GLOBAL ANALYSIS OF QUASILINEAR WAVE EQUATIONS ON ASYMPTOTICALLY KERR-DE SITTER SPACES

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1 GLOBAL ANALYSIS OF QUASILINEAR WAVE EQUATIONS ON ASYMPTOTICALLY KERR-DE SITTER SPACES PETER HINTZ AND ANDRAS VASY Astract. We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space; in particular the trapped geodesics form a normally hyperolic invariant manifold. We prove the gloal existence and decay, to constants for the actual wave equation, of solutions. The key new ingredient compared to earlier work y the authors in the semilinear case [3] and y the first author in the non-trapping quasilinear case [29] is the use of the Nash-Moser iteration in our framework.. Introduction We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space. An important feature is that, as in perturations of Kerr-de Sitter space, the trapped geodesics form a normally hyperolic invariant manifold. We prove the gloal existence and decay of solutions; this means decay to constants for the actual wave equation. This result is part of a new framework for solving quasilinear wave equations with normally hyperolic trapping, which extends the semilinear framework developed y the two authors [3] and the non-trapping quasilinear theory developed y the first author [29]. The main new tool introduced here is a Nash-Moser iteration necessitated y the loss of derivatives in the linear estimates at the normally hyperolic trapping. To our knowledge, this is the first gloal result for the forward prolem for a quasilinear wave equation on either a Kerr or a Kerr-de Sitter ackground. We remark, however, that Dafermos, Holzegel and Rodnianski [9] have constructed ackward solutions for Einstein s equations on the Kerr ackground; for ackward constructions the trapping does not cause difficulties. For concreteness, we state our results first in the special case of Kerr-de Sitter space, ut it is important to keep in mind that the setting is more general. By adding an ideal oundary at infinity in the standard description of Kerr-de Sitter space, the region of Kerr-de Sitter space we are interested in can e considered a (non-compact) 4-dimensional manifold with oundary M. The interior M is equipped with a Lorentzian metric g 0, recalled elow, depending on three parameters Λ > 0 (the cosmological constant), M > 0 (the lack hole mass) and a (the Date: April 4, 204, revised May 0, Mathematics Suject Classification. 35L72, 35L05, 35P25. Key words and phrases. Quasilinear waves, Kerr-de Sitter space, -pseudodifferential operators, Nash-Moser iteration, resonances, asymptotic expansion. 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.

2 2 PETER HINTZ AND ANDRAS VASY angular momentum), though we usually drop this in the notation. This Lorentzian metric has a specific structure at M, i.e. infinity, called a totally characteristic, or -, structure. Here recall that on any n-dimensional manifold with oundary M, the Lie algera of smooth vector fields tangent to the oundary is denoted y V (M); in local coordinates (x, y,..., y n ), with x a oundary defining function, these are linear cominations of x x and yj with C (M) coefficients. Just as a dual metric is a linear comination of symmetric tensor products of coordinate vector fields, a dual metric in this totally characteristic setting, also called a dual -metric, is a linear comination of x x x x, 2 (x x yj + yj x x ), 2 ( y i yj + yj yi ). One can think of this as a symmetric ilinear form; then a Lorentzian dual -metric is a non-degenerate ilinear form of signature (, n ). The corresponding wave operator is thus a totally characteristic, or -, operator, Diff 2 (M), i.e. is the sum of products of up to two factors of elements of V (M), with C (M) coefficients. The actual metric is then a linear comination of dx x dx x, 2( dx x dy j + dy j dx x ), 2 (dy i dy j + dy j dy i ). We denote linear cominations of these tensors over a point p y Sym 2 Tp M. In order to set up our prolem, see Figure for an illustration, we consider two functions t j, j =, 2, with forward, resp. ackward, time-like differentials near their respective 0-set H j, which are linearly independent at their joint 0-set, and let Ω = t ([0, )) t 2 ([0, )), with Ω compact, so Ω is a compact manifold with corners with three oundary hypersurfaces H, H 2 and X = M, all intersected with Ω. We are interested in solving the forward prolem for wave-like equations in Ω, i.e. imposing vanishing Cauchy data at H, which we assume is disjoint from X; initial value prolems with general Cauchy data can always e converted into an equation of this type. In order to compress notation for elements of V (M) applied to a function u, it is convenient to introduce the notation du = (x x u) dx x + j ( yj u)dy j in terms of local coordinates. This is a re-interpretation of the differential du of u in terms of the -forms dx x and dy j dual to the vector fields x x and yj, thus it is in fact invariantly defined. Note that when one writes e.g. a(u, du), one could instead, at least locally, write a(u, x x u, y u,..., yn u); the du notation is more concise and invariant. One calls linear cominations of dx x and dy j over a point p elements of T p M. We note that d preserves reality. The wave equations we consider include those of the form g(u, du)u = f + q(u, du), We will always assume that Λ, M and a are such that the non-degeneracy condition [47, (6.2)] holds, which in particular ensures that the cosmological horizon lies outside the lack hole event horizon.

3 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 3 Figure. Setup for the discussion of the forward prolem on Kerr-de Sitter space. Indicated are the ideal oundary X, the Cauchy hypersurface H and the hypersurface H 2, which has two connected components which lie eyond the cosmological horizon and eyond the lack hole event horizon, respectively. The horizons at X themselves are the projections to the ase of the (generalized) radial sets L ±, discussed elow, each of which has two components, corresponding to the two horizons. The projection to the ase of the icharacteristic flow is indicated near a point on L + ; near L, the directions of the flowlines are reversed. Lastly, Γ is the trapped set, and the projection of a trapped trajectory approaching Γ within Γ = Γ + Γ, discussed elow, is indicated. where g(0, 0) = g 0, and for each p M, g p (v 0, v) : R T p M Sym 2 T p M, depending smoothly on p, and 2 q(u, du) = a j u ej N j= N j k= X jk u, e j, N j N 0, N j + e j 2, with a j C (M), X jk V (M). (.) Our central result in the form which is easiest to state, without reference to the natural Soolev spaces, is: Theorem. On Kerr-de Sitter space with angular momentum a M, for α > 0 sufficiently small and f Cc (M ) with sufficiently small H 4 -norm, the wave equation g(u, du)u = f + q(u, du), with q as aove with N j for all j, has a unique smooth (in M ) gloal forward solution of the form u = u 0 + ũ, x α ũ ounded, u 0 = cχ, χ C (M) identically near M. Further, the analogous conclusion holds for the Klein-Gordon operator m 2 with m > 0 sufficiently small, without the presence of the u 0 term, i.e. for α > 2 Here aj is only relevant if N j = 0.

4 4 PETER HINTZ AND ANDRAS VASY 0, m > 0 sufficiently small, if f Cc (M ) has sufficiently small H 4 -norm, ( g(u, du) m 2 )u = f + q(u, du) has a unique smooth gloal forward solution u x α L (Ω). In fact, for Klein-Gordon equations one can also otain a leading term, analogously to u 0, which now has the form cx iσ χ, σ the resonance of g(0) m 2 with the largest imaginary part; thus Im σ < 0, so this is a decaying solution. The only reason the assumption a M is made is due to the possile presence (to the extent that we do not disprove it here) of resonances in Im σ 0, apart from the 0-resonance with constants as the resonant state, for larger a. Below, in Section 2, we give a general result in a form that makes it clear that this is the only remaining item to check indeed, this even holds in natural vector undle settings. In order to state the natural gloal regularity assumptions, we now discuss the Soolev spaces corresponding to our setting: We measure regularity with respect to V (M), and for non-negative integer s, one lets H s (M) e the set of (complexvalued) u L 2 (M) such that V... V j u L 2 (M) for j s and V,..., V j V (M). Here, L 2 is the L2 -space with respect to any -metric, such as the Kerrde Sitter metric, which is thus in local coordinates given y a density which is a positive smooth multiple of x dx dy... dy n. Further, one introduces the weighted Soolev spaces H s,α (M) = xα H s (M); Hs,α (M; R) denotes the real-valued elements of these spaces. Sections of vector undles in H s,α are defined y local trivializations; the Soolev spaces on Ω are defined y restriction. We then relax (.) to a j C (M) + H (M), X jk (C + H )V (M), (.2) in our assumptions. Generalizing the forcing as well, and making the conclusion more precise, the more natural version of Theorem is, with further generalization given in Theorems 3 and 4: Theorem 2. On Kerr-de Sitter space with angular momentum a M, for α > 0 sufficiently small and f H,α with sufficiently small H 4,α -norm, the wave equation g(u, du)u = f + q(u, du), with q as aove with N j for all j, has a unique, smooth in M, gloal forward solution of the form u = u 0 + ũ, ũ H,α, u 0 = cχ, χ C (M) identically near M. Further, the analogous conclusion holds for the Klein-Gordon equation m 2 with m > 0 sufficiently small, without the presence of the u 0 term, i.e. for α > 0, m > 0 sufficiently small, if f H,α (Ω) has sufficiently small H 4,α -norm, ( g(u, du) m 2 )u = f + q(u, du) has a unique, smooth in M, gloal forward solution u H,α (Ω). For the proofs, we refer to Corollaries 5.3 and 5.6, which are special cases of Theorems 5.0 and 5.5. For any finite amount of regularity of the solution, our arguments only require a finite numer of derivatives: Indeed, for sufficiently large s 0, C R and for s s 0, it is sufficient to assume f H Cs,α, with small H 4,α -norm, to ensure the existence of a unique gloal forward solution u with H s,α -regularity, i.e. with ũ Hs,α in the case of wave equations, u H s,α in the case of Klein-Gordon equations; see Remark 5.2 for details. We now discuss previous results on Kerr-de Sitter space and its perturations. There seems to e little work on non-linear equations in Kerr-de Sitter type settings; indeed the only paper the authors are aware of is the earlier paper [3] of the authors

5 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 5 in which the semilinear Klein-Gordon equation was studied (with small data wellposedness shown) with non-linearity depending on u only, so that the losses due to the trapping could still e handled y a contraction mapping argument. In addition, the same paper also analyzed non-linearities depending on du provided these had a special structure at the trapped set. There is more work on the linear equation on perturations of de Sitter-Schwarzschild and Kerr-de Sitter spaces: a rather complete analysis of the asymptotic ehavior of solutions of the linear wave equation was given in [47], upon which the linear analysis of the present paper is ultimately ased. Previously in exact Kerr-de Sitter space and for small angular momentum, Dyatlov [2, 20] has shown exponential decay to constants, even across the event horizon; see also the more recent work of Dyatlov [9]. Further, in de Sitter- Schwarzschild space (non-rotating lack holes) Bachelot [3] set up the functional analytic scattering theory in the early 990s, while later Sá Barreto and Zworski [4] and Bony and Häfner [6] studied resonances and decay away from the event horizon, Dafermos and Rodnianski in [4] showed polynomial decay to constants in this setting, and Melrose, Sá Barreto and Vasy [38] improved this result to exponential decay to constants. There is also physics literature on the suject, starting with Carter s discovery of this space-time [8, 7], either using explicit solutions in special cases, or numerical calculations, see in particular [50], and references therein. We also refer to the paper of Dyatlov and Zworski [24] connecting recent mathematical advances with the physics literature. While it received more attention, the linear, and thus the non-linear, equation on Kerr space (which has vanishing cosmological constant) does not fit directly into our setting; see the introduction of [47] for an explanation and for further references and [5] for more ackground and additional references. Some of the key works in this area include the polynomial decay on Kerr space which was shown recently y Tataru and Tohaneanu [44, 43] and Dafermos, Rodnianski and Shlapentokh- Rothman [0,, 6], after pioneering work of Kay and Wald in [34] and [48] in the Schwarzschild setting. Andersson and Blue [] proved a decay result for the Maxwell system on slowly rotating Kerr spaces; see also the earlier work of Bachelot [2] in the Schwarzschild setting. The crucial normal hyperolicity of the trapping, corresponding to null-geodesics that do not escape through the event horizons, in Kerr space was realized and proved y Wunsch and Zworski [49]; later Dyatlov extended and refined the result [22, 23]. Note that a stronger version of normal hyperolicity is a notion that is stale under perturations. On the non-linear side, Luk [35] estalished gloal existence for forward prolems for semilinear wave equations on Kerr space under a null condition, and Dafermos, Holzegel and Rodnianski [9] constructed ackward solutions for Einstein s equations on Kerr space. (There was also recent work y Marzuola, Metcalfe, Tataru and Tohaneanu [37] and Tohaneanu [46] on Strichartz estimates, which are applied to the study of semilinear wave equations with power non-linearities, and y Donninger, Schlag and Soffer [8] on L estimates on Schwarzschild lack holes, following L estimates of Dafermos and Rodnianski [3, 2], of Blue and Soffer [5] on nonrotating charged lack holes giving L 6 estimates, and of Finster, Kamran, Smoller and Yau [25, 26] on Dirac waves on Kerr.) In the next section, Section 2, we explain the ingredients of the proof of Theorem 2, and we also state natural generalizations. At the end of that section we provide a detailed roadmap through this paper.

6 6 PETER HINTZ AND ANDRAS VASY The authors are very grateful to Semyon Dyatlov for providing a preliminary version of his manuscript [23] and for discussions aout it, as well as for pointing out the reference [32]. They are also very grateful to Maciej Zworski for comments that improved the exposition. They are also thankful to Gunther Uhlmann, Richard Melrose and Rafe Mazzeo for comments and interest in this project. 2. Overview of the proof and the more general results Having stated the result, we now explain why it holds. Before doing this we recall some notation. The description of V (M) in the introduction in terms of local coordinates shows that it is the space of all C sections of a vector undle, T M, with local asis x x, yj. The dual undle of T M is denoted y T M; it has a local asis of dx x, dy j. A -metric is a non-degenerate symmetric ilinear form on the fiers of T M smoothly depending on the ase point; a Lorentzian -metric is one of signature (, n ). We point out that the -differential d, defined locally y du = (x x u) dx x + j ( yj u)dy j maps H s,α (M) to H s,α (M; T M). In order to start the explanation, it is est to egin with the underlying linear equation; after all, the non-linearity is just a rather serious perturation! In general, the analysis of -differential operators (locally finite sums of finite products of elements of V (M)), such as g Diff 2 (M), has two ingredients, corresponding to the two orders, smoothness and decay, of the Soolev spaces: () -regularity analysis. This provides the framework for understanding PDE at high -frequencies, which in non-degenerate situations involves the - principal symol and perhaps a suprincipal term. This is sufficient in order to control solutions u in H s,r modulo H s,r, s < s, i.e. modulo a space with higher regularity, ut no additional decay. Since for the inclusion H s,r H s,r to e compact one needs oth s > s and r > r, this does not control the prolem modulo relatively compact errors. (2) Normal operator analysis. This provides a framework for understanding the decay properties of solutions of the PDE. The normal operator is otained y freezing coefficients of the differential operator L at M to otain a dilation-invariant -operator N(L). One then Mellin transforms the normal operator in the normal variale to otain a family of operators ˆL(σ), depending on the Mellin-dual variale σ. The -regularity analysis, in non-degenerate situations, gives control of this family ˆL(σ) in a Fredholm sense, uniformly as σ with Im σ ounded. However, in any such strip, ˆL(σ) will still typically have finitely many poles σ j ; these poles, called resonances, dictate the asymptotic ehavior of solutions of the PDE. In order to have a Fredholm operator L, one needs to work in spaces such as H s,r, where r is such that there are no resonances σ j with Im σ j = r. One can also work in slightly more general spaces, such as C H s,r, r > 0, identified with a space of distriutions via u = u 0 +ũ, ũ H,α, u 0 = cχ, corresponding to (c, ũ) C H s,r. Now, the -regularity analysis for our non-elliptic equation involves the (null)- icharacteristic flow. In view of the version of Hörmander s theorem on propagation of singularities in this setting, and in view of the a priori control on Cauchy data at

7 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 7 H, what one would like is that all icharacteristics tend to TH M in one direction. Moreover, for the purposes of the adjoint prolem, which effectively imposes Cauchy data at H 2, one would like that the icharacteristics tend to TH 2 M in the other direction. Unfortunately, icharacteristics within TX M can never leave this space, and thus will not tend to TH M. This is mostly resolved, however, y the conormal undle of the horizons at X, which give rise to a undle of saddle points for the icharacteristic flow. Since the flow is homogeneous, it is convenient to consider it in S M = ( T M \ o)/r +. The characteristic set in S M has two components Σ ±, with Σ forward-oriented (i.e. future oriented time functions increase along nullicharacteristics in Σ ), Σ + ackward oriented. Then the images of the conormal undles of the horizons in the cosphere undle are sumanifolds L ± Σ ± of SX M, with one-dimensional stale ( )/unstale (+) manifold L ± transversal to SX M. (The flow within L ± need not e trivial; if it is, one has radial points, as in the a = 0 de Sitter-Schwarzschild space. However, for simplicity we refer to the L ± estimates as radial point estimates in general.) The realistic ideal situation, called a non-trapping one, then is if all (null-)icharacteristics in SΩ M (Σ + \ L + ) tend to SH 2 M L + in the ackward direction, and SH M L + in the forward direction, with a similar statement for Σ, with ackward and forward interchanged. 3 In this non-trapping setting the only sutlety is that the propagation estimates through L ± require that the differentiaility order s and the decay order r e related y s > 2 + βr for a suitale β > 0 (dictated y the Hamilton dynamics at L ±), i.e. the more decay one wants, the higher the regularity needs to e. This is still not the case in Kerr-de Sitter space, though it is true for neighorhoods of the static patch in de Sitter space, and its perturations. The additional ingredient for Kerr-de Sitter space is normally hyperolic trapping, introduced in this context y Wunsch and Zworski [49], given y smooth sumanifolds Γ ± Σ ±. Here Γ ± are invariant sumanifolds for the Hamilton flow, given y the transversal intersection of locally defined smooth, Hamilton flow invariant, Γ ± = Γ ± + Γ ±, with Γ ± Σ transversal to SX M Σ, and Γ± + SX M Σ. Comining results of [22, 23] (which would work directly in a dilation invariant setting) and [30] we show that for r > 0 sufficiently small, one can propagate H s,r estimates through Γ ±. This suffices to complete the -regularity setup if the non-trapping requirement is replaced y: All (null-)icharacteristics in SΩ M (Σ + \ (L + Γ + )) tend to either SH 2 M L + Γ + in the ackward direction, and SH M L + Γ + in the forward direction, with the tending to Γ + allowed in only one of the forward and ackward directions, with a similar statement for Σ, with ackward and forward interchanged. Finally, this is satisfied in Kerr-de Sitter space, and also in its -perturations (the whole setup is perturation stale). Next, one needs to know aout the resonances of the operator. For the wave operator, the only resonance with non-negative imaginary part is 0, with the kernel of ˆL(σ) one dimensional, consisting of constants. Since strips can only have finitely many resonances, there is r > 0 such that in Im σ r the only resonance is 0; then H s,r C works for our Fredholm setup. For the Klein-Gordon equation with 3 Notice that due to the assumption on the one-dimensional stale/unstale manifold eing transversal to SX M, there cannot e non-trivial icharacteristics in S M tending to L + in oth the forward and the ackward direction, since a icharacteristic is either completely in SX M, or completely outside it.

8 8 PETER HINTZ AND ANDRAS VASY m > 0 small, the m = 0 resonance at 0 moves to σ = σ (m) inside Im σ < 0, see [2, 3]. Thus, one can either work with H s,r where r is sufficiently small (depending on m), or with H s,r C, though with C now identified with cx iσ χ. We now discuss the non-linear terms. Here the asic point is that H s,0 is an algera if s > n/2, and thus for such s, products of elements of H s,r possess even more decay if r > 0, ut they ecome more growing if r < 0. Thus, one is forced to work with r 0. First, with the simplest semilinear equation, with no derivatives in the nonlinearity q (so N j 2 is replaced y N j = 0), the regularity losses due to the normally hyperolic trapping are in principle sufficiently small to allow for a contraction mapping principle (Picard iteration) ased argument. However, for the actual wave equation on Kerr-de Sitter space, the 0-resonance prohiits this, as the iteration maps outside the space H s,r C. Thus, it is the semilinear Klein- Gordon equation that is well-ehaved from this perspective, and this was solved y the authors in [3]. On the other hand, if derivatives are allowed, with an at least quadratic ehavior in du, then the non-linearity annihilates the 0-resonance. Unfortunately, since the normally hyperolic estimate loses + ɛ derivatives, as opposed to the usual real principal type/radial point loss of one derivative, the solution operator for g will not map q(u, du) ack into the desired Soolev space, preventing a non-linear analysis ased on the contraction mapping principle. Fortunately, the Nash-Moser iteration is designed to deal with just such a situation. In this paper we adapt the iteration to our requirements, and in particular show that semilinear equations of the kind just descried are in fact solvale. In particular, we prove that all the estimates used in the linear prolems are tame. Here we remark that Klainerman s early work on gloal solvaility involved the Nash-Moser scheme [32], though this was later removed y Klainerman and Ponce [33]. In the present situation the loss of derivatives seems much more serious, however, due to the trapping, so it seems unlikely that the solution scheme can e made more classical. However, we are also interested in quasilinear equations. Quasilinear versions of the aove non-trapping scenario were studied y the first author [29], who showed the solvaility of quasilinear wave equations on perturations of de Sitter space. The key ingredient in dealing with quasilinear equations is to allow operators with coefficients with regularity the same kind as what one is proving for the solutions, in this case H s,r -regularity. All of the smooth linear ingredients (microlocal elliptic regularity, propagation of singularities, radial points) have their analogue for H s,r coefficients if s is sufficiently large. Thus, in [29] a Picard-type iteration, u k+ = g(u k ) (f + q(u k, du k )) was used to solve the quasilinear wave equations on de Sitter space. Notice that g(uk ) has non-smooth coefficients; indeed, these lie in a weighted -Soolev space. In our Kerr-de Sitter situation there is normally hyperolic trapping. However, notice that as we work in decaying Soolev spaces modulo constants, g(u) differs from a Kerr-de Sitter operator with smooth coefficients, g(c), y one with decaying coefficients. This means that one can comine the smooth coefficient normally hyperolic theory, as in the work of Dyatlov [22], with a tame estimate in H s,r with r < 0; the sign of r here is a crucial gain since for r < 0 the propagation estimates through normally hyperolic trapped sets ehave in exactly the same way as real principal type estimates. In comination this provides the required tame estimates

9 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 9 for Kerr-de Sitter wave equations, and Nash-Moser iteration completes the proof of the main theorem. We emphasize that our treatment of these quasilinear equations is systematic and general. Thus, quasilinear equations which at X = M are modelled on a finite dimensional family L = L(v 0 ), v 0 C d small corresponding to the zero resonances (thus the family is 0-dimensional without 0-resonances!), of smooth -differential operators on a vector undle with scalar principal symol which has the icharacteristic dynamics descried aove (radial sets, normally hyperolic trapping, etc.) fits into it, provided two conditions hold for the normal operator (i.e. the dilation invariant model associated to L at M). 4 () First, the resonances for the model L(v 0 ) have negative imaginary part, or if they have 0 imaginary part, the non-linearity annihilates them. (2) Second, the normally hyperolic trapping estimates of Dyatlov [22] hold for ˆL(σ) (as Re σ ) in Im σ > r 0 for some r 0 > 0. In the semiclassical rescaling, with σ = h z, h = σ, this is a statement aout ˆL h,z = h m ˆL(h z), Im z > r 0 h. By Dyatlov s recent result 5 [23] this indeed is the case if ˆL h,z satisfies that at Γ its skew-adjoint part, 2i (ˆL h,z ˆL h,z ) hdiff (X), for z R has semiclassical principal symol ounded aove y hν min /2 for some ɛ > 0, where ν min is the minimal expansion rate in the normal directions at Γ; see [23, Theorem ] and the remark elow it (which allows the non-trivial skew-adjoint part, denoted y Q there, microlocally at Γ). It is important to point out that in view of the decay of the solutions either to 0 if there are no real resonance, or to the space of resonant states corresponding to real resonances, the conditions must e checked for at most a finite dimensional family of elements of the smooth algera Ψ (M), and moreover there is no need to prove tame estimates, deal with rough coefficients, etc., for this point, and one is in a dilation invariant setting, i.e. can simply Mellin transform the prolem. Thus, in principle, solving wave-type equations on more complicated undles is reduced to analyzing these two aspects of the associated linear model operator at infinity. Concretely, we have the following two theorems: Theorem 3. Let M e a Kerr-de Sitter space with angular momentum a < 3 2 M that satisfies [47, (6.3)], 6 E a vector undle over it with a positive definite metric k on E, and let L g(u, du) Diff 2 (M; E) have principal symol G = g (u, du) (times the identity), and suppose that L 0 = L g(0,0) satisfies that () the large parameter principal symol of 2i σ (L 0 L 0), with the adjoint taken relative to k dg, at the trapped set Γ is < ν min /2 as an endomorphism of E, (2) ˆL 0 (σ) has no resonances in Im σ 0. 4 The differential operator needs to e second order, with principal symol a Lorentzian dual metric near the Cauchy hypersurfaces if the latter are used; otherwise the order m of the operator is irrelevant. 5 This could presumaly also e seen from the work of Nonnenmacher and Zworski [40] y checking that this extension goes through without significant changes in the proof. 6 This condition on Λ, M and a ensures non-trapping classical dynamics for the null-geodesic flow.

10 0 PETER HINTZ AND ANDRAS VASY Then for α > 0 sufficiently small, there exists 7 d > 0 such that the following holds: If f H,α (Ω) has a sufficiently small H 2d-norm, then the equation L g(u, du)u = f + q(u, du) has a unique, smooth in M, gloal forward solution u H,α (Ω). In particular, the conditions at Γ for the theorem hold if a M, E = Λ M, L g(u, du) = g(u, du) the differential form d Alemertian, or indeed if L g(u, du) g(u, du) is a 0th order operator, since hyperolicity is shown in [47] in the full stated range of a, while for a = 0, 2i (L 0 L 0) can e computed explicitly at Γ, with k eing the Riemannian metric of the form α 2 d x 2 + h near the projection of Γ, where g has the form α 2 d x 2 h, x an appropriate oundary defining function on M strictly away from the horizons. Thus, in this case the only assumption in the theorem remaining to e checked is the second one, concerning resonances. Theorem 4. Let M e a Kerr-de Sitter space with angular momentum a < 3 2 M that satisfies [47, (6.3)], E a vector undle over it with a positive definite metric k on E, and let L g(u, du) Diff 2 (M; E) have principal symol G = g (u, du) (times the identity). Suppose that L 0 = L g(0,0) is such that ˆL 0 (σ) has a simple resonance at 0, with resonant states spanned y u 0,,..., u 0,d, and no other resonances in Im σ 0. Consider the family ˆL g(u0, du 0)(σ), u 0 Span{u 0,,..., u 0,d } with small enough norm. Suppose that () this family only has resonances at 0 in Im σ 0, and these are given y Span{u 0,,..., u 0,d }, (2) Γ is uniformly normally hyperolic for ˆL g(u0, du 0)(σ) for u 0 of small norm, (3) the large parameter principal symol of 2i σ (L 0 L 0), with the adjoint taken relative to k dg, at the trapped set Γ is < ν min /2, (4) q(u 0, du 0 ) = 0 for u 0 Span{u 0,,..., u 0,d }. Then for α > 0 sufficiently small, there exists 8 d > 0 such that the following holds: If f H,α has a sufficiently small H 2d,α -norm, then the equation L g(u, du)u = f + q(u, du) has a unique, smooth in M, gloal forward solution of the form u = u 0 + ũ, ũ H,α, u 0 = χ d j= c ju 0,j, χ C (M) identically near M. Here uniformly normally hyperolic in the theorem means that one has a smooth family Γ = Γ u0 of trapped sets, with a smooth family of stale/unstale manifolds, with uniform ounds (within this family) on the normal expansion rates for the flow, which ensures that the normally hyperolic estimates are uniform within the family (for small u 0 ); see the discussion around (4.27) for details. Again, the conditions at Γ for the theorem hold if a M, E = Λ M, if L g(u, du) g(u, du) is a 0th order operator, g(u, du) the differential form d Alemertian, since the structurally stale r-normally hyperolic statement is shown in [47] (which implies the uniform normal hyperolicity required in the theorem), while for a = 0, 2i (L 0 L 0) can e computed explicitly at Γ, as mentioned aove, and upper ounds on this are stale under perturations. The uniform normal hyperolicity condition at Γ holds if a < 3 2 M, E = Λ M, L g(u, du) = g(u, du) the differential form d Alemertian, with g(u 0, du 0 ) eing a Kerr-de Sitter metric for u 0 Span{u 0,,..., u 0,d } with small norm since 7 See the proof of this theorem in Section 5.4, in particular (5.27), for the value of d. 8 The value of d is given in (5.27) in the course of the proof of this theorem in Section 5.4.

11 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER the hyperolicity of Γ was shown in this generality in [47]. However, the computation of 2i (L 0 L 0) is more involved. The plan of the rest of this paper is the following. In Section 3 we show that the non-smooth pseudodifferential operators of [29] facilitate tame estimates (operator ounds, composition, etc.), with Section 4 estalishing tame elliptic estimates in Section 4., tame real principal type and radial point estimates in Section 4.2 and tame estimates at normally hyperolic trapping in Section 4.3 for r < 0. In Section 4.4, we adapt Dyatlov s analysis at normally hyperolic trapping given in [23] to our needs. Finally, in Section 5 we solve our quasilinear equations y first showing that the microlocal results of Section 4 comine with the high energy estimates for the relevant normal operators following from the discussion of Section 4.4 to give tame estimates for the forward propagator in Section 5., and then showing in Section 5.2 that the Nash-Moser iteration indeed allows for solving our wave equations. Section 5.3 then explains the changes required for quasilinear Klein-Gordon equations. Finally, in Section 5.4 we show how our methods apply in the general settings of Theorems 3 and Tame estimates in the non-smooth operator calculus In this section we prove the asic tame estimates for the H -coefficient, or simply non-smooth, -pseudodifferential operators defined in [29]. 3.. Mapping properties. We start with the tame mapping estimate, Proposition 3., which essentially states that for non-smooth pseudodifferential operators A, a high regularity norm of Au can e estimated y a high regularity norm of A times a low regularity norm of u, plus a low regularity norm of A times a high regularity norm of u. This is stronger than the a priori continuity estimate one gets from the ilinear map (A, u) Au, which would require a product of high norms of oth. In case A is a multiplication operator, this is essentially a -version of a (weak) Moser estimate, see Corollary 3.2. We work on the half space R n + with coordinates z = (x, y) [0, ) R n ; the coordinates in the fier of the -cotangent undle are denoted ζ = (λ, η), i.e. we write -covectors as λ dx x + η dy. Recall from [29] the symol class with the norm S m;0 H s := {a(z, ζ): ζ m a(z, ζ) H s L ζ } a S m;0 H s = ξ s â(ξ, ζ) ζ m L ζ L 2 ξ where â denotes the Mellin transform in x and Fourier transform in y of a. Left quantizations of such symols, denoted Op(a) Ψ m;0 H s, act on u C c (R n +) y Op(a)u(z) = e izζ a(z, ζ)û(ζ) dζ. Also recall S m;k H s = {a S m;0 H s : α ζ a S m α ;0 H s, α k}. and Ψ m;k H s = Op Sm;k H s. For revity, we will use the following notation for Soolev, symol class and operator class norms, with the distinction etween symolic and -Soolev norms eing clear from the context: u s := u H s, u s,r := u H s,r,,

12 2 PETER HINTZ AND ANDRAS VASY a m,s := a S m;0 H s, a (m;k),s := a S m;k H s, A m,s := A Ψ m;0 H s, A (m;k),s := A Ψ m;k H s. If A is a -operator acting on an element of a weighted -Soolev space with weight r (which will e apparent from the context), then A m,s is to e understood as x r Ax r m,s, similarly for A (m;k),s. Lastly, for A H sψm, we write A H s, Ψm y an ause of notation, for an unspecified H sψm -seminorm of A. Recall the notation x + = max(x, 0) for x R. Proposition 3.. (Extension of [29, Proposition 3.9].) Let s R, A = Op(a) Ψ m;0 H s, and suppose s R is such that s s m, s > n/2 + (m s ) +. Then A defines a ounded map H s, and for all fixed µ, ν with Hs m µ > n/2 + (m s ) +, ν > n/2 + (m s ) + + s s, there is a constant C > 0 such that Au s m C( A m,µ u s + A m,s u ν ). (3.) Oserve that y the assumptions on s and s, the intervals of allowed µ, ν are always non-empty (since they contain µ = s and ν = s ). Estimates of the form (3.), called tame estimates e.g. in [28, 42], are crucial for applications in a Nash- Moser iteration scheme. Proof of Proposition 3.. We compute Au 2 s m = ζ 2(s m) Âu(ζ) 2 dζ ( ζ 2(s m) â(ζ ξ, ξ)û(ξ) dξ) 2 dζ. We split the inner integral into two pieces, corresponding to the domains of integration ζ ξ ξ and ξ ζ ξ, which can e thought of as splitting up the action of A on u into a low-high and a high-low frequency interaction. We estimate ( 2 ζ 2(s m) â(ζ ξ, ξ)û(ξ) dξ) dζ ζ ξ ξ ( ) ζ 2(s m) ξ 2m ζ ξ ξ ζ ξ 2µ dξ (3.2) ξ 2s ( ζ ξ 2µ â(ζ ξ, ξ) 2 ) ξ 2m ξ 2s û(ξ) 2 dξ dζ, and we claim that the integral which is the first factor on the right hand side is uniformly ounded in ζ: Indeed, if s m 0, then we use ζ 2 ξ on the domain of integration, thus ζ ξ ξ ζ ξ ξ ζ 2(s m) dξ ζ ξ 2µ ξ 2(s m) ζ ξ 2µ dξ L ζ, since µ > n/2; if, on the other hand, s m 0, then ξ ζ ξ + ζ gives ξ 2(m s ) dξ ζ ξ 2µ ζ 2(m s ) ζ ξ + 2(µ (m s )) ζ ξ 2µ dξ L ζ,

13 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 3 since µ > n/2 + (m s ); hence, from (3.2), the H s m norm of the low-high frequency interaction in Au is ounded y C µ a m,µ u s. We estimate the norm of high-low interaction in a similar way: We have ( 2 ζ 2(s m) â(ζ ξ, ξ)û(ξ) dξ) dζ ξ ζ ξ ( ) ζ 2(s m) ξ 2m ξ ζ ξ ζ ξ 2s dξ (3.3) ξ 2ν ( ζ ξ 2s â(ζ ξ, ξ) 2 ) ξ 2m ξ 2ν û(ξ) 2 dξ dζ. If s m 0, the first inner integral on the right hand side is ounded y dξ ζ ξ 2(s s +m) ξ 2(ν m) ξ dξ, 2(s s +ν) ξ ζ ξ where we use s s m, and this integral is finite in view of ν > n/2 + s s; if s m 0, then ζ 2(m s ) dξ dξ, ζ ξ 2s ξ 2(ν m) ξ 2(ν m+s) ξ ζ ξ which is finite in view of ν > n/2 + m s. In summary, we need ν > n/2 + max(m, s ) s = n/2 + (m s ) + + s s and can then ound the H s m norm of the high-low interaction y C ν a m,s u ν. The proof is complete. Using H s S0;0 H s, we otain the following weak version of the Moser estimate for the product of two -Soolev functions: Corollary 3.2. Let s > n/2, s s. If u H s, v Hs, then uv Hs, and one has an estimate uv s C( u µ v s + u s v ν ) for fixed µ > n/2 + ( s ) +, ν > n/2 + s + s. In particular, for u, v H s, for fixed µ > n/2. uv s C( u µ v s + u s v µ ) 3.2. Operator compositions. We give a tame estimate for the norms of expansion and remainder terms arising in the composition of two non-smooth operators: Proposition 3.3. Suppose s, m, m R, k, k N 0 are such that s > n/2, s s k and k m + k. Suppose P = p(z, D) Ψ m;k H s, Q = q(z, D) Ψ m ;0 H s. Put E j := β! ( β ζ p Dz β q)(z, D), β =j R := P Q 0 j<k Then E j Ψ m+m j;0 H s and R k ;0 Ψm H s, and for µ > n/2 fixed, E j Ψ m+m j;0 H s R Ψ m k ;0 H s C( P Ψ m;j H µ Q Ψ m ;0 H s+j C( P Ψ m;k H µ Q Ψ m ;0 H s+k E j. + P Ψ m;j H s Q Ψ m ;0 H µ+j ), ). + P Ψ m;k H s Q Ψ m ;0 H µ+k

14 4 PETER HINTZ AND ANDRAS VASY Proof. The statements aout the E j follow from Corollary 3.2. For the purpose of proving the estimate for R, we define p 0 = k ζ p S m k;0 H s, D k z q S m ;0 H s k, where we write ζ k = ( β ζ ) β =k, similarly for Dz k. Notice that in particular p 0 S 0;0 H s. Then R = r(z, D) with ( ) ˆr(η; ζ) p 0 (η ξ; ζ + tξ) dt q 0 (ξ; ζ) dξ 0 y Taylor s formula, hence η 2s ˆr(η; ζ) 2 dη ζ 2m ( ) η 2s η ξ ξ η ξ 2µ dξ ξ 2s ( ( ) ξ η ξ 2µ p 0 (η ξ, ζ + tξ) 2 2s q 0 (ξ; ζ) 2 dt 0 ζ 2m ( ) η 2s + ξ η ξ η ξ 2s dξ ξ 2µ ( ( ) ξ η ξ 2s p 0 (η ξ, ζ + tξ) 2 2µ q 0 (ξ; ζ) 2 dt 0 ζ 2m ) dξ dη ) dξ dη, which implies the claimed estimate for k = 0. For k > 0, we use a trick of Beals and Reed [4] as in the proof of Theorem 3.2 in [29] to reduce the statement to the case k = 0: Recall that the idea is to split up q(z, ζ) into a trivial part q 0 with compact support in ζ and n parts q i, where q i has support in ζ i, and then writing P Q i = c jk P D k j z i k j=0 ( D j z i q i )(z, D) D k z i for some constants c jk R using the Leiniz rule; then what we have proved aove for k = 0 can e applied to the j-th summand on the right hand side, which we expand to order k j, giving the result Reciprocals of and compositions with H s functions. We also need sharper ounds for reciprocals and compositions of -Soolev functions on a compact n-dimensional manifold with oundary. Localizing using a partition of unity, we can simply work on R n +. Proposition 3.4. (Extension of [29, Lemma 4.].) Let s > n/2 +, u, w H s, a C, and suppose that a + u c 0 near supp w. Then w/(a + u) H s, and one has an estimate w a + u C( u µ, a C N )c 0 max(c s 0, ) ( w s + w µ ( + u s ) ). (3.4) s for any fixed µ > n/2 + and some s-dependent N N.

15 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 5 Proof. Choose ψ 0, ψ C such that ψ 0 on supp w, ψ on supp ψ 0, and such that moreover a + u c 0 > 0 on supp ψ. Then we have w/(a + u) 0 c 0 w 0. We now iteratively prove higher regularity of w/(a + u) and an accompanying tame estimate: Let us assume w/(a + u) H s for some s s. Let Λ s = λ s ( D) Ψ s e an operator with principal symol ζ s. Then w ψ 0 w ψ 0 w Λ s ( ψ)λ s + ψλ s a + u 0 a + u 0 a + u 0 w + c w 0 ψ(a + u)λ s a + u 0 a + u 0 ( c 0 w 0 + c w ) (3.5) 0 ψλ s w 0 + ψ[λ s, a + u] a + u 0 ( c 0 w s + w w + ψ[λ s, u] ), a + u s a + u 0 where we used that the support assumptions on ψ 0 and ψ imply ( ψ)λ s ψ 0 Ψ, and ψ[λ s, a] Ψ s. Hence, in order to prove that w/(a + u) H s, it suffices to show that [Λ s, u]: H s H 0. Let v Hs. Since (Λ s uv) (ζ) = λ s (ζ)û(ζ ξ)ˆv(ξ) dξ (uλ s v) (ζ) = û(ζ ξ)λ s (ξ)ˆv(ξ) dξ, we have, y taking a first order Taylor expansion of λ s (ζ) = λ s (ξ +(ζ ξ)) around ζ = ξ, ([Λ s, u]v) (ζ) = ( ) β ζ λ s (ξ + t(ζ ξ)) dt ( Dz β u) (ζ ξ)ˆv(ξ) dξ, β = 0 thus, writing u = D z u H s, ( û ([Λ s, u]v) (ζ) ξ + t(ζ ξ) dt) s (ζ ξ) ˆv(ξ) dξ. 0 To otain a tame estimate for the L 2 ζ norm of this expression, we again use the method of decomposing the integral into low-high and high-low components: The low-high component is ounded y ( ) sup 0 t ξ + t(ζ ξ) 2(s ) dξ ζ ξ ξ ζ ξ 2(µ ) ξ 2(s ) ( ζ ξ 2(µ ) û (ζ ξ) ) 2 ξ 2(s ) ˆv(ξ) 2 dξ dζ; the first inner integral, in view of s, so the sup is ounded y ξ 2(s ), which cancels the corresponding term in the denominator, is finite for µ > n/2 +. For the high-low component, we likewise estimate ( ) sup 0 t ξ + t(ζ ξ) 2(s ) dξ ξ ζ ξ ζ ξ 2(s ) ξ 2ν ( ζ ξ 2s û (ζ ξ) ) 2 ξ 2ν ˆv(ξ) 2 dξ dζ,

16 6 PETER HINTZ AND ANDRAS VASY and the first inner integral on the right hand side is ounded y ζ ξ 2(s s ) dξ ξ 2ν ξ dξ 2(s s +ν) ξ ζ ξ ecause of s s, which is finite for ν > n/2 + s s. We conclude that s [Λ s, u]v 0 C µν ( u µ v s + u s v ν ), for µ > n/2 +, ν > n/2 + s s. Plugging this into (3.5) yields ( ) w c 0 w s + ( + u µ ) w a + u a + u + u s w, s a + u where the implicit constant in the inequality is independent of c 0, w and u. Using the areviations q σ := w/(a+u) σ, u σ = u σ, w σ = w σ and fixing µ > n/2+, this means q s c 0 (w s + ( + u µ)q s + u s q ν ), ν > n/2 + s s, with the implicit constant eing independent of c 0, w, a, u, µ. We will use this for s γ := n/2 + with ν = s, and for s > γ, we will take ν = γ, thus otaining a tame estimate for q s. In more detail, for s γ, we have q s c 0 (w s + ( + u s )q s ), which gives, with C 0 = max(, c 0 ), γ q γ c 0 w γ (c 0 ( + u γ)) j + (c 0 ( + u γ)) γ q 0 c 0 Cγ 0 w γ( + u γ ) γ j=0 using the ound q 0 c 0 w 0 c 0 w γ. For γ < s s, we have q s c0 (w s + u s q γ + ( + u µ )q s ), thus for integer k with γ + k s, k q γ+k c 0 (w s + u s q γ ) (c 0 ( + u µ)) j + (c 0 ( + u µ)) k q γ j=0 c 0 Ck 0 ( + u µ ) k (w s + ( + u s )q γ ) c 0 Cγ+k 0 ( + u µ ) γ+k (w s + ( + u s )w γ ), where we used µ > γ in the last inequality, thus proving the estimate (3.4) in case s is an integer; in the general case, we just use q γ q γ for γ < γ, in particular for γ = s s γ, and use the aove with q γ+k replaced y q γ +k. As in [29], one thus otains regularity results for compositions, ut now with sharper estimates. To illustrate how to otain these, let us prove an extension of [29, Proposition 4.5]. Let M e a compact n-dimensional manifold with oundary, s > n/2 +, α 0. Proposition 3.5. Let u H s,α (M). If F : Ω C, F (0) = 0, is holomorphic in a simply connected neighorhood Ω of the range of u, then F (u) H s,α (M), and ν F (u) s,α C( u µ,α )( + u s,α ) (3.6) for fixed µ > n/2 +. Moreover, there exists ɛ > 0 such that F (v) H s,α (M) depends continuously on v H s,α (M), u v s,α < ɛ.

17 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 7 Proof. Oserve that u(m) is compact. Let γ C denote a smooth contour which is disjoint from u(m), has winding numer around every point in u(m), and lies within the region of holomorphicity of F. Then, writing F (z) = zf (z) with F holomorphic in Ω, we have F (u) = 2πi γ u F (ζ) ζ u dζ, Since γ ζ u/(ζ u) H s,α (M) is continuous y Proposition 3.4, we otain, using the estimate (3.4), F (u) s,α C( u µ ) ( u s,α + u µ,α ( + u s ) ), which implies (3.6) in view of α 0. The continuous (in fact, Lipschitz) dependence of F (v) on v is a consequence of Proposition 3.4 and Corollary 3.2. We also study compositions F (u) for F C (R; C) and real-valued u. Proposition 3.6. (Extension of [29, Proposition 4.7].) Let F C (R; C), F (0) = 0. Then for u H s,α (M; R), we have F (u) Hs,α(M), and one has an estimate F (u) s,α C( u µ,α )( + u s,α ) (3.7) for fixed µ > n/2 +. In fact, F (u) depends continuously on u. Proof. The proof is the same as in [29], using almost analytic extensions, only we now use the sharper estimate (3.4) to otain (3.7). Proposition 3.7. (Extension of [29, Proposition 4.8].) Let F C (R; C), and u C (M; R), u H s,α (M; R); put u = u + u. Then F (u) C (M) + H s,α (M), and one has an estimate F (u) F (u ) s,α C( u C N, u µ,α )( + u s,α ) for fixed µ > n/2 + and some N N. In fact, F (u) depends continuously on u. Proof. The proof is the same as in [29], ut now uses the sharper estimate (3.4). 4. Microlocal regularity: tame estimates When stating microlocal regularity estimates (like elliptic regularity, real principal type propagation, etc.) for operators with coefficients in H s (Rn +), we will give two quantitative statements, one for low regularities σ n/2, in which we will not make use of any tame estimates estalished earlier, and one for high regularities n/2 σ s, in which the tame estimates will e used. To concisely write down tame estimates, we use the following notation: The right hand side of a tame estimate will e a real-valued function, denoted y L, of the form L(p l,..., p l a; p h,..., p h ; u l,..., u l c; u h,..., u h d) d c = c j (p l,..., p l a)u h j + c jk (p l,..., p l a)p h j u l k j= j= k= (4.) here, the c j and c jk are continuous functions. In applications, p l/h j will e a low/high regularity norm of the coefficients of a non-smooth operator, and u l/h j will e a low/high regularity norm of a function that an operator is applied to. The

18 8 PETER HINTZ AND ANDRAS VASY important feature of such functions L is that they are linear in the u l/h j, and all p h j, uh j, corresponding to high regularity norms, only appear in the first power. 4.. Elliptic regularity. Concretely, we have the following quantitative elliptic estimate: Proposition 4.. (Cf. [29, Theorem 5.].) Let m, s, r R and ζ 0 T R n + \ o. Suppose P = p (z, D) H sψm (Rn +) has a homogeneous principal symol p m. Moreover, let R Ψ m ;0 H s (R n +). Let P = P + R, and suppose p m p m is elliptic at ζ 0. Let s R e such that s s and s > n/2 + + ( s) +, and suppose that u H s+m,r (R n +) satisfies P u = f H s,r (Rn +). Then there exists B Ψ 0 (Rn +) elliptic at ζ 0 such that Bu H s+m, and for s n/2 + t, t > 0, the estimate Bu s+m,r C( P (m;),n/2++( s)++t, R m,n/2+( s)++t) ( u s+m,r + f s,r ) holds. For s > n/2, ɛ > 0, there is a tame estimate Bu s+m,r L( P (m;),n/2++ɛ, R m,n/2+ɛ ; P (m;),s, R m,s ; (4.2) u n/2+m +ɛ,r, f n/2 +ɛ,r ; u s+m,r, f s,r ). (4.3) Let us point out that in our application of such an estimate to the study of nonlinear equations it will e irrelevant what exactly the low regularity norms in (4.3) are; in fact, it will e sufficient to know that there is some tame estimate of the general form (4.3), and this in turn is in fact clear without any computation, namely it follows directly from the fact that we have tame estimates for all non-smooth operations involved in the proof of this proposition. The same remark applies to all further tame microlocal regularity results elow. The only point where the precise numerology does matter is when one wants to find an explicit ound on the numer of required derivatives for the forcing term in Theorems 2, 3 and 4, as we will do. Proof of Proposition 4.. We can assume that r = 0 y conjugating P y x r. Choose a 0 S 0 elliptic at ζ 0 such that p m is elliptic 9 on supp a 0. Let Λ m Ψ m e a -ps.d.o with full symol λ m (ζ) independent of z, whose principal symol is ζ m, and define q(z, ζ) := a 0 (z, ζ)λ m (ζ)/p m (z, ζ) S 0; H s, then y Proposition 3.4 and Corollary 3.2, we have Q = q(z, D), Q (0;k),σ C( P (m;k),n/2++ɛ )( + P (m;k),σ ), σ > n/2 +, ɛ > 0. (4.4) Put B = a 0 (z, D)Λ m, then Q P = B + R with R Ψ m ;0 H s ; y Proposition 3.3, we have for n/2 < σ s R m,σ Q (0;),µ P (m;),σ+ + Q (0;),σ P (m;),µ+, µ > n/2. (4.5) 9 And non-vanishing, which only matters near the zero section.

19 QUASILINEAR WAVE EQUATIONS ON KERR-DE SITTER 9 Now, since Bu = QP u R u = Qf QRu R u, we need to estimate the H s -norms of Qf, QRu and R u, which we will do using Proposition 3.. In the low regularity regime, we have, for t > 0 and s n/2 + t, using (4.4) and (4.5): Qf s Q 0,n/2+( s)++t f s C( P m,n/2++( s)++t) f s, R u s R m,n/2+( s)++t u s+m C( P (m;),n/2++( s)++t) u s+m, QRu s C( P m,n/2++( s)++t) R m,n/2+( s)++t u s+m, giving (4.2). In the high regularity regime, in fact for 0 s s, we have, for ɛ > 0, Qf s Q 0,n/2+ɛ f s + Q 0,s f n/2 +ɛ C( P m,n/2++ɛ )( f s + ( + P m,s ) f n/2 +ɛ ), R u s R m,n/2+ɛ u s+m + R m,s u n/2+m +ɛ C( P (m;),n/2++ɛ )( u s+m + ( + P (m;),s ) u n/2+m +ɛ ), QRu s L( P m,n/2++ɛ, R m,n/2+ɛ ; P m,s, R m,s ; giving (4.3). The proof is complete. u n/2+m +ɛ ; u s+m ), There is a similar tame microlocal elliptic estimate for operators of the form P = P + P + R with P, R as aove and P Ψ m, as in part (2) of [29, Theorem 5.], where the tame estimate now also involves the C N -norm of the smooth part P of the operator for some (s-dependent) N Real principal type propagation; radial points. Tame estimates for real principal type propagation and propagation near radial points can e deduced from a careful analysis of the proofs of the corresponding results in [29]. The main oservation is that the regularity requirements, given in the footnotes to the proofs of these results in [29], indicate what regularity is needed to estimate the corresponding terms: For example, an operator in A Ψ m;0 H s with m 0 maps Hm/2 to under the condition s > n/2 + m/2, which is to say that one has a ound H m/2 Aũ m/2 A m,n/2+m/2+ɛ ũ m/2, ɛ > 0. This means that the only places where one needs to use tame operator ounds for operators with coefficients of regularity s are those where the condition for mapping properties etc. to hold reads s σ where σ is the regularity of the target space, i.e. where σ is comparale to the regularity s of the coefficients. We again only prove the tame real principal type estimate in the interior; the estimate near the oundary is proved in the same way, see also the discussion at the end of Section 4.. Proposition 4.2. (Cf. [29, Theorem 6.6].) Let m, r, s, s R. Suppose P m H s Ψm (Rn +) has a real, scalar, homogeneous principal symol p m, and let P m 0 Since in our application P will only depend on finitely many complex parameters, there is no need to prove an estimate which is also tame with respect to the C N -norm of P ; however, this could easily e done in principle.