = e 1/t for t > 0, χ 0 (t) = 0 for t 0, χ Cc ([0, )) e identically 1 near 0 with χ 0, and indeed with χ χ = χ 2 1, χ 1 0, χ 1 Cc ([0, )), and let ψ Cc (R) e identically 1 near 0. Let a = χ 0 (φ 2 + φ 2 + κ)χ(φ 2 +)ψ(p), κ > 0 small. Notice that on supp a, if χ is supported in [0, R], φ 2 + R, φ 2 φ κ = R + κ, so a is localized near Γ if R and κ are taken sufficiently small. Then 1 4 H p(a 2 ) = (c 2 +φ c 2 φ 2 )(χ 0 χ 0)(φ 2 + φ 2 + κ)χ(φ 2 +) 2 ψ(p) 2 c 2 +φ 2 +(χ χ)(φ 2 +)χ 0 (φ 2 + φ 2 + κ) 2 ψ(p) 2. Now χ 0 0, so the two terms have opposite signs. Let 6 and then Here a ± = φ ± (χ 0 χ 0 )(φ2 + φ 2 + κ)χ(φ 2 +)ψ(p), e = c + φ + χ 1 (φ 2 +)χ 0 (φ 2 + φ 2 + κ)ψ(p); 1 4 H p(a 2 ) = c 2 +a 2 + c 2 a 2 + e 2. (2.3) supp e supp a, supp e Γ + =, 6 If φ± is not defined gloally, a ± are not defined as stated. (The term e 2 need not have a sign, so this issue does not arise for it; see the Weyl quantization argument elow.) However, a ± need not e real elow, so as long as one can choose ψ ± complex valued with ψ ± 2 = φ 2 ±, replacing the first factor of φ ± with ψ ± in the definition of a ± allows one to complete the argument in general.

7 NON-TRAPPING ESTIMATES 7 with the last statement following from φ 2 + taking values away from 0 on supp χ 1 ; see Figure 1. Figure 1. Supports of the commutant a and the error term e in the positive commutator argument of the non-trapping estimate near the trapped set Γ, Theorem 1.1. The support of a is indicated in light gray; on supp a\supp e, darker colors correspond to larger values of a. Also shown are the forward, resp. ackward, trapped set Γ, resp. Γ +, and the icharacteristic flow neary. The figure already suggests that H p (a 2 ) is non-positive away from supp e, and actually negative away from supp e Γ; see equation (2.3). One then takes A Ψ h (X) with principal symol a, and with WF h(a) supp a, A ± Ψ h (X) with principal symols of a ±, and with WF h(a ± ) supp a ±, C ± have symol c ± and with WF h(c ± ) supp c ± ; one similarly lets E Ψ h (X) have principal symol e, and wave front set in the support of e. This gives that i 4h [P, A A] = (C + A + ) (C + A + ) (C A ) (C A ) + E E + hf, (2.4) for some F Ψ h (X) with WF h(f ) supp a. Thus i 4h [P, A A]u, u = C + A + u 2 C A u 2 + E u 2 + h F u, u. Expanding the left hand side gives P A Au, u A AP u, u = Au, AP u AP u, Au + (P P )A Au, u. As we are assuming that P P is O(h 2 ) near Γ, we may also assume that this holds on supp a, thus the last term is O(h 2 ) u 2. Thus, C + A + u 2 + C A u 2 E u 2 + h 1 AP u, Au + C 1 h u 2. (2.5)

8 8 PETER HINTZ AND ANDRAS VASY Now, y the duality of H h,γ and H h,γ relative to the L2 inner product, AP u, Au AP u H h,γ Au Hh,Γ hɛ 2 Au 2 H h,γ + 1 2hɛ AP u 2 H h,γ. (2.6) Further, for ɛ > 0 small, ɛ Q + Au 2 can e estimated in terms of C + A + u 2 + O(h) u 2, as can e seen y comparing the principal symols, in particular using the ellipticity of C + on supp a. One can thus asor ɛ 2 Au 2 H h,γ into the left hand side of (2.5). This shows C + A + u 2 + C A u 2 C E u 2 + Ch 2 AP u 2 H h,γ + Ch u 2. Now, since the region supp e is disjoint from Γ +, it is ackward non-trapped, and thus the standard propagation of singularities with complex asorption (see e.g. [4, Lemma 5.1]) implies that E u is controlled y P u microlocalized off Γ +, hence y Q + P u, modulo higher order (in h) terms in P u. This proves the first part of Theorem 1.1 since A ± is an elliptic multiple of Q ± microlocally near Γ. Thus, if we have a ound u C h 1 s P u L 2, 0 < s < 1/2, and thus h u 2 C h 1 2s P u 2 L 2 C h 1 2s P u 2 H h,γ, this implies a non-trapping estimate: u Hh,Γ Ch 1 P u H h,γ. This completes the proof of Theorem 1.1. In fact, as mentioned earlier, a slight change of point of view proves Theorem 1.1 directly. To see this, we use the Weyl quantization 7 when choosing a, a ±, c ±, e ; since we are on a manifold, this requires identifying functions with half-densities via trivialization of the half-density undle y the Riemannian metric; this identification preserves self-adjointness. We also write P h,z as the Weyl quantization of p 0 + hp 1 with p 0, p 1 real modulo O(h 2 ). Then the principal symol calculation aove holds with p 0 in place of p, and with p 1 included it yields additional terms 1 4 H p(a 2 ) = (c 2 +φ c 2 φ 2 hφ + H p1 φ + + hφ H p1 φ ) Now, (2.4) ecomes (χ 0 χ 0)(φ 2 + φ 2 + κ)χ(φ 2 +) 2 ψ(p) 2 (c 2 +φ 2 + hφ + H p1 φ + )(χ χ)(φ 2 +)χ 0 (φ 2 + φ 2 + κ) 2 ψ(p) 2. i 4h [P, A A] = (C + A + ) (C + A + ) (C A ) (C A ) + h(a +G + + G +A + + A G + G A ) + E + h 2 F, with G ± eing the Weyl quantization of g ± = ± 1 2 (H p 1 φ ± ) (χ 0 χ 0 )(φ2 + φ 2 + κ)χ(φ 2 +)ψ(p), (2.7) and with F Ψ h (X) with WF h(f ) supp a. 7 The Weyl quantization is actually irrelevant. It is straightforward to see that if A Ψh (X) and if the principal symol of A is real then the real part of the suprincipal symol is defined independently of choices. This is all that is needed for the argument elow.

9 NON-TRAPPING ESTIMATES 9 Correspondingly, (2.5) ecomes C + A + u 2 + C A u 2 Eu, u + h 1 AP u, Au + 2h A + u G + u + 2h A u G u + C 1 h 2 u 2. (2.8) The terms with G ± on the right hand side can e estimated y ɛ A + u 2 + ɛ 1 h 2 G + u 2 + ɛ A u 2 + ɛ 1 h 2 G u 2, and for ɛ > 0 sufficiently small, the A ± u 2 terms can now e asored into the left hand side of (2.8). Proceeding as aove yields C + A + u 2 + C A u 2 C Eu, u + Ch 2 AP u 2 H h,γ + Ch2 u 2. (2.9) Together with the non-trapping for the E term this gives the gloal estimate u 2 H h,γ Ch 2 P u 2 H h,γ + Ch2 u 2, and now the last term on the right hand side can e asored into the left hand side for sufficiently small h, giving the estimate (1.5). Notice that this also directly gives a weaker version of the Wunsch-Zworski estimate (1.2), namely u L 2 Ch 2 P u L 2, in view of the continuity of the inclusions H h,γ h 1/2 L 2 and h 1/2 L 2 H h,γ. Remark 2.1. If one is interested in a fixed operator, rather than in a parameterdependent family of operators, one can naturally strengthen the estimates (1.4) (1.5) y adding h 1 ˆP 0 u to the left hand sides, where ˆP 0 is any elliptic multiple of P. A more natural way of phrasing such an improvement is to use coisotropic, normally isotropic spaces H h,γ and H h,γ, where the squared norm on H h,γ is defined y u 2 Hh,Γ = Q 0 u 2 + Q + u 2 + Q u 2 + h 1 ˆP 0 u 2 + h u 2, which strengthens the space and therefore weakens its dual. Using these spaces instead in (2.6), one otains an additional term from h Au 2 Hh,Γ, namely ˆP 0 Au 2, which is ounded y C(h 1 AP u 2 H + h u 2 ), and thus the remainder of the h,γ second proof goes through. 3. Non-trapping estimates in non-dilation invariant settings We now transfer Theorem 1.1 into the -setting; the discussion in the previous section is essentially the dilation invariant special case of this, as we will explain elow, though in the -setting there is additional localization near the oundary. One main application of the -estimate is in the analysis of linear and non-linear waves on asymptotically Kerr-de Sitter spaces; see [11, 7] for details Notation and definitions. For a general reference for -analysis, see Melrose [8]. Let M e an n-dimensional compact manifold with oundary X.

10 10 PETER HINTZ AND ANDRAS VASY Let V (M) e the Lie algera of -vector fields on M, i.e. of vector fields on M which are tangent to X. Elements of V (M) are sections of a natural vector undle on M, namely the -tangent undle T M; in local coordinates (τ, x) near X, the fiers of T M are spanned y τ τ and x. The fiers of the dual undle T M, called -cotangent undle, are spanned y dτ τ and dx. It is often convenient to consider the fier compactification T M of T M, where the fiers are replaced y their radial compactification. The new oundary of T M at fier infinity is the -cosphere undle S M; it still possesses the compactification of the old oundary T XM, see Figure 2. S M is naturally the quotient of T M \o y the R + -action of dilation in the fiers of the cotangent undle. Many sets that we will consider elow are conic susets of T M \ o, and we will often view them as susets of S M. Figure 2. The radially compactified cotangent undle T M near T XM; the cosphere undle S M, which is the oundary at fier infinity of T M, is also shown, as well as the zero section o M T M and the zero section over the oundary o X T XM. For a C ( T M), we say a S m ( T M) if a satisfies α z β ζ a(z, ζ) C αβ ζ m β for all multiindices α, β in any coordinate chart, where z are coordinates in the ase and ζ coordinates in the fier; more precisely, in local coordinates (τ, x) near X, we take ζ = (σ, ξ), where we write -covectors as σ dτ τ + j ξ j dx j. We define the quantization Op(a) of a, acting on smooth functions u supported in a coordinate chart, y ( ) Op(a)u(τ, x) = (2π) n e i(τ τ ) σ+i(x x τ τ )ξ φ τ a(τ, x, τ σ, ξ)u(τ, x ) dτ dx d σ dξ, where the τ -integral is over [0, ), and φ Cc (( 1/2, 1/2)) is identically 1 near 0. 8 For general u, define Op(a)u using a partition of unity. We write Op(a) Ψ m (M). We say that a is a symol of Op(a). The equivalence class of a in S m ( T M)/S m 1 ( T M) is invariantly defined on T M and is called 8 The cutoff φ ensures that these operators lie in the small -calculus of Melrose, in particular that such quantizations act on weighted -Soolev spaces, defined elow.

11 NON-TRAPPING ESTIMATES 11 the principal symol of Op(a). We will tacitly assume that all our operators have homogeneous principal symols. If A Ψ m1 (M) and B Ψm2 (M), then [A, B] Ψm1+m2 1 (M), and its principal symol is 1 i H a 1 i {a, }, where the Hamilton vector field H a of the principal symol a of A is the extension of the Hamilton vector field from T M \ o to T M \ o, which is a homogeneous degree m 1 vector field on T M \ o tangent to the oundary TX M. In local coordinates (τ, x, σ, ξ) on T M as aove, this has the form H a = ( σ a)(τ τ ) (τ τ a) σ + j ( ( ξj a) xj ( xj a) ξj ). (3.1) We define icharacteristics completely analogously to the semiclassical setting. The microsupport WF (A) T M \ o of A = Op(a) Ψ m (M) is the complement of the set of all ρ T M \ o such that a is rapidly decaying in a conic neighorhood around ρ. Note that WF (A) is conic, hence we will also view it as a suset of S M. Fix a -density on M, which is locally of the form a dτ τ dz, a > 0. Define the -Soolev space H k(m) for k Z 0 y H k (M) = {u L 2 (M): X 1 X k u L 2 (M), X 1,..., X j V (M)}, and for general k R y duality and interpolation. Moreover, define the weighted -Soolev spaces H s,α (M) := τ α H s (M) for s, α R, where τ is a oundary defining function, i.e. τ = 0 at X and dτ 0 there. Every - pseudodifferential operator A Ψ m (M) is a ounded operator A: Hs,α (M) H s m,α (M), s, α R. For A Ψ m (M) with principal symol a Sm ( T M), we say that A is elliptic at ρ T M \o if there is a constant C > 0 such that a(z, ζ) C ζ m for (z, ζ) in a conic neighorhood of ρ. The characteristic set of A is the complement (in T M \ o) of the set of all ρ at which A is elliptic. For u H,α (M), define its H s,α wave front set WF s,α (u) T M\o as the complement of the set of all ρ T M \o for which there exists a S 0 ( T M) elliptic at ρ such that Op(a)u H s,α (M). In particular, WFs,α (u) = if and only if u H s,α (M). Microlocal elliptic regularity states that if Au = f with A Ψ m (M), u, f H,α (M), ρ / WF s m,α (f) and A is elliptic at ρ, then ρ / WF s,α (u). If A Ψ m (M) has a principal symol with non-positive imaginary part, u, f H,α (M), Au = f, moreover ρ / WF s,α (u) and γ ρ([0, T ]) WF s m+1,α (f) = for some T > 0, then the propagation of singularities states that γ ρ (T ) / (u). WF s,α 3.2. Setup, statement and proof of the result. Suppose P Ψ m (M), P P Ψ m 2 (M). Let p e the principal symol of P, which is thus a homogeneous degree m function on T M \ o, which we assume to e real-valued. Let ρ denote a homogeneous degree 1 defining function of S M. Then the rescaled Hamilton vector field V = ρ m 1 H p

12 12 PETER HINTZ AND ANDRAS VASY is a C vector field on T M away from the 0-section, and it is tangent to all oundary faces. The characteristic set Σ is the zero-set of the smooth function ρ m p in S M. We will, somewhat imprecisely, refer to the flow of V in Σ S M as the Hamilton, or (null)icharacteristic flow; its integral curves, the (null)icharacteristics, are reparameterizations of those of the Hamilton vector field H p, projected y the quotient map T M \ o S M. We first work microlocally near the trapped set, namely assume that (1) Γ Σ SX M is a smooth sumanifold disjoint from the image of T X \ o (so τd τ is elliptic near Γ), (2) Γ + is a smooth sumanifold of Σ SX M in a neighorhood U 1 of Γ, (3) Γ is a smooth sumanifold of Σ transversal to Σ SX M in U 1, (4) Γ + has codimension 2 in Σ, Γ has codimension 1, (5) Γ + and Γ intersect transversally in Σ with Γ + Γ = Γ, (6) the rescaled Hamilton vector field V = ρ m 1 H p is tangent to oth Γ + and Γ, and thus to Γ. Figure 3. An exemplary situation with trapping: Shown are the (projection from S M to the ase M of the) trapped set Γ, the - cosphere undle over X as well as a forward icharacteristic starting at a point ρ Γ. We assume that Γ + is ackward trapped for the Hamilton flow (i.e. icharacteristics in Γ + near Γ tend to Γ as the parameter goes to ), i.e. is the unstale manifold of Γ, while Γ is forward trapped, i.e. is the stale manifold of Γ, see Figure 3; indeed, we assume a quantitative version of this. (There is a completely analogous statement if Γ + is forward trapped and Γ is ackward trapped: replacing P y P preserves all assumptions, ut reverses the Hamilton flow.) To state this, let φ e a defining function of Γ, and let φ + C ( S M) e a defining function of Γ + in SX M; thus Γ + is defined within S M y τ = 0, φ + = 0. Notice that V eing to tangent to SX M (due to (3.1)) implies that V τ is a multiple of τ; we assume that, near Γ, V τ = c 2 τ, c > 0; (3.2) this is consistent with the staility of Γ. By the tangency requirement, with ˆp 0 = ρ m p,

13 NON-TRAPPING ESTIMATES 13 V φ = α φ + ν ˆp 0, α smooth; notice that changing φ y a smooth nonzero multiple f gives V (fφ ) = α fφ + ν f ˆp 0 + (V f)φ, so α depends on the choice of φ. On the other hand, the tangency requirement gives V φ + = α + φ + +β + τ +ν + ˆp 0. For the sake of conciseness, rather than stating the assumptions on the Hamilton flow as in [13], we assume directly that φ ± satisfy V φ = c 2 φ + ν ˆp 0, V φ + = c 2 +φ + + β + τ + ν + ˆp 0, (3.3) with c ± > 0 smooth near Γ, β +, ν ± smooth near Γ and {φ +, φ } > 0 (3.4) near Γ. Let U 0 U 0 U 1 e a neighorhood of Γ such that the Poisson racket in (3.4) as well as c ± have positive lower ounds. Now, given a oundary defining function τ, SX M \ S X (where S X is the image of T X under the R + -dilation quotient) can e identified with ( dτ τ + T X) + T X) (which in turn can e identified with two copies of T X) since ( dτ τ TX M = Span{ } dτ τ T X, and each R + -orit outside T X intersects ( dτ τ +T X) ( dτ τ + T X) in a unique point. This provides the connection etween the - and the semiclassical perspectives, i.e. analysis on TX M and that of T X. In fact, if p is a homogeneous degree m function, then ρ m p, where ρ can e taken as the reciprocal of the asolute value of the symol of τd τ in this region (which is welldefined, independent of choices), gives a function on {±1} T X; this is exactly the semiclassical rescaling, with ρ m p the semiclassical principal symol (depending on the parameter ±1) of the rescaled operator family (cf. [11, 2.1]). Notice that if we merely assume the normal hyperolicity within SX M, in the sense of this identification with T X, as in [13, 1.2], then [13, Lemma 4.1], as corrected in [12], actually gives such defining functions φ 0 ± within SX M (i.e. letting τ = 0); taking an aritrary extension in case of φ +, and an extension which is a defining function in case of Γ, all the requirements aove are satisfied. In particular, Kerr and Kerr-de Sitter spaces satisfy these assumptions, as do their perturations when the angular momentum a is small; see [13, Proposition 2.1] for the Kerr setting and [11, 6] for the Kerr-de Sitter one. Indeed, in the Kerr case the full range of a < M, M the lack hole mass, satisfies the hypotheses, as shown y Dyatlov [5]. There is an asymmetry etween the roles of φ ± and τ, and thus we consider the paraolic defining function ρ + = φ Mτ for Γ +, M > 0, to e chosen. Then, near Γ, ˆρ + = V ρ + = 2c 2 +φ β + φ + τ + 2ν + φ + ˆp 0 Mc 2 τ = 2c 2 +φ 2 + (Mc 2 2β + φ + )τ + 2ν + φ + ˆp 0 c 2 +ρ + + 2ν + φ + ˆp 0, c + > 0, (3.5) if M > 0 is chosen sufficiently large, consistently with the forward trapped nature of Γ. (Here the term with ˆp 0 is considered harmless as one essentially restricts to the characteristic set, ˆp 0 = 0.) Also, note that one can use 9 the reciprocal 9 Indeed, in the semiclassical setting, after Mellin transforming this prolem, σ 1 plays the role of the semiclassical parameter h, which in that case commutes with the operator.

14 14 PETER HINTZ AND ANDRAS VASY ρ = σ 1 of the principal symol σ of τd τ as the local defining function of S M as fier-infinity in T M near Γ; then V ρ = α ρτ (3.6) for some α smooth in view of (3.1). Similar to the normally isotropic spaces in the semiclassical setting, we introduce spaces which are normally isotropic at Γ. 10 Concretely, let Q ± Ψ 0 (M) have principal symol φ ± as efore, ˆP0 Ψ 0 (M) have principal symol ˆp 0 and let Q 0 Ψ 0 (M) e elliptic, with real principal symol for convenience, on U 0 c (and thus neary). Define the (gloal) -normally isotropic spaces at Γ of order s, H,Γ s, y the norm u 2 H = Q 0u 2 s,γ H + Q +u 2 s H + Q u 2 s H + τ 1/2 u 2 s H + ˆP s 0 u 2 H + s u 2, H s 1/2 (3.7) and let H, s,γ e the dual space relative to L2, which is thus 11 Q 0 H s + Q + H s + Q H s + τ 1/2 H s + ˆP 0 H s + H s+1/2. Note that microlocally away from Γ, H,Γ s is just the standard Hs space while H, s,γ is H s since at least one of Q 0, Q ± and τ is elliptic. Moreover, Ψ k (M) A : H,Γ s Hs k,γ is continuous since [Q +, A] Ψ k 1 (M) etc.; the analogous statement also holds for the dual spaces. Further, the last term in (3.7) can e replaced y u 2 as i[q H s 1 +, Q ] = B B + R, B Ψ 1/2 (M), R Ψ 2 (M), using the same argument as in the semiclassical setting (however, it cannot e dropped altogether unlike in the semiclassical setting!). Remark 3.1. The notation H,Γ s (M) is justified for the space is independent of the particular defining functions φ ± chosen; near Γ any other choice would replace φ ± y smooth non-degenerate linear cominations plus a multiple of τ and of ˆp 0, denote these y φ ±, and thus the corresponding Q ± can e expressed as B + Q + +B Q +B τ + ˆB ˆP 0 +B 0 Q 0 +R, B ±, B 0, B, ˆB Ψ 0 (M), R Ψ 1 (M), so the new norm can e controlled y the old norm, and conversely in view of the non-degeneracy. Our result is then: Theorem 3.2. With P, H,Γ s, H,s,Γ as aove, for any neighorhood U of Γ and for any N there exist B 0 Ψ 0 (M) elliptic at Γ and B 1, B 2 Ψ 0 (M) with WF (B j ) U, j = 0, 1, 2, WF (B 2 ) Γ + = and C > 0 such that B 0 u H s,γ B 1 Pu H,s m+1,γ + B 2 u H s + C u H N, (3.8) 10 Note that T M is not a symplectic manifold (in a natural way) since the symplectic form on TM M does not extend smoothly to T M. Thus, the word normally isotropic is not completely justified; we use it since it reflects that in the analogous semiclassical setting, see [13], the set Γ is symplectic, and the origin in the symplectic orthocomplement (T αγ) of T αγ, which is also symplectic, is isotropic within (T αγ). 11 We refer to [9, Appendix A] for a general discussion of the underlying functional analysis. In particular, Lemma A.3 there essentially gives the density of C (M) in H,Γ s (M): one can simply drop the suscript e in the statement of that lemma to conclude that H (M) (so in particular H s(m)) is dense in Hs,Γ (M), and then the density of C (M) in H s (M) for any s completes the argument. The completeness of H,Γ s (M) follows from the continuity of Ψ0 (M) on Hs 1/2 (M).

15 NON-TRAPPING ESTIMATES 15 i.e. if all the functions on the right hand side are in the indicated spaces: B 1 Pu H,s m+1,γ, etc., then B 0 u H,Γ s, and the inequality holds. The same conclusion also holds if we assume WF (B 2 ) Γ = instead of WF (B 2 ) Γ + =. Finally, if r < 0, then, with WF (B 2 ) Γ + =, (3.8) ecomes B 0 u H s,r B 1 Pu H s m+1,r while if r > 0, then, with WF (B 2 ) Γ =, B 0 u H s,r B 1 Pu H s m+1,r + B 2 u H s,r + B 2 u H s,r + C u H N,r, (3.9) + C u H N,r, (3.10) Remark 3.3. Note that the weighted versions (3.9)-(3.10) use standard weighted -Soolev spaces; this corresponds to non-trapping semiclassical estimates if the suprincipal symol has the correct, definite, sign at Γ. Proof. We may assume that U U 0 is disjoint from a neighorhood of WF (Q 0 ), and thus ignore Q 0 in the definition of H,Γ s elow. We first prove that there exist B 0, B 1, B 2 as aove and B 3 Ψ 0 (M) with WF (B 3 ) U such that B 0 u H s,γ B 1 Pu H,s m+1,γ + B 2 u H s + B 3 u H s 1 + C u H N. (3.11) An iterative argument will then prove the theorem. The proof is a straightforward modification of the construction in the semiclassical setting aove, replacing φ 2 + y φ Mτ, M > 0 large, in accordance with (3.5). We start y pointing out that for any B 0 Ψ 0 (M) and any B 3 Ψ 0 (M) elliptic on WF ( B 0 ), we have ˆP 0 B0 u H s C B 0 Pu H s m + C B 3 u H s 1, (3.12) y simply using that ˆP 0 is an elliptic multiple of P modulo Ψ 1 C B 0 Pu H,s m,γ B 0 Pu H s m thus automatically controlled. (M). Since, the ˆP 0 contriution to B 0 u H s,γ in (3.11) is So let χ 0 (t) = e Ϝ/t for t > 0, χ 0 (t) = 0 for t 0, with Ϝ > 0 (large) to e specified, χ Cc ([0, )) e identically 1 near 0 with χ 0, and indeed with χ χ = χ 2 1, χ 1 0, χ 1 Cc ([0, )), and let ψ Cc (R) e identically 1 near 0. As we use the Weyl quantization, 12 we write P as the Weyl quantization of p = p 0 + ρp 1, with ρp 1 of order m 1. Let a = ρ s+(m 1)/2 χ 0 (ρ + φ 2 + κ)χ(ρ + )ψ( ρ m p), (3.13) κ > 0 small. Notice that on supp a, if χ is supported in [0, R], ρ + R, φ 2 ρ + + κ = R + κ, so a is localized near Γ if R and κ are taken sufficiently small. In particular, the argument of χ 0 is ounded aove y R+κ, so given any M 0 > 0 one can take Ϝ > 0 large so that χ 0χ 0 M 0 χ 2 0 = 2 χ 0χ 0, 12 Again, the Weyl quantization is irrelevant: If A Ψ m (X) and the principal symol of A is real then the real part of the suprincipal symol is defined independently of choices, which suffices elow.

16 16 PETER HINTZ AND ANDRAS VASY with 1/2, C, on the range of the argument of χ 0. In fact, we also need to regularize, namely introduce a ɛ = (1 + ɛ ρ 1 ) 2 a, ɛ [0, 1], (3.14) which is a symol of order s (m 1)/2 2 for ɛ > 0, and is uniformly ounded in symols of order s (m 1)/2 as ɛ varies in [0, 1]. In order to avoid more cumersome notation elow, we ignore the regularizer and work directly with a; since the regularizer gives the same kind of contriutions to the commutator as the weight ρ s+(m 1)/2, these contriutions can e dominated in exactly the same way. Then, with p = p 0 + ρp 1 as aove, W = ρ m 2 H ρp1, which is a smooth vector field near S M as ρp 1 is order m 1, noting W ρ = α 1 τ ρ similarly to (3.6), and W τ = α,1 τ y the tangency of W to τ = 0, 1 4 H p(a 2 ) = ( ˆρ + /2 + c 2 φ 2 + ν φ ˆp 0 ρφ + (W φ + ) ρmα,1 τ + ρφ (W φ )) ρ 2s (χ 0 χ 0)(ρ + φ 2 + κ)χ(ρ + ) 2 ψ( ρ m p) ( 2s + m 1) ρ 2s ( α + ρ α 1 )τχ 0 (ρ + φ 2 + κ) 2 χ(ρ + ) 2 ψ( ρ m p) ρ 2s (ˆρ + + ρw ρ + )(χ χ)(ρ + )χ 0 (ρ + φ 2 + κ) 2 ψ( ρ m p) 2 + m 2 ( α + ρ α 1) ρ 2s ( ρ m p)τχ 0 (ρ + φ 2 + κ) 2 χ(ρ + ) 2 (ψψ )( ρ m p). (3.15) A key point is that the second term on the right hand side, given y the weight ρ 2s+m 1 eing differentiated, can e asored into the first y making Ϝ > 0 large so that ˆρ + χ 0(ρ + φ 2 + κ) dominates 2s + m 1 α τχ 0 (ρ + φ 2 + κ) on supp a, which can e arranged as 2s + m 1 α τ is ounded y a sufficiently large multiple of ˆρ + there. Thus, 1 4 H p(a 2 ) = c 2 +a 2 + c 2 a 2 a 2 +2g + a + +2g a +e+ẽ+2a + j + p+2a j p (3.16) with a ± = ρ s φ ± (χ 0 χ 0 )(ρ + φ 2 + κ)χ(ρ + )ψ( ρ m p), a = ρ s τ 1/2( (M(c 2 /2) β + φ + ρma,1 )(χ 0 χ 0)(ρ + φ 2 + κ) 1 4 ( 2s + m 1)( α + ρ α 1)χ 0 (ρ + φ 2 + κ) 2) 1/2 χ(ρ+ )ψ( ρ m p), g ± = ± 1 2 ρ s+1 ((W φ ± ) ν ± ρ m 1 p 1 ) (χ 0 χ 0 )(ρ + φ 2 + κ)χ(ρ + )ψ( ρ m p), e = 1 2 ρ 2s (ˆρ + + ˆρW ρ + )χ 1 (ρ + ) 2 χ 0 (ρ + φ 2 + κ) 2 ψ( ρ m p) 2, ẽ = m 2 ρ 2s ( ρ m p)( α + ρ α 1 )τχ 0 (ρ + φ 2 + κ) 2 χ(ρ + ) 2 (ψψ )( ρ m p), j ± = ± 1 2 ν ± ρ s+m (χ 0 χ 0 )(ρ + φ 2 + κ)χ(ρ + )ψ( ρ m p); the square root in a is that of a non-negative quantity and is C for M large (so that β + φ + can e asored into M(c 2 /2)) and Ϝ large (so that a small multiple of

17 NON-TRAPPING ESTIMATES 17 χ 0 can e used to dominate χ 0 ), as discussed earlier. Moreover, supp e supp a, supp e Γ + =, supp ẽ supp a, supp ẽ Σ =. This gives, with the various operators eing Weyl quantizations of the corresponding lower case symols, i 4 [P, A A] = (C + A + ) (C + A + ) (C A ) (C A ) A A + G +A + + A +G + + G A + A G + E + Ẽ + A +J + P + P J +A + + A J P + P J A + F (3.17) where now A Ψ s (m 1)/2 (M), A ±, A Ψ s (M), G ± Ψ s 1 (M), E Ψ 2s (M), Ẽ Ψ 2s (M), J ± Ψ s m (M), F Ψ 2s 2 (M) with WF (F ) supp a. After this point the calculations repeat the semiclassical argument: First using P P Ψ m 2 (M), C + A + u 2 + C A u 2 + A u 2 Eu, u + Ẽu, u + APu, Au + 2 A +u G + u + 2 A u G u + 2 J + Pu, A + u + 2 J Pu, A u + C 1 F 1 u 2 H s 1 + C 1 u 2, H N (3.18) where we took F 1 Ψ 0 (M) elliptic on WF (F ) and with WF ( F 1 ) near Γ. Noting that WF (Ẽ) Σ =, the elliptic estimates give Ẽu, u C B 1Pu 2 H s m + C u 2 H N if B 1 Ψ 0 (M) is elliptic on supp ẽ. Let Λ Ψ(m 1)/2 (M) e elliptic with real principal symol λ, and let Λ Ψ (m 1)/2 (M) e a parametrix for it so that ΛΛ Id = R 0 Ψ (M). Then APu, Au Λ APu, Λ Au + R 0 APu, Au 1 2ɛ Λ APu 2 + ɛ H,0,Γ 2 Λ Au 2 H + C u 2 0,Γ H N As Λ A Ψ s (M), for sufficiently small ɛ > 0, ɛ 2 Λ Au 2 can e asored H 0,Γ into 13 C + A + u 2 + C A u 2 + A u 2 plus B 0 ˆP0 u 2 H and as discussed aove,, s the latter already has the control required for (3.11). On the other hand, taking B 1 Ψ 0 (M) elliptic on WF (A), as Λ A Ψ s m+1 (M), Λ APu 2 C B H,0 1 Pu 2,Γ H,s m+1,γ + C u 2. H N 13 The point eing that A + C+ C +A + ɛa ΛQ + Q +Λ A has principal symol c 2 + a2 + ɛa2 φ 2 + λ2 which can e written as the square of a real symol for ɛ > 0 small in view of the main difference in vanishing factors in the two terms eing that χ 0 in a2 + is replaced y χ 0 in a, and thus the corresponding operator can e expressed as C C for suitale C, modulo an element of Ψ 2s 2 (M), with the latter contriuting to the H s 1 error term on the right hand side of (3.11).

18 18 PETER HINTZ AND ANDRAS VASY Similarly, to deal with the J ± terms on the right hand side of (3.18), one writes J ± Pu, A ± u 1 ( ) B 1 Pu 2 + C u 2 + ɛ 2ɛ H s m H N 2 A ±u 2 L 2 1 ( ) B 1 Pu 2 + C u 2 + ɛ 2ɛ H,s m,γ H N 2 A ±u 2 L 2, while the G ± terms can e estimated y ɛ A + u 2 + ɛ 1 G + u 2 + ɛ A u 2 + ɛ 1 G u 2, and for ɛ > 0 sufficiently small, the A ± u 2 terms in oth cases can e asored into the left hand side of (3.18) while the G ± into the error term. This gives, with F 2 having properties as F 1, C + A + u 2 + C A u 2 + A u 2 Eu, u + C B 1 Pu 2 H,s m+1,γ + C 2 F 2 u 2 H s 1 + C 2 u 2. H N By the remark efore the statement of the theorem, if B 0 Ψ 0 (M) is such that χ 0 (ρ + φ 2 + κ)χ(ρ + )ψ(p) > 0 on WF (B 0 ), B 0 u 2 can e added to the H s 1/2 left hand side at the cost of changing the constant in front of F 2 u 2 H s 1 + u 2 H N on the right hand side. Taking such B 0 Ψ 0 (M), and B 1 elliptic on WF (A) as efore, B 2 Ψ 0 (M) elliptic on WF (E) ut with WF (B 2 ) disjoint from Γ +, we conclude that B 0 u 2 H s,γ C B 1Pu 2 H,s m+1,γ + C B 2 u 2 H s + C F 2 u 2 H s 1 + C u 2, H N proving (3.11), up to redefining B j y multiplication y a positive constant. Recall that unless one makes sufficient a priori assumptions on the regularity of u, one actually needs to regularize, ut as mentioned after (3.14), the regularizer is handled in exactly the same manner as the weight. Now in general, with χ as efore, ut supported in [0, 1] instead of [0, R], let χ R = χ( /R) and write a = a R,κ to emphasize its dependence on these quantities. When R and κ are decreased, supp a R,κ also decreases in Σ in the strong sense that 0 < R < R and 0 < κ < κ imply that a R,κ is elliptic on supp a R,κ within Σ, and indeed gloally if the cutoff ψ is suitaly adjusted as well. Thus, if u H N, say, one uses first (3.11) with s = N + 1, and with B j given y the proof aove, so the B 3 u term is a priori ounded, to conclude that B 0 u H,Γ s and the estimate holds, so in particular, u is in H N+1/2 microlocally near Γ (concretely, on the elliptic set of B 0 ). Now one decreases κ and R y an aritrarily small amount and applies (3.11) with s = N + 3/2; the B 3 u term is now a priori ounded y the microlocal memership of u in H N+1/2, and one concludes that B 0 u H N+3/2,Γ, so in particular u is microlocally in H N+1. Proceeding inductively, one deduces the first statement of the theorem, (3.8). If one reverses the role of Γ + and Γ in the statement of the theorem, one simply reverses the roles of ρ + = φ Mτ and φ 2 in the definition of a in (3.13). This reverses the signs of all terms on the right hand side of (3.15) whose sign mattered elow, and thus the signs of the first three terms on the right hand side of (3.17), which then does not affect the rest of the argument. In order to prove (3.9), one simply adds a factor τ 2r to the definition of a in (3.13). This adds a factor τ 2r to every term on the right hand side of (3.17), as

19 NON-TRAPPING ESTIMATES 19 well as an additional term r 2 τ 2r ρ 2s c 2 χ 0 (ρ + φ 2 + κ) 2 χ(ρ + ) 2 ψ(p) 2, which for r < 0 has the same sign as the terms whose sign was used aove, and indeed can e written as the negative of a square. Thus (3.16) ecomes 1 4 H p(a 2 ) = c 2 +a 2 + c 2 a 2 a 2 a 2 r (3.19) + 2g + a + + 2g a + e + ẽ + 2j + a + p + 2j a p with r a r = 2 τ r ρ s c χ 0 (ρ + φ 2 + κ)χ(ρ + )ψ(p), and all other terms as aove apart from the additional factor of τ r in the definition of a ±, etc. Since a r is actually elliptic at Γ when r 0, this proves the desired estimate (and one does not need to use the improved properties given y the Weyl calculus!). When the role of Γ + and Γ is reversed, there is an overall sign change, and thus r > 0 gives the advantageous sign; the rest of the argument is unchanged. Remark 3.4. As in the semiclassical setting, see Remark 2.1, the estimate (3.8) can e strengthened y adding the term B 0 ˆP0 u H s+1 to the left hand side, which is controlled y elliptic regularity, likewise for (3.9) (3.10). A more natural way of phrasing such an improvement is to use coisotropic, normally isotropic spaces H,Γ s,s and H,Γ in the estimate (3.8), where the squared norm on H,Γ s is defined y u 2 Hs,Γ = Q 0 u 2 H s + Q +u 2 H s + Q u 2 H s + τ 1/2 u 2 H s + ˆP 0 u 2 H s+1/2 + u 2, H s 1/2 i.e. strengthening the norm of ˆP 0 u y a half, which strengthens the space and weakens its dual. To otain the necessary elliptic estimate (3.12) with the strengthened norms on the terms involving B 0, ut keeping the norm on B 3 u (which is required for the iterative argument at the end of the proof), one can choose B 0 with WF (I B 0 ) Γ = so that B 3 can e chosen to e microsupported away from Γ, and thus B 3 u H s 1/2 C B 3 u H s 1/2,Γ on B 1 Pu eing B 1 Pu H,s m+1/2,γ measured in H s 3/2 H s 1 is controlled using (3.8), with the norm C B 1 Pu H,s m+1,γ, as required. References, and the error term eing [1] Jean-François Bony, Nicolas Burq, and Thierry Ramond. Minoration de la résolvante dans le cas captif. Comptes Rendus Mathematique, 348(23 24): , [2] Kiril Datchev and András Vasy. Gluing semiclassical resolvent estimates via propagation of singularities. Int. Math. Res. Notices, 2012(23): , [3] Kiril Datchev and András Vasy. Propagation through trapped sets and semiclassical resolvent estimates. Annales de l Institut Fourier, 62(6): , [4] Kiril Datchev and András Vasy. Semiclassical resolvent estimates at trapped sets. Annales de l Institut Fourier, 62(6): , [5] Semyon Dyatlov. Asymptotics of linear waves and resonances with applications to lack holes. Preprint, arxiv: , [6] Semyon Dyatlov. Spectral gaps for normally hyperolic trapping. Preprint, arxiv: , [7] Peter Hintz and András Vasy. Gloal analysis of nonlinear wave equations on asymptotically Kerr-de Sitter spaces. Preprint, arxiv: , 2014.

20 20 PETER HINTZ AND ANDRAS VASY [8] Richard B. Melrose. The Atiyah-Patodi-Singer index theorem, volume 4 of Research Notes in Mathematics. A K Peters Ltd., Wellesley, MA, [9] Richard B. Melrose, András Vasy, and Jared Wunsch. Diffraction of singularities for the wave equation on manifolds with corners. Astérisque, 351, vi+136pp, [10] Stéphane Nonnenmacher and Maciej Zworski. Decay of correlations for normally hyperolic trapping. Preprint, arxiv: , [11] András Vasy. Microlocal analysis of asymptotically hyperolic and Kerr-de Sitter spaces. With an appendix y S. Dyatlov. Inventiones Math., 194: , [12] Jared Wunsch and Maciej Zworski. Erratum to Resolvent estimates for normally hyperolic trapped sets. Posted on http: // www. math. northwestern. edu/ ~ jwunsch/ erratum_ wz. pdf. [13] Jared Wunsch and Maciej Zworski. Resolvent estimates for normally hyperolic trapped sets. Ann. Henri Poincaré, 12(7): , [14] Maciej Zworski. Semiclassical Analysis. Graduate studies in mathematics. American Mathematical Society, Department of Mathematics, Stanford University, CA , USA address: phintz@math.stanford.edu address: andras@math.stanford.edu