Existence and asymptotics of nonlinear Helmholtz eigenfunctions
Jesse Gell-Redman, Andrew Hassell, Jacob Shapiro, Junyong Zhang
EEXISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZEIGENFUNCTIONS
JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG
Abstract.
We prove the existence and asymptotic expansion of a large class of solutionsto nonlinear Helmholtz equations of the form(∆ − λ ) u = N [ u ] , where ∆ = − (cid:80) j ∂ j is the Laplacian on R n with sign convention that it is positive asan operator, λ is a positive real number, and N [ u ] is a nonlinear operator that is a sumof monomials of degree ≥ p in u , u and their derivatives of order up to two, for some p ≥
2. Nonlinear Helmholtz eigenfunctions with N [ u ] = ±| u | p − u were first consideredby Guti´errez [10]. Such equations are of interest in part because, for certain nonlinearities N [ u ], they furnish standing waves for nonlinear evolution equations, that is, solutionsthat are time-harmonic.We show that, under the condition ( p − n − / > k > ( n − /
2, for every f ∈ H k +2 ( S n − ) of sufficiently small norm, there is a nonlinear Helmholtz functiontaking the form u ( r, ω ) = r − ( n − / (cid:16) e − iλr f ( ω ) + e + iλr g ( ω ) + O ( r − (cid:15) ) (cid:17) , as r → ∞ , (cid:15) > , for some g ∈ H k ( S n − ). Moreover, we prove the result in the general setting of asymp-totically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces X s, l ± , Y s, l ± , defined by Vasy [32], between which the Helmholtz operator ∆ − λ actsinvertibly. These spaces have a variable spatial weight l ± , varying in phase space anddistinguising between the two ‘radial sets’ corresponding to incoming oscillations, e − iλr ,and outgoing oscillations e + iλr . Our spaces have, in addition, module regularity withrespect to two different ‘test modules’, and have algebra (or pointwise multiplication)properties which allow us to treat nonlinearities N [ u ] of the form specified above. Introduction
In this article we prove the existence and asymptotic expansion of a large class of solutionsto nonlinear Helmholtz equations of the form(1.1) (∆ − λ ) u = N [ u ] , where ∆ = − (cid:80) j ∂ j is the Laplacian on R n with the sign convention that it is positive as anoperator, λ is a positive real number, and N [ u ] is a nonlinear operator that is a monomialin u , u and their derivatives of order up to two. Such equations are of interest in partbecause, for certain nonlinearities N [ u ], they furnish standing waves for nonlinear evolutionequations, that is, solutions that are time-harmonic. Indeed this is the case whenever N [ e iθ u ] = e iθ N [ u ], for all θ ∈ R . For example, if N [ u ] = α | u | q u , then Ψ( z, t ) = u ( z ) e iλ t solves the nonlinear Schr¨odinger equation(1.2) − i∂ t Ψ = ∆Ψ − α | Ψ | q Ψ , while if N [ u ] = |∇ u | u , then v ( z, t ) = u ( z ) e iλt solves the nonlinear wave equation(1.3) ( ∂ t + ∆) v = |∇ v | v. In this article, we will study the existence and asymptotic behaviour of ‘small’ solutionsto equation (1.1). Moreover, we shall do this not just for the standard Laplacian on R n butfor potential and/or metric perturbations of this Laplacian, and even more generally for the We thank Andr´as Vasy for helpful conversations. This work was supported by the Australian ResearchCouncil through grant DP180100589. J. Shapiro was supported by an AMS-Simons travel grant. J. Zhangwas supported by NSFC Grants (11771041, 11831004) and H2020-MSCA-IF-2017(790623). a r X i v : . [ m a t h . A P ] A ug JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG
Laplacian on asymptotically conic manifolds. However, in this introduction we shall mostlydiscuss the flat Euclidean case, as our results are new even in this setting.Since the linearization of this equation at u = 0 is just the standard Helmholtz equation,(1.4) (∆ − λ ) u = 0 , it is intuitively clear that ‘small’ nonlinear eigenfunctions should behave similarly to linearHelmholtz eigenfunctions. The structure of these is well known. The space of Helmholtzeigenfunctions of polynomial growth is parametrized by distributions on the ‘sphere at in-finity’, S n − . We shall only consider those eigenfunctions associated to smooth functionson the sphere at infinity. Given f ∈ C ∞ ( S n − ), there is a unique Helmholtz eigenfunctionsatisfying (in standard polar coordinates, r = | z | , ω = z/ | z | )(1.5) (∆ − λ ) u = 0 , u = r − ( n − / (cid:16) e − iλr f ( ω ) + e + iλr g ( ω ) + O ( r − ) (cid:17) , as r → ∞ , where g ∈ C ∞ ( S n − ) is determined by f . (In fact, in the simple case of the flat Laplacian, g ( ω ) = i ( n − / f ( − ω ), but in the presence of metric or potential perturbations, g is notso explicit, and is indeed related to the scattering matrix of the perturbed operator.) Wecall f the ‘incoming data’ or ‘incoming radiation pattern’ for the eigenfunction u , while g is referred to as the ‘outgoing data’ or ‘outgoing radiation pattern’. It is an arbitrarychoice whether to parametrize eigenfunctions by their incoming or their outgoing data; eachdetermines the other.1.1. Main results.
Our main result, at least as it applies to the flat Laplacian on R n , isthat ‘small’ nonlinear eigenfunctions can be parametrized in a similar way. We state ourresult first for the equation(1.6) (∆ − λ ) u = α | u | q u. Theorem 1.1 (Main Theorem, Euclidean case) . Let q ∈ N , p = 2 q + 1 , and assume that (1.7) ( p − n − > . Let k be an integer greater than ( n − / . There exist (cid:15), (cid:15) (cid:48) > sufficiently small, such thatfor every f ∈ H k +2 ( S n − ) with (cid:107) f (cid:107) H k +2 ( S n − ) < (cid:15) , there is a solution u to equation (1.6) ,satisfying (1.8) u = r − ( n − / (cid:16) e − iλr f ( ω ) + e + iλr g ( ω ) + O ( r − (cid:15) (cid:48) ) (cid:17) , as r → ∞ , for some g ∈ H k ( S n − ) .Moreover, uniqueness holds in the following sense. Fix a C ∞ function χ ( r ) equal to zerofor r small and for r large, and let (cid:96) = − / − δ for any δ satisfying < δ ≤ (4 p ) − . Let u − = χ ( r ) r − ( n − / e − iλr f ( ω ) . Then given f with (cid:107) f (cid:107) H k +2 ( S n − ) sufficiently small, there isexactly one nonlinear eigenfunction u of the form (1.8) , with the property that u − u − hassmall norm in the Hilbert space H ,(cid:96) ;1 ,k + defined in (2.34) .Remark . The solution is a scattering type solution, not a L solution. From the Pohozaevidentity, it is known that the sign of α plays an important role in the existence of finite energysolution to (1.6), while the sign of α plays no role in Theorem 1.1. Remark . As mentioned above, Ψ( z, t ) = u ( z ) e iλ t is a global-in-time solution whichsolves (1.2) but it is time-periodic without any decay. This is quite different from theclassical finite-energy solution to (1.2).Our proof of Theorem 1.1 principally makes use of the asymptotically conic structure of R n near infinity; in particular it uses neither the translation symmetries of R n nor exactformulae for resolvent kernels. The more general version of our main result is valid in thesetting of asymptotically conic manifolds. To prepare for the definition of such spaces, letus recall that, given a compact Riemannian manifold ( N , g N ), the metric cone over N isthe Riemannian manifold N × (0 , ∞ ) r with metric of the form dr + r g N . This space is XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 3 incomplete as r →
0; it can be completed topologically by adding a single point at r = 0(the ‘cone point’), but it is usually not a manifold then; and if it is, the metric is usuallysingular at r = 0. The sole exception is when N is the sphere S n − with its standardmetric, in which case the cone over ( N , g N ) is R n minus the origin, and adding the conepoint recovers the missing point.We define an asymptotically conic manifold to be the interior M ◦ of a compact manifoldwith boundary M , with Riemannian metric g taking a particular form near the boundary. Tospecify this, let x be a boundary defining function for ∂M (that is, the boundary ∂M is givenby x = 0, where x vanishes to first order at ∂M and x > M ◦ ) and let y = ( y , . . . , y n − )be local coordinates on ∂M extended to a collar neighbourhood { x ≤ c } of the boundary,where c > g has theproperty that, near any point on ∂M , there are coordinates ( x, y , . . . , y n − ) as above suchthat, in this coordinate patch, g takes the form(1.9) g = dx x + h ( x, y, dy ) x , where h is a smooth (0 , ∂M . This definition is betterunderstood by passing to the variable r = 1 /x , which goes to infinity at the boundary of M . The metric then takes the form(1.10) g = dr + r h ( 1 r , y, dy ) . If h is independent of x for small x , then this is precisely a conic metric for large r , where k = h (0 , y, dy ). More generally, it is asymptotic to this conic metric (smoothness of h in x is equivalent to having an asymptotic expansion in powers of 1 /r as r → ∞ .) Inparticular, the metric is always complete, as the boundary is at r = ∞ which is infinitedistance from any interior point. Thus, we can think of an asymptotically conic manifoldas a complete noncompact Riemannian manifold that is asymptotic, at infinity, to the‘large end of a cone’, but having no conic singularity (as a true cone usually does at r =0). Such spaces have curvature tending to zero at infinity, and local injectivity radiustending to infinity, so balls of a fixed size are asymptotically Euclidean as their centre tendsto infinity. For this reason, they are sometimes called ‘asymptotically Euclidean spaces’,although ‘asymptotically locally Euclidean’ would perhaps be a better term, as the globalstructure of ‘infinity’, that is, the boundary ∂M of the compactification M , can be quitedifferent from S n − .Particular instances of asymptotically conic manifolds include flat Euclidean space, orany compact metric perturbation of the flat metric on Euclidean space. In this case, M isthe radial compactification of R n , given by the union of R n with the ‘sphere at infinity’, S n − . Connected sums of such manifolds are also asymptotically conic. The topology andgeodesic dynamics on such manifolds can be intricate. For example, any convex co-compacthyperbolic manifold can have its metric modified near infinity to be asymptotically conic;while this is an artificial construction, it provides a very large class of asymptotically conicspaces with complicated topology and hyperbolic trapped set. Theorem 1.4 (Main Theorem, asymptotically conic case) . Let ( M ◦ , g ) be an asymptoticallyconic manifold of dimension n , and let V be a smooth function on M vanishing to secondorder at ∂M (that is, in the ‘noncompact’ picture, V is O ( r − ) as r → ∞ with an expansionat infinity in negative powers of r , obeying symbolic derivative estimates). Let H = ∆ g + V where ∆ g is the Laplace-Beltrami operator on ( M ◦ , g ) . Let N ( u, u, ∇ u, ∇ u, ∇ (2) u, ∇ (2) u ) bea sum of monomial terms, each of which has degree not less than p in u and u and theirderivatives up to order two, with coefficients smooth on M , and assume that p satisfies (1.7) .Let k be an integer greater than ( n − / . There exist (cid:15), (cid:15) (cid:48) > sufficiently small, such thatfor every f ∈ H k +2 ( ∂M ) with (cid:107) f (cid:107) H k +2 ( ∂M ) < (cid:15) , there is a function u on M ◦ satisfying ( H − λ ) u = N ( u, u, ∇ u, ∇ u, ∇ (2) u, ∇ (2) u ) JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG with asymptotics (1.11) u = r − ( n − / (cid:16) e − iλr f ( ω ) + e + iλr g ( ω ) + O ( r − (cid:15) (cid:48) ) (cid:17) , as r → ∞ , for some g ∈ H k ( ∂M ) . Moreover, uniqueness holds in same sense as in Theorem 1.1.Remark . We first clarify the meaning of a “monomial of degree not less than p in u and u and their derivatives up to order two, with coefficients smooth on M ”. These derivatives areunderstood to be taken with respect to a frame of vector fields that are uniformly boundedwith respect to the metric g . Thus, as r → ∞ we could take ∂ r and r − ∂ y j , for example;these are the natural analogues of the gradient in the Euclidean sense, written with respectto polar coordinates. For example, if p = 3, then on Euclidean R n the nonlinear term N could take the form | u | u + |∇ u | u + |∇ (2) u | u + ∂ u∂z ∂u∂z u + u . Remark . The first result along the lines of Theorem 1.1 was obtained by Guti´errez [10].Curiously, the set of pairs ( n, p ) treated in that paper is almost disjoint to ours: it coversthe case n = 3 , p = 3 for example, but higher n and p are excluded, while our methodworks most easily with large n and p . In fact, in view of the condition (1.7) in our twotheorems we can treat p ≥ n = 2, p ≥ n = 3, p ≥ n = 4 , p ≥ n ≥
6. We discuss previous literature more fully below.1.2.
Strategy of the proof.
The basic strategy of our proof of Theorem 1.1 is fixed pointargument which is similar to [10]. Given incoming data f , Guti´errez formed the lineareigenfunction u and showed that the map(1.12) Φ : u (cid:55)→ u + (∆ − ( λ + i ) − α | u | p − u is a contraction map on some Banach space, provided that the norm of u is sufficiently small.Guti´errez used L q spaces, for example L when p = 3 and n = 3 ,
4. Given u ∈ L , it is clearthat the cubic term | u | u lies in L / , while uniform resolvent bounds of Kenig-Ruiz-Sogge[19] and the extension restriction estimates of Stein-Tomas [27] are used to show that theoutgoing resolvent maps L / back to L . The fixed point of Φ is a nonlinear eigenfunction,as one sees by applying ∆ − λ to both sides, and it has the same incoming data as u .In our approach, we use polynomially weighted L -based Sobolev spaces, with an aniso-tropic weight. Vasy [31] has shown how to construct two families of Hilbert spaces betweenwhich ∆ − λ maps as a bounded invertible operator :(1.13) ∆ − λ : X s, l ± −→ Y s − , l ± +1 . In (1.13), the space Y s, l ± = H s, l ± is a variable order L -based Sobolev space. The index s ∈ R is a regularity parameter, specifying how many derivatives are locally in L , while l ± is a variable spatial weight, which varies ‘microlocally’, i.e. in phase space T ∗ R n . Theweight l + is chosen so that u ∈ X s, l + , localized in frequency close to the incoming radialoscillation e − iλr , decays at least as r − ( n − / − δ with δ > e iλr , slower decay, as r − ( n − / δ , is permitted. The weight l − has the opposite property: the decay must be faster than r − ( n − / near the outgoing radialoscillation, but can be slower near the incoming radial oscillation.This means that, for the + sign, the ‘outgoing’ expansion at infinity typical of generalizedeigenfunctions is permitted, while the ‘incoming’ expansion is not, while for the − sign, thesituation is reversed. This is consistent with the statement that the inverse map to (1.13)is, for the + sign, the outgoing resolvent (∆ − ( λ + i ) − , and for the − sign, the incomingresolvent (∆ − ( λ − i ) − , meaning that solutions (∆ − λ ) u = f ∈ C ∞ c ( R n ) with u ∈ X s, l ± admit asymptotic expansions of the form u = r − ( n − / e ± iλr ∞ (cid:88) j =0 r − j v j , v j ∈ C ∞ ( S n − ) . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 5
The domain of (1.13) is defined by an the priori regularity condition (1.14) X s, l ± := { u ∈ H s, l ± : (∆ − λ ) u ∈ H s − , l ± +1 } . The exponents ( s − , l ± +1) reflect the order (2 ,
0) of the operator P , as well as the ellipticityof P at fibre-infinity and the fact that P is of real principal type at spatial-infinity, leadingto a loss of one order of decay in the spatial regularity l ± . It is a tautology that ∆ − λ is abounded operator from X s, l ± to Y s − , l ± +1 . What is not obvious is that this is an invertiblemap, a result due to Vasy [32] with methods going back to Melrose [23], and which we givea detailed proof of below. The inverse operator depends on the choice of sign ± (the choicegiving either the incoming or outgoing resolvent), and as in the work of Guti´errez, althougha choice must be made, the only effect of this choice is to determine whether one prescribesthe incoming data f in the main theorems, or the outgoing data g .One thus obtains an inverse mapping R ( λ + i
0) : H s − , l +1 −→ H s, l , but this is not enoughto solve nonlinear problems. Given that our nonlinear term is assumed to be polynomial, weneed to work with spaces of functions with good algebra (or multiplicative) properties. Thespaces X s, l ± and Y s, l = H s, l ± are not suitable for this purpose, even (surprisingly) for large s . Recall that, for s > n/ H s, , the standard Sobolev space of order s , forms an algebra,i.e. H s, · H s, ⊂ H s, . If we include spatial weights, then (at least for constant weights),these combine additively, in the sense that we have for r , r ∈ R , H s,r · H s,r ⊂ H s,r + r .However, our weights are typically negative – indeed, they are forced to be so to obtainbijectivity of ∆ − λ – so this will not lead to a mapping Φ on a fixed space, as in (1.12);indeed, the nonlinear operation must gain one order of spatial decay to account for the lossof one order in the action of the resolvent inverting (1.13).To do this, we work with spaces with additional regularity with respect to the differentialoperators with coefficients that grow linearly at infinity but which annihilate the outgo-ing oscillation e iλr . These are generated by the operators r ( ∂ r − iλ ) and purely angulardifferential operators ∂ ω . This type of regularity condition is precisely the “module regu-larity” introduced by the second author together with Melrose and Vasy in [12], and usedby the first author with Haber and Vasy in [8] to solve a semilinear wave equation. Thusfor s, (cid:96) ∈ R , κ, k ∈ N with (cid:96) < − / , κ ≥
1, we define L -based Sobolev spaces H s,(cid:96) ; κ,k + inwhich s is the order of differentiability in the usual sense, i.e. relative to constant coefficientvector fields, (cid:96) is the decay rate relative to L , κ is the order of “module” differentiabilityjust described, and k is the order of differentiability tangential in the “angular” direction.(Provided κ ≥
1, we can take the spatial weight (cid:96) here to be a constant slightly less than − /
2, as the module regularity itself — which is asymmetric with respect to the incomingand outgoing oscillations, e ± iλr — enforces additional vanishing of the incoming oscilla-tions.) We arrive at a refinement of the mapping property (1.13), namely we obtain aninvertible map(1.15) ∆ − λ : X s,(cid:96) ; κ,k + −→ Y s − ,(cid:96) +1; κ,k + , where Y s,(cid:96) ; κ,k + = H s,(cid:96) ; κ,k + and, analogously to (1.14), the X s,(cid:96) ; κ,k + are given by X s,(cid:96) ; κ,k + .. = { u ∈ H s,(cid:96) ; κ,k + : (∆ − λ ) u ∈ H s − ,(cid:96) +1; κ,k + } . See Theorem 2.6 below. The inverse map to (1.15) we continue to denote by R ( λ + i Y s − ,(cid:96) +1; κ,k + within an appropriate choiceof Y s − , l + +1 . For κ ≥ k ≥ ( n − /
2, these spaces satisfy improved multiplicativeproperties. For example, we have (cid:16) H s,(cid:96) ; κ,k + (cid:17) p ⊂ H s,p(cid:96) +( p − n/ − κ ; κ,k + ;when κ = 1, we gain ( p − n/ − H s,(cid:96) ;1 ,k + from the combination of applying the nonlinear operator followed by the resolvent. With (cid:96) = l + = − / − δ , where δ can be taken arbitrarily small, this leads to the condition (1.7). JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG
For κ = 1 and sufficiently large k , we obtain our nonlinear eigenfunctions using a con-traction map on the space H s,(cid:96) ;1 ,k + . However, the nonlinear eigenfunction, or even the lineareigenfunction u , does not lie in this space as its incoming oscillations do not have the re-quired decay. To deal with this, we decompose u , the linear eigenfunction with incomingdata f , into two terms, u = u + + u − , where u − contains the leading incoming oscillation(which is the obstruction to membership in H s,(cid:96) ;1 ,k + ). Consequently, the term u + lies in H s,(cid:96) ;1 ,k + but u − does not. (Indeed, one can think of u + as a sum of purely outgoing termsplus the lower-order incoming terms, with additional decay, comprising u .) We seek anonlinear eigenfunction satisfying u = u + (∆ − ( λ + i ) − N [ u ] , where N ( u ) is the nonlinear term. Notice that, since the resolvent gains us two orders ofsmoothness, according to (1.15), N can involve derivatives of u up to order 2. Subtracting u − from both sides we have the equivalent equation u − u − = u + + (∆ − ( λ + i ) − N [ u ] , and now defining w = u − u − we obtain w = u + + (∆ − ( λ + i ) − N [ u − + w ] . Thus, it suffices to show that the map(1.16) Φ( w ) := u + + (∆ − ( λ + i ) − N [ u − + w ]is a contraction on H s,(cid:96) ;1 ,k + when the norm of w in this space is sufficiently small, which weshow provided the norm of f in H k +2 , k > ( n − /
2, is sufficiently small.1.3.
Previous literature.
Standing wave solutions to nonlinear Schr¨odinger equationshave been studied for a long time. The first studies were on finite-energy solutions, wherethe linearization at u = 0 is the operator ∆ + λ with λ >
0; this problem is of a differentcharacter, as the linearization at u = 0 is an invertible operator. See [1, 2, 25] for classicalwork on this subject on Euclidean space, and [4, 22] for more recent works on hyperbolicand rotationally symmetric manifolds. The more recent literature is vast and we make noattempt to review it.The first paper to study nonlinear Helmholtz eigenfunctions seems to be [10] by Guti´errez,already discussed earlier in this introduction. She was able to show that, for the cubic non-linearity and in dimensions 3 and 4, that there are nonlinear eigenfunctions with arbitrarysmall incoming data f ∈ L ( S n − ). We note in passing that the restriction and uniformSobolev estimates of [3, 9] allow one to extend Guti´errez’ method to all asymptotically conicmanifolds.There result of [10] is a perturbative result from the zero solution, as is ours here. Non-perturbative were found by Evequoz and Weth [6], who used mountain pass techniques tofind solutions far from the zero solution. These approaches have been extended in variousways in [21, 20]. In [5] the topology of the zero level sets of bounded real solutions to(∆ − u + u = 0 are studied.In the microlocal analysis literature, the underlying theory of real principal type propa-gation in the setting of ‘scattering’ pseudodifferential operators, was developed by Melrosein [23]. The scattering calculus itself appeared earlier (at least on Euclidean space) in workof H¨ormander and Parenti, see for example [26]. A Fredholm theory for nonelliptic oper-ators was developed by Vasy [31] on anisotropic Sobolev spaces. This is elaborated andexplained in detail in his lecture notes [32]. His method applies to operators that are ofreal principal type, except for manifolds of radial points which have a particular structure.The first author with Haber and Vasy [8] used this Fredholm framework to study the Feyn-man propagator on asymptotically Minkowski spaces and showed that the semilinear waveequation with polynomial nonlinear is solvable for small data, using a setup very similar tothat considered here. This latter result is an extension to a more fundamentally microlocalsetting of a previous result of Hintz and Vasy [13]. Indeed, the latter two authors have XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 7 developed a robust microlocal analysis framework which they use to study quasilinear waveequations in various noncompact settings, see in particular [14, 15, 18]. In a recent series ofpapers [28, 29, 30], Vasy considers ‘second-microlocal’ regularity for the Helmholtz opera-tors, both at a fixed finite energy and near zero energy, which is very similar to our moduleregularity here. He proves mapping properties for the resolvent that overlap our result onthe invertibility of the Helmholtz operator on spaces with module regularity in Theorem 2.6below.1.4.
Outline of this paper.
In Section 2 we review the theory of pseudodifferential oper-ators with variable order and define anisotropic Sobolev spaces. We discuss the geometryof the bicharacteristic flow of ∆ − λ at spatial infinity and define the radial sets. We alsodiscuss module regularity and define the corresponding spaces of functions. Finally, weconsider algebra properties of these spaces with sufficient module regularity.In Section 3 we prove the invertibility of ∆ − λ acting between spaces as in (1.15). Theproof of this is at least implicitly contained in works of Vasy, particularly his lecture notes[32], but it is not explicitly written out for this operator. Since, in addition, this is quiterecently developed technology and not standard, we have decided to give at least some ofthe details, to make the paper more self-contained.In Section 4 we prove the main theorems, using the technical preparation of the previoustwo sections. 2. Scattering calculus
In this section, we discuss the technical tools that we need for the proof of the maintheorems. We begin by discussing the pseudodifferential operators – the scattering calculus– used in the proof, on R n , and extend this in the following subsection to asymptoticallyconic manifolds. We refer to [32] and [23] for more detailed treatment of the scatteringcalculus.2.1. The scattering calculus on R n . Throughout this paper, we denote Euclidean coor-dinates on R n by z = ( z , . . . , z n ), and their dual coordinates by ζ = ( ζ , . . . , ζ n ). We usethe Japanese bracket (cid:104) z (cid:105) to denote (1 + | z | ) / . The Fourier transform, with H¨ormander’snormalization, will be denoted F , with inverse F − :(2.1) F f ( ζ ) = (cid:90) e − iz · ζ f ( z ) dz, F − ˜ f ( z ) = (2 π ) − n (cid:90) e iz · ζ ˜ f ( ζ ) dζ, We denote − i∂/∂z j by D z j , and use multi-index notation D αz , α = ( α , . . . , α n ) ∈ N n forhigher-order derivatives, in the standard way.Pseudodifferential operators on R n are defined via their symbols, which are functionson T ∗ R n . For sufficiently decaying symbols, say a ( z, ζ ) ∈ S ( T ∗ R n ), the correspondingpseudodifferential operator (defined by left quantization) is the operator with kernel(2.2) Op( a )( z, z (cid:48) ) := (2 π ) − n (cid:90) e i ( z − z (cid:48) ) · ζ a ( z, ζ ) dζ. This definition is extended to a larger class of symbols by integration by parts. The scatteringcalculus is obtained by letting a lie in a (scattering) symbol class S s,(cid:96) ( T ∗ R n ). For fixed realnumbers s and r this symbol class is defined by the estimates(2.3) ∀ α, β ∈ N n , ∃ C α,β < ∞ such that (cid:12)(cid:12)(cid:12) D αz D βζ a ( z, ζ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (cid:104) z (cid:105) (cid:96) −| α | (cid:104) ζ (cid:105) s −| β | . This is a rather restrictive class of symbols in which z and ζ are treated symmetrically:differentiation in ζ leads to decay in ζ and differentiation in z leads to decay in z . It is inthe H¨ormander class of symbols [17, Section 18.4] relative to the slowly varying metric dz (cid:104) z (cid:105) + dζ (cid:104) ζ (cid:105) . JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG
The class of pseudodifferential operators of order ( s, r ) is by definition the class of operatorsobtained from symbols a ∈ S s,(cid:96) ( T ∗ R n ) as above, and is denoted Ψ s,(cid:96) sc ( R n ). These pseudodif-ferential operators form a bi-filtered algebra; concretely, the composition of an operator inΨ s ,(cid:96) sc ( R n ) with an operator in Ψ s ,(cid:96) sc ( R n ) is an operator in Ψ s + s ,(cid:96) + (cid:96) sc ( R n ). The symbolof the composition Op( a ) ◦ Op( b ) is given by c ( z, ζ ) = e iD y · D η a ( z, η ) b ( y, ζ ) (cid:12)(cid:12)(cid:12) y = z,η = ζ , and has an asymptotic expansion(2.4) c ( z, ζ ) ∼ (cid:88) α i | α | D αζ a ( z, ζ ) D αz b ( z, ζ ) /α !Given this formula, and the decay of derivatives from (2.15), it is clear that the principalsymbol , which for a ∈ S s,(cid:96) ( T ∗ R n ) is its equivalence class in S s,(cid:96) ( T ∗ R n ) /S s − ,(cid:96) − ( T ∗ R n ),is multiplicative under composition. Notice that, unlike in the usual pseudodifferentialcalculus, here the principal symbol is well defined up to symbols decaying (a full integerorder) faster in z , as well as decaying (a full integer order) faster in ζ . That means that theprincipal symbol is, in effect, completely well-defined at infinity for all finite frequencies ζ ,and not just asymptotically as | ζ | → ∞ , at least in the case of classical symbols (discussedbelow).We elaborate on this point. It is convenient in the scattering calculus to view symbolson the compactification of T ∗ R n . We have already mentioned in the Introduction the radialcompactification R n of R n . This is obtained via the diffeomorphism ϕ : R n −→ B n from R n to its unit ball, given by z (cid:55)→ ϕ ( z ) = z (cid:104) z (cid:105) ∈ B n . The closure of the image of this map is obviously the closed unit ball B n , and the map re-alizes R n as the interior of this compact manifold with boundary. In keeping with standardnotation we write R n (cid:39) B n where the notation indicates that we keep in mind the identi-fication between points in R n with points in the ball B n . We similarly radially compactifythe fibre copy of R n . Thus, we may understand the behavior of symbols by pulling themback via ϕ − × ϕ − to R n × R n . This is particularly helpful for classical symbols, whichby definition take the form (cid:104) z (cid:105) (cid:96) (cid:104) ζ (cid:105) s C ∞ ( R n × R n ) (such functions automatically satisfy thesymbol estimates (2.15)). The class of such symbols is denoted S s,(cid:96) cl ( T ∗ R n ). In particular,for classical symbols of order (0 , R n × R n ,and the principal symbol can be viewed as the boundary value of this symbol. Notice thatthis has two ‘components’, one a function at fibre-infinity, that is, on R n × S n − , and oneat ‘spatial infinity’, that is, at S n − × R n . More generally, for an operator A with classicalsymbol a of order ( s,
0) (such as our Helmholtz operator ∆ − λ ), the principal symbol isconveniently viewed as the combination of a fibre component, σ fiber , , ( A )( z, ζ ), which ishomogeneous in ζ of degree s (and is hence determined by ζ restricted to any sphere), and abase (or spatial) component, σ base , , ( A )( ω, ζ ) where ω is the limiting value of z/ | z | on thesphere at infinity, and ζ ∈ R n . These have an obvious compatibility relation at the ‘corner’where | z | and | ζ | are both infinite. In particular, for the Helmholtz operator, the principalsymbol is given by σ fiber , , (∆ − λ )( z, ζ ) .. = (cid:88) i,j g ij ζ i ζ j ,σ base , , (∆ − λ )( ω, ζ ) .. = (cid:88) i,j g ij ζ i ζ j − λ . It is important to understand that the base symbol need not be homogenous, and indeed isnot homogeneous for the Helmholtz operator.Suppose A ∈ Ψ s,(cid:96) sc ( R n ) has classical symbol. The elliptic set of A , Ell s,(cid:96) ( A ) = Ell( A ), isthe open subset of ∂ ( R n × R n ) consisting of those points near which the principal symbolis at least as big as c (cid:104) z (cid:105) (cid:96) (cid:104) ζ (cid:105) s for some c >
0. Its complement in ∂ ( R n × R n ) is called the XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 9 characteristic variety, Σ s,(cid:96) ( A ) = Σ( A ). The Helmholtz operator ∆ − λ is elliptic at fibre-infinity, thus the characteristic variety is contained in the component at spatial infinity, andis given by(2.5) Σ(∆ − λ ) = { ( ω, ζ ) ∈ S n − × R n | | ζ | = λ } . We also define the operator wavefront set or microlocal support , WF (cid:48) ( A ) of A , to be thecomplement of the set of points q ∈ ∂ ( R n × R n ) such that, in a neighborhood U of q , thefull symbol a ( z, ζ ) satisfies (2.3) for all s, (cid:96) ∈ R . Thus, intuitively speaking, A is microlocallyof order −∞ in both the fibre and base senses away from WF (cid:48) ( A ).Returning to the composition formula (2.4), it is straightforward to see from this thatthe commutator of two pseudodifferential operators A ∈ Ψ s ,(cid:96) sc ( R n ) and B ∈ Ψ s ,(cid:96) sc ( R n )is an operator [ A, B ] in Ψ s + s − ,(cid:96) + (cid:96) − ( R n ), with principal symbol given by the Poissonbracket of the symbols a and b of these operators:(2.6) σ pr ([ A, B ]) = { a, b } mod S s + s − ,(cid:96) + (cid:96) − ( T ∗ R n ) . We also recall that the Poisson bracket is given in terms of the Hamilton vector fields by(2.7) { a, b } = H a ( b ) = − H b ( a ) , H a = (cid:88) j (cid:16) ∂a∂ζ j ∂∂z j − ∂a∂z j ∂∂ζ j (cid:17) . This is conceptually important for us in relation to the Fredholm estimates in Section 3. Inthe elliptic region, these Fredholm estimates are easy to obtain, but in a neighbourhood ofthe characteristic variety of ∆ − λ , they are obtained from positive commutator estimates,that is, from operators whose commutator with ∆ − λ has positive principal symbol mi-crolocally. Equation (2.7) shows that this amounts to finding symbols b such that H p ( a ) ispositive, where p = | ζ | − λ is the symbol of the Helmholtz operator. This then motivatesconsidering the properties of the Hamilton vector field of p , and its flow lines (known asbicharacteristics), within the characteristic variety Σ(∆ − λ ).The Hamiltonian vector field H p , for p = | ζ | − λ the symbol of the Euclidean Helmholtzoperator, is given by ˙ z = 2 ζ, ˙ ζ = 0 . We would like to view this on the compactification R n × R n , and investigate its behaviourin a neighbourhood of Σ(∆ − λ ). To do this, we use polar coordinates, ( r, ω ), as before,and then choose arbitrary local coordinates y on a patch of the sphere S n − . We also write x = r − , which serves as a boundary defining function for spatial infinity. Let ( ν, η ) be thedual coordinates to ( r, y ). In these coordinates, the full symbol p of ∆ − λ takes the form p ( r, y, ν, η ) = ν − i ( n − r − ν + r − b k η k + r − h jk η j η k − λ , where h jk = h jk ( y ) is the dual metric corresponding to the standard round metric h = h jk ( y ), on S n − , b k = D y j h jk + h jk D y j log( (cid:112) | det h | ), and we use the summation convention.We now make the change of variables to µ j = r − η j as these quantities have uniformlybounded length as r → ∞ . In terms of ( µ, ν ) the full symbol is p ( r, y, ν, µ ) = ν − i ( n − r − ν + r − b k µ k + h jk µ j µ k − λ , and we see that the principal symbol at spatial infinity is(2.8) ν + h jk µ j µ k − λ = ν + | µ | y − λ , where | µ | y := h jk µ j µ k is the metric function on T ∗ S n − . Thus the characteristic set Σsatisfies(2.9) Σ = { x = 0 , ν + | µ | y = λ } . R − R + l + = − − δ l + = − + δ µν Figure 1.
The bicharacteristic flow in the characteristic setΣ( P ) = { x = 0 , ν + | µ | y = λ } with P = ∆ − λ .In the canonical coordinates ( r, y ; ν, η ) we easily compute the Hamilton vector field of theprincipal symbol:(2.10) ˙ r = 2 ν, ˙ y l = 2 r − h lk η k , ˙ ν = 2 r − h jk η j η k , ˙ η l = − r − ∂h jk ∂y l η j η k . Changing to the variable µ , and writing x = r − , the equations become(2.11) ˙ x = − νx , ˙ y j = 2 xh jk µ k , ˙ ν = 2 xh jk µ j µ k , ˙ µ l = − xνµ l − x ∂h jk ∂y l µ j µ k . It is clear that this vector field vanishes to first order as x →
0. Dividing by x we obtain arescaled Hamilton vector field that we denote by H p , taking the form(2.12) ˙ x = − νx, ˙ y j = 2 h jk µ k , ˙ ν = 2 h jk µ j µ k , ˙ µ l = − νµ l − ∂h jk ∂y l µ j µ k . In the coordinates ( x, y, ν, µ ) this is a smooth vector field on the compactification. We canwrite it using derivative notation as follows:(2.13) H p = − ν ( x∂ x + R µ ) + 2 | µ | y ∂ ν + H S n − , where H S n − is the Hamilton vector of the round metric | µ | y on T ∗ S n − and R µ = µ · ∂ µ is the radial vector field on the fibers of T ∗ S n − . In these coordinates, and on Σ, we have H p = 2 νR µ − | µ | y ∂ ν + H S n − . We can check directly that H p ( ν + | µ | y ) = 0 and that theprecisely on the two ‘radial sets’(2.14) R ± .. = {| µ | y = 0 = x, ν = ± λ } . Remark . Notice that the incoming radial set R − is a source, and the outgoing radial set R + a sink, for the rescaled Hamilton vector field H p . Note, also, that the coefficient of x∂ x in H p is ± λ at R ± , hence always nonzero. This nonvanishing has the important consequencethat we can find operators with positive commutators at R ± , despite H p vanishing there.In this sense the radial sets are ‘nondegenerate’.Up to this point, we have taken the spatial weight (cid:96) to be constant. To consider variableorder spaces, we allow the spatial weight to itself be a classical symbol l of order (0 , XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 11 (variable weights will always be written in boldface). Choosing an arbitrary small positivenumber δ , we define the symbol class S s, l δ ( T ∗ R n ) by the estimates(2.15) ∀ α, β ∈ N n , ∃ C α,β < ∞ such that (cid:12)(cid:12)(cid:12) D αz D βζ a ( z, ζ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (cid:104) z (cid:105) l − (1 − δ ) | α | + δ | β | (cid:104) ζ (cid:105) s −| β | . These are symbol estimates of type (1 − δ , δ ) in the z variable (in the sense of H¨ormander),which are slightly ‘worse’ than the standard estimates of type (1 , (cid:104) z (cid:105) l (cid:104) ζ (cid:105) s times a C ∞ function on R n × R n , and these symbols incur logarithmic losses whendifferentating the l function.Changing to these symbol classes makes essentially no difference since we can take δ arbitrary small, while pseudodifferential calculus, as is well known, works with inessentialchanges provided δ < /
2. The only differences are that the principal symbol takes values in S s,(cid:96) ( T ∗ R n ) /S s − δ ,(cid:96) − δ ( T ∗ R n ) instead of S s,(cid:96) ( T ∗ R n ) /S s − ,(cid:96) − ( T ∗ R n ), and the commuta-tor [ A, B ] above will have order Ψ s + s − δ ,(cid:96) + (cid:96) − δ sc ( R n ) instead of Ψ s + s − ,(cid:96) + (cid:96) − ( R n ).2.2. The scattering calculus on asymptotically conic manifolds.
We now work inthe setting of asymptotically conic manifolds. Thus, let M be a compact manifold withboundary, and M ◦ its interior. Let x be a boundary defining function for M (meaning ∂M = { x = 0 } , x vanishes simply at ∂M , and x > M ◦ ), and y coordinates on a patch O of ∂M , extended to a collar neighbourhood { x < c } of ∂M , where c > g on M ◦ , we call ( x, y ) as an ‘adapted coordinate system’ near aboundary point (0 , y ) with y ∈ O provided that g takes the form (1.9) in this coordinatesystem on the patch O . This condition determines a metric h on the boundary ∂M , suchthat g is asymptotic to the conic metric dr + r h as r → ∞ , where r := 1 /x , as is clearfrom the equivalent expression (1.10).We now define scattering pseudodifferential operators of order ( s, (cid:96) ) on M ◦ . We do this bymimicking the behaviour of scattering symbols of order ( s, (cid:96) ) on R n . To do this, we choosea diffeomorphism χ from a small open set O ⊂ ∂M to an open set O (cid:48) ⊂ S n − , where S n − is viewed as the set of vectors of unit length in R n . We then consider the diffeomorphism(2.16) ( x, y ) (cid:55)→ χ ( y ) x = rχ ( y ) ∈ R n . One can check that the norm of the derivative of this map is uniformly bounded, where wemeasure with respect to the metric g on M ◦ and with respect to the Euclidean metric on R n . We now define suitable cotangent variables that are uniformly bounded (with respectto the dual metric g ∗ ). Let ( ν, η ) be the dual variables to coordinates ( r, y ), and define µ = r − η = xη as we did in the previous section. Then ν is the symbol of D r = x D x and µ j is the symbol of r − D y j = xD y j ; since ( x ∂ x , x∂ y j ) clearly form a uniformly bounded,uniformly nondegenerate frame of functions with respect to the metric g , these are uniformlybounded and uniformly nondegenerate linear coordinates on the cotangent bundle, withrespect to the dual metric g ∗ . We define scattering symbols of order ( s, (cid:96) ) on T ∗ M ◦ tobe functions a satisfying usual symbolic estimates of order s away from ∂M , and near theboundary satisfies(2.17) (cid:12)(cid:12)(cid:12) ( xD x ) j D αy D kν D βµ a ( x, y, ν, µ ) (cid:12)(cid:12)(cid:12) ≤ C j,k,α,β x − (cid:96) (cid:104) ( ν, µ ) (cid:105) s − k −| β | for all α, β ∈ N n − . We will denote the class of such symbols by S s,(cid:96) ( sc T ∗ M ). Scatteringpseudodifferential operators A ∈ Ψ s,(cid:96) sc ( M ◦ ) are defined as follows: using the local diffeomor-phism (2.16), the symbol is mapped (using the induced map on the cotangent bundle) toa symbol of order ( s, (cid:96) ) on T ∗ R n ; we then quantize to a pseudodifferential operator when( z, z (cid:48) ) are in the range of this diffeomorphism and pull back to M ◦ by the same map (2.16).By covering M ◦ with a finite number of coordinate charts and using a partition of unity, weget a globally defined operator. This quantization depends on the choice of charts, partitionof unity, etc, but all choices lead to the same operator modulo an operator in Ψ s − ,(cid:96) − ( M ◦ )so this is of no importance. To complete the picture, we include in Ψ s,(cid:96) sc ( M ◦ ) all kernels T ∗ M sc T ∗ ∂M M = { x = 0 } sc S ∗ MM × { }R − , l + = − / δ R + , l + = − / − δ Figure 2.
The radial sets in the compactified cotangent bundle. In thecase M ◦ = R n , sc T ∗ ∂M M = ∂ R n × R n and sc S ∗ M = R n × ∂ R n .K ( z, z (cid:48) ) that are smooth and rapidly decreasing, with all derivatives, as the distance be-tween z and z (cid:48) tends to infinity in M ◦ . This definition is equivalent to the definition of thescattering pseudodifferential operators defined in [23] using the ‘scattering double space’.We see in (2.17) that there are two types of vector fields (in terms of their behaviour nearthe boundary) that play a role in M . First, there are the b-vector fields , which by definitionare smooth vector fields that at the boundary are tangent to M . These are generatedover C ∞ ( M ) by x∂ x and ∂ y j near the boundary, and govern the regularity of scatteringsymbols in the spatial coordinates ( x, y ) (this is called conormal regularity in the microlocalliterature). Second, there are the scattering vector fields , which are just x times b-vectorfields, so generated by x ∂ x and x∂ y j . These have the property of generating, over C ∞ ( M ),all smooth vector fields on M that are uniformly bounded with respect to g . Scatteringdifferential operators of order ( k,
0) are precisely differential operators of order k that, nearthe boundary, can be written in terms of scattering vector fields with C ∞ ( M )-coefficients.In the case that ( M ◦ , g ), we can take the constant coefficient vector fields ∂ z j as generatorsof the scattering vector fields. Both will play a role in our analysis; the s parameter inour pseudodifferential calculus is regularity with respect to scattering vector fields, whileb-vector fields define module regularity (measured by the κ and k parameters, as discussedin the Introduction).Similarly to the Euclidean case, we can compactify the cotangent bundle T ∗ M ◦ in a waythat mimics the compactification R n × R n above. We have (by assumption) a compactifica-tion M of M ◦ . In the interior of M ◦ , we can compactify each cotangent fibre radially. It onlyremains to say how the fibres are compactified in the limit as we approach the boundary.We define a scattering cotangent bundle , denoted sc T ∗ M over M , which over the interior isnaturally isomorphic to the usual cotangent bundle, and has the property that, near ∂M ,using adapted coordinate system ( x, y ), the corresponding coordinates ( ν, µ ) (as definedabove) are linear coordinates on the fibres of this bundle that remain valid uniformly up tothe boundary ∂M . Compactifying each fibre radially gives us a compactification, denoted sc T ∗ M , analogous to the square in Figure 2. This is a manifold with corners of codimensiontwo. Clearly x is a boundary defining function at spatial infinity. Let ρ denote a boundarydefining function for fibre-infinity — we may take ρ = (cid:104) ( ν, µ ) (cid:105) − when x is small. We will call( x, y, ν, µ ) adapted coordinates on the scattering cotangent bundle over the neighbourhood { x < c, y ∈ O } of (0 , y ) ∈ ∂M .We then can consider the subspace of operators A with ‘classical’ symbols a of order( s, (cid:96) ) that take the form x − (cid:96) ρ − s times a smooth function on sc T ∗ M . The class of suchsymbols will be denoted S s,(cid:96) cl ( sc T ∗ M ). For such operators, the principal symbol can bedefined similarly to the classical case on R n . Supposing for simplicity that (cid:96) = 0, wehave a symbol at fibre-infinity, σ fiber ,s, ( A ), which is a function on sc T ∗ M homogeneous ofdegree s on each fibre, and a symbol at spatial infinity, σ base ,s, ( A ), which is the symbol a restricted to x = 0, a function on the scattering cotangent bundle restricted to ∂M (which XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 13 we denote sc T ∗ ∂M M ). These functions are well-defined, that is, they depend only on A ,not on the particular quantization procedure (that is, the precise way of relating symbols a and operators A ). The two symbols have the compatibility condition that σ base ,s, ( A ) isasymptotically homogeneous of degree s , that is, ρ s σ base ,s, ( A ) has a limit at ρ = 0, andagrees at the corner, x = ρ = 0, with the limiting value of ρ s σ fiber ,s, ( A ).Now let P denote the operator ∆ g + V − λ on M ◦ , where ∆ g is the (positive) Laplacianwith respect to g and V ∈ x C ∞ ( M ) is a real potential, vanishing to second order at ∂M .(The positive real number λ will be fixed throughout.) Then the principal symbols of P , inadapted coordinates ( x, y, ν, µ ) near the boundary, are(2.18) σ fiber , , ( P ) = ν + h jk ( y ) µ j µ k , and(2.19) σ base , , ( P ) = ν + h jk ( y ) µ j µ k − λ . This looks very similar to the form of the base symbol of the flat Laplacian on R n —compare (2.8). It follows that the characteristic variety is given by (2.9), just as in the flatcase. Moreover, exactly the same computation can be made as in the flat case to deducethat the Hamilton vector field of p , the symbol of P , takes the form(2.20) H p = − ν ( x∂ x + R µ ) + 2 | µ | y ∂ ν + H S n − + xW, where W is a b-vector field (that is, tangent to x = 0). That is, the Hamilton vector fieldtakes the same form as (2.13), up to a error xW . In particular, the radial sets, where theHamilton vector field vanishes, take the same form (2.14) as in the Euclidean case. Thismeans that the microlocal analysis of the operator P is no more complicated than that of theflat Laplacian on R n and means that, from this point of view, Theorem 1.4 is a very naturalgeneralization of Theorem 1.1.Our last topic to discuss is variable order scattering pseudodifferential operators. This iscompletely analogous to the case of variable order operators on R n . We allow the spatialweight r to be a classical symbol l of order (0 ,
0) on sc T ∗ M , and allow a δ loss in the symbolestimates. Thus, (2.17) is replaced by(2.21) (cid:12)(cid:12)(cid:12) ( xD x ) j D αy D kν D βµ a ( x, y, ν, µ ) (cid:12)(cid:12)(cid:12) ≤ C j,k,α,β x − l − δ ( j + k + | α | + | β | ) (cid:104) ( ν, µ ) (cid:105) s − k −| β | and the rest of the theory proceeds as in the Euclidean case.2.3. Sobolev spaces of variable order.
We begin with the Euclidean case. The Sobolevspaces H s,(cid:96) ( R n ) are the usual weighted Sobolev spaces defined by H s,(cid:96) ( R n ) = { f ∈ S (cid:48) ( R n ) | (cid:104) D (cid:105) s (cid:104) z (cid:105) (cid:96) f ∈ L ( R n ) } . Here (cid:104) D (cid:105) s is the Fourier multiplier, given by F − (cid:104) ζ (cid:105) s F . These can be equivalently definedby the condition that f ∈ H s,(cid:96) ( R n ) if and only if Af ∈ L for all A ∈ Ψ s,(cid:96) sc ( R n ); it is enoughto require this for just one A that is ‘totally elliptic’, that is, both its symbol at fibre-infinity and at spatial infinity are everywhere elliptic, or equivalently, Ell( A ) = ∂ ( R n × R n ).We use this characterization to define the Sobolev spaces for variable orders: we say that f ∈ H s, l ( R n ) if Af ∈ L for all A ∈ Ψ s, l sc ( R n ); again, it is enough to require this for onetotally elliptic operator. If in addition, A is invertible, that is, A − ∈ Ψ − s, − l sc ( R n ), then thenorm of f in H s, l ( R n ) can be taken to be (cid:107) Af (cid:107) L .The Sobolev spaces for an asymptotically conic manifold are defined analogously: we say f ∈ H s, l ( M ◦ ) if Af ∈ L for all A ∈ Ψ s, l sc ( M ◦ ); it is enough to require this for one totallyelliptic operator. If in addition, A is invertible, that is, A − ∈ Ψ − s, − l sc ( M ◦ ), then the normof f in H s, l ( M ◦ ) can be taken to be (cid:107) Af (cid:107) L .Pseudodifferential operators of variable order act on Sobolev spaces with variable orderin the expected way: if A ∈ Ψ s, l sc ( M ◦ ), then A is a bounded linear map from H s (cid:48) , l (cid:48) ( M ◦ ) to H s (cid:48) − s, l (cid:48) − l ( M ◦ ). Also, the dual space of H s, l ( M ◦ ) is H − s, − l ( M ◦ ). The duality between these two spaces can be realized by choosing any invertible A ∈ Ψ s, l sc ( M ◦ ). Then for u ∈ H s, l ( M ◦ ), v ∈ H − s, − l ( M ◦ ) we define ( u, v ) .. = (cid:104) Au, A − v (cid:105) L , It is easy to check that this pairing is independent of the particular invertible operator A ∈ Ψ s, l sc ( M ◦ ).We now define the spaces X s, l + and Y s, l + . We have already discussed in the Introduc-tion that Y s, l + is precisely the variable order Sobolev space H s, l + as defined above, for avariable spatial weight l + with specific properties on Σ( P ) (the behaviour away from Σ( P )is unimportant). First, we require that, for some small δ > l + takes values in [ − / − δ, − / δ ],and is equal to − / ∓ δ in a neighbourhood of R ± .This ensures elements of Y s, l + + are permitted to have outgoing oscillations of the form r − ( n − / e iλr but not incoming oscillations of the form r − ( n − / e − iλr . Second, we requirethat(2.23) l + is nonincreasing along the Hamilton flow of P within Σ( P ).Since bicharacteristics within Σ( P ) start at R − and end at R + , these two conditions arecompatible. We also define(2.24) l − = − − l + . This automatically means that l − has analogous properties to l + with the incoming andoutgoing radial sets swapped. In particular, we have(2.25) l − takes values in [ − / − δ, − / δ ],is equal to − / ± δ in a neighbourhood of R ± ,and is nondecreasing along the Hamilton flow of P within Σ( P ). Remark . Condition (2.23) is imposed so that regularity of approximate solutions of
P u = 0 can be propagated from the incoming radial set R − towards the outgoing radial set R + ( λ ), in the following section.We then define the spaces X s, l ± by (2.26) X s, l ± .. = { u ∈ H s, l ± | P u ∈ H s − , l ± +1 } , with norm(2.27) (cid:107) u (cid:107) X s, l ± = (cid:107) u (cid:107) H s, l ± + (cid:107) P u (cid:107) H s − , l ± +1 . Test modules and Sobolev spaces with module regularity.
We next introducethe ‘test modules’ with respect to which we will assume further differentiability. A testmodule M , as defined in [12], is a subspace of Ψ , ( R n ), or Ψ , ( M ◦ ) in the general case,that is closed under commutators, that contains the identity and is a module over Ψ , .(Here we adapt the definition of [12] slightly to allow order 1 in the fibre as well as thespatial slot, as is convenient here.) We shall also work only with finitely generated modules M , which have the form M = (cid:110) N (cid:88) j =0 C j A j | C j ∈ Ψ , ( M ◦ ) (cid:111) , for some fixed finite set A = Id , A , . . . A N ⊂ Ψ , ( M ◦ ), the generators of the module,which should be closed under taking commutators in the sense that[ A j , A k ] = N (cid:88) l =0 E jkl A l E jkl ∈ Ψ , ( M ◦ ) . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 15
The most important modules for us will be the modules M ± λ defined by the characteristiccondition(2.28) M ± := M ± λ = { A ∈ Ψ , : R ± ⊂ Σ , ( A ) } , where R ± are the radial sets in (2.14). Before discussing these in detail, we note that theyare examples of what will become (for us) a useful general class of modules M γ also definedby a characteristic condition given in terms of a real parameter γ . Recalling the coordinates( x, y, ν, µ ) defined near the boundary ∂M = { x = 0 } with respect to local coordinates y onan open set O ⊂ ∂M , with ( x, y ) adapted coordinates on M , let R γ = { x = 0 , | µ | y = 0 , ν = γ } . Then we define(2.29) M γ := { A ∈ Ψ , : R γ ⊂ Σ , ( A ) } . Writing Diff , ⊂ Ψ , for the subspace of differential operators, one can then choose agenerating set for these modules containing three types of operators(i) A N ∈ Diff , ( M ) and, A N = r ( D r − γ ) = − xD x − γx , on x ≤ c, where x ≤ c on our fixed collar neighborhood of ∂M .(ii) A j ∈ Diff , ( M ), j = 1 , . . . , N which are purely angular in the sense that they arein the C ∞ ( M ) linear span of the D y p for some coordinates ( y p ) on ∂M and adaptedcoordinates ( x, y ) on M , and lastly(iii) A (cid:48) k ∈ Diff ( M ◦ ), k = 1 , . . . , N which are supported in { x > c } , so in particular A (cid:48) k ∈ Ψ , ∞ sc ,and using these one has(2.30) M γ .. = (cid:110) C + N (cid:88) j =1 C j A j + N (cid:88) k =1 C (cid:48) k A (cid:48) k + C N A N | C j , C (cid:48) k ∈ Ψ , ( M ◦ ) (cid:111) . Here N + N + 1 = N , i.e. there are N + 1 total generators including A . (To be overlyconcrete, one can cover the boundary with m coordinate charts O m with coordinates y ( q ) i , i =1 , . . . , n − , q = 1 , . . . m , with the ( y ( q ) i ) i coordinates on O q , and let A j = A i,q = χ q ∂ y ( q ) i for the χ q a partition of unity subordinate to O , . . . , O m , and then choose the A (cid:48) k be anyfinite family of vector fields for which, for all q ∈ T ∗ M ◦ with x ( q ) > c , there is a A (cid:48) k with σ ( A (cid:48) k )( q ) (cid:54) = 0.) In the case γ = 0, these generators form a basis (over C ∞ ( M )) of theb-vector fields, that is, all vector fields tangent to the boundary of M (in the case of R n ,this includes all constant coefficient vector fields times a factor r ).We note that in particular the A j and A (cid:48) k are all elements of M γ (this is not a requirementfor test modules in other contexts, e.g. [11]). The operators A N in (i) and A j in (ii)taken together have the feature that for any point q ∈ Σ , ( P ) \ R + there is an elementin the module A = (cid:80) CA N + (cid:80) N j =1 C j A j such that σ , ( A )( q ) (cid:54) = 0, i.e. q ∈ Ell , ( A ).Indeed, using adapted coordinates ( x, y, ν, µ ) for one of our coordinate charts O q and writing q = (0 , y, ν, µ ), ν + | µ | = λ , if µ (cid:54) = 0 then we can choose a vector field V := χc i D y i with σ ∂M ( V )( q ) (cid:54) = 0 where this is the standard symbol of a vector field on a closed manifold and χ is supported in O q and c i ∈ R . Then σ , ( V ) = σ , ( r − V ) = ( | ν | + | µ | ) − χ (cid:80) c i µ i , andthus σ base , , ( V ) = χc i µ i , in particular σ base , , ( V )( q ) (cid:54) = 0. Similarly σ base , , ( A N ) = ν − λ, so A N is elliptic on the whole of R − .In the case that γ = ± λ , the test module M ± λ will be abbreviated M ± , as in (2.28).These test modules can be characterized as all scattering pseudodifferential operators of order (1 ,
1) that are characteristic at the radial set R ± ( λ ) given by (2.14). Analytically,the significance of these two modules is that all the generators annihilate the correspondingradial oscillation, e ± iλr .For any γ ∈ R , we can then define the weighted Sobolev spaces H s,(cid:96) ; κ M γ with moduleregularity of order κ with respect to M γ . These consist of functions in H s,(cid:96) that remain inthis space under the composition of any κ elements in M γ . For γ = ± λ , these spaces willbe denoted H s,(cid:96) ; κ ± for brevity. A distribution u will lie in H s,(cid:96) ; κ M γ if and only if u | x>c ∈ H s + κ and for any adapted coordinate system ( x, y ) we have( r ( D r − γ )) j D βy u ∈ H s,(cid:96) whenever j + | β | ≤ κ. To impart the structure of a Hilbert space to H s,(cid:96) ; κ M γ we use the generators A j , j = 0 , . . . , N of M γ , where the A j run over all the A j and A (cid:48) k in the definition of M γ above. Then, usingstandard multi-index notation A α = A α . . . A α N N , α ∈ N N +1 , we define(2.31) H s,(cid:96) ; κ M γ .. = (cid:8) u ∈ H s,(cid:96) : A α u ∈ H s,(cid:96) whenever | α | ≤ κ (cid:9) , (cid:107) u (cid:107) H s,(cid:96) ; κ M γ .. = (cid:88) | α |≤ κ (cid:107) A α u (cid:107) H s,r . In particular, when γ = ± λ , we define(2.32) H s,(cid:96) ; κ ± .. = H s,(cid:96) ; κ M ± λ . As well as the modules M γ , we shall need to consider the smaller module N ⊂ M γ (for any γ ) generated only by the purely angular and purely interior derivatives, i.e. inthe notation preceding (2.30), only the generators A = Id, the purely angular A j for j = 1 , . . . , N and the interior A (cid:48) k , k = 1 , . . . , N ,(2.33) N = (cid:110) C + N (cid:88) j =1 C j A j + N (cid:88) k =1 C (cid:48) k A (cid:48) k | C j , C (cid:48) k ∈ Ψ , ( M ◦ ) (cid:111) . In direct analogy with H s,(cid:96) ; κ M γ , writing the generators of M γ as A j , j = 0 , . . . , N and thoseof N as B k , k = 0 , . . . , N −
1, we define u ∈ H s,(cid:96) ; κ,k M γ if an only if u | x>c ∈ H s + κ + k , and forany adapted coordinate system ( x, y ) we have(2.34) H s,(cid:96) ; κ,k M γ .. = { u ∈ H s,(cid:96) : A α B β u ∈ H s,(cid:96) , | α | ≤ κ, | β | ≤ k } , (cid:107) u (cid:107) H s,(cid:96) ; κ,k M γ .. = (cid:88) | α |≤ κ, | β |≤ k (cid:107) A α B β u (cid:107) H s,(cid:96) . In particular, for γ = ± λ , we put(2.35) H s,(cid:96) ; κ,k ± .. = H s,(cid:96) ; κ,k M ± λ . Notice that we have the simple relation between these spaces:
Lemma 2.3.
Let γ, γ (cid:48) ∈ R . (2.36) H s,(cid:96) ; κ,k M γ (cid:48) = e i ( γ (cid:48) − γ ) r H s,(cid:96) ; κ,k M γ . Proof.
This follows directly from the relation ( D r − γ (cid:48) ) e i ( γ (cid:48) − γ ) r u = e i ( γ (cid:48) − γ ) r ( D r − γ ) u . (cid:3) We also note without proof the simple mapping property of scattering pseudodifferentialoperators on these spaces:
Lemma 2.4.
Let A ∈ Ψ m,(cid:96) (cid:48) sc ( M ) . Then A is a bounded operator (2.37) A : H s,(cid:96) ; κ,k ± → H s − m,(cid:96) − (cid:96) (cid:48) ; κ,k ± . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 17
The module N enjoys an important vanishing property at the radial sets, which wedescribe now. Returning to the general situation for a moment, let M be any test module,generated by A = Id , A , . . . , A N , and let P be the Helmholtz operator on R n or thegeneralized Helmholtz operator on M ◦ . We shall say that M is P -critical at the subset S ⊂ Σ( P ) if there exist C jk ∈ Ψ , ( M ) with(2.38) ix − [ A j , P ] = N (cid:88) k =0 C jk A k with σ , ( C jk ) = 0 on S for all j, k . Lemma 2.5.
The module N is P -critical at both radial sets R + ∪ R − .Proof. Consider the commutators with generators of N . The condition is trivially satisfiedfor generators microsupported away from the boundary of M , since then (2.38) is vacuousfor S = R ± , and the commutator is a differential operator of order 2, so can certainly bewritten in the form (2.38).It remains to check the commutator with tangential derivatives D y j near the boundary.The operator P takes the form near the boundary, in adapted coordinates ( x, y ),(2.39) ( x D x ) + i ( n − x ( x D x ) + x Q + x ˜ Q − λ , where Q is a differential operator of order 2 in the tangential derivatives ∂ y j with coefficientssmooth on M (in fact, it is precisely the tangential Laplacian, ∆ h ( x ) for ∂M with metric h ( x )), and ˜ Q ∈ Diff , (in fact, it has a purely radial component, obtained by differentiating h ( x ) in x ). The commutator with D y j is an operator of the form ix − [ D y j , P ] = ix [ D y j , Q ] + ix [ D y j , ˜ Q ] . The term ix [ D y j , Q ] is a second order differential operator in the tangential variables times x , therefore takes the form of a sum of terms of the form C jk A k , for 1 ≤ k ≤ n −
1, where C jk is equal to (cid:80) l b l x∂ y l , where b l ∈ C ∞ ( M ). Since the symbol of x∂ y l vanishes at R + ,this satisfies the conditions of P -criticality. The term ix [ D y j , ˜ Q ] is a scattering differentialoperator of order (1 ,
0) times x , which also satisfies the condition for P -criticality. (cid:3) By analogy with the spaces X s, l + and Y s, l + , we define(2.40) Y s,(cid:96) ; κ,k + = H s,(cid:96) ; κ,k + and (2.41) X s,(cid:96) ; κ,k + .. = { u ∈ H s,(cid:96) ; κ,k + | P u ∈ H s − ,(cid:96) +1; κ,k + } with norm(2.42) (cid:107) u (cid:107) X s,(cid:96) ; κ,k + = (cid:107) u (cid:107) H s,(cid:96) ; κ,k + + (cid:107) P u (cid:107) H s − ,(cid:96) +1; κ,k + . The main technical result of this paper is the following mapping property of the Helmholtzoperator P on these spaces with module regularity: Theorem 2.6.
Let s, (cid:96) ∈ R , and assume (cid:96) ∈ ( − / , − / . For any natural numbers κ ≥ , k ≥ , the map (2.43) P : X s,(cid:96) ; κ,k ± −→ Y s − ,(cid:96) +1; κ,k ± is an isomorphism of Hilbert spaces. In particular the inverse map, i.e., the outgoing ( + ),resp. incoming ( − ), resolvent is bounded as a map (2.44) R ( λ ± i
0) : H − s,(cid:96) +1; κ,k ± −→ H s,(cid:96) ; κ,k ± . We defer the proof to Section 3.
Multiplicative properties of weighted Sobolev spaces with module regular-ity.
We prove multiplicative properties of weighted Sobolev spaces on R n , or more generallyon asymptotically conic manifolds, with additional module regularity. We use the module M generated by b-vector fields, as that gives us the best multiplicative properties, anddeduce more general multiplicative properties as a corollary. Lemma 2.7.
Let (cid:96), (cid:96) (cid:48) ∈ R , s, κ, k ∈ N . If κ ≥ and k ≥ ( n − / , multiplication on C ∞ c ( M ) extends to a bounded bilinear map (2.45) H s,(cid:96) ; κ,k M · H s,(cid:96) (cid:48) ; κ,k M −→ H s,(cid:96) + (cid:96) (cid:48) + n/ κ,k M . Before the proof, we make some remarks concerning spaces of distributions on R n whoseregularity in an L -based Sobolev sense is a given order, κ , with some additional order k ofregularity only in certain directions. Write R n = R d × R n − d , z = ( z (cid:48) , z (cid:48)(cid:48) ), where z (cid:48) ∈ R d and z (cid:48)(cid:48) ∈ R n − d . Define(2.46) Y κ,kd ( R d × R n − d ) = { u : (cid:104) ζ (cid:105) κ (cid:104) ζ (cid:48)(cid:48) (cid:105) k ˆ u ∈ L } , with ζ = ( ζ (cid:48) , ζ (cid:48)(cid:48) ). Thus distributions in Y κ,k have κ total derivatives in L and an additional k derivatives in z (cid:48)(cid:48) in L . For our purposes, below we extend this definition to the case whereone factor is a closed manifold N of dimension n − d ; for κ, k ∈ N , and dµ N a measure on N , Y κ,kd ( R dz × N ) = { u ∈ L ( R dz × N , dz dµ N ) | D αz Au ∈ L ( R dz × N , dz dµ N ) , for all | α | ≤ κ, A ∈ Diff k ( N ) } , and below we will take d = 1 and N = ∂M .We will use the following lemma, which is proven in [13, Lemma 4.4] Lemma 2.8.
Let κ, k ∈ R . If κ > d/ and k ≥ ( n − d ) / then Y κ,kd ( R dz × N ) is an algebra.Proof of Lemma 2.7. First, we suppose (cid:96) = (cid:96) (cid:48) = s = 0. Let u, v ∈ C ∞ c ( M ◦ ), and let χ = χ ( x ) ∈ C ∞ ( M ) be a cutoff function, identically one near the boundary, and supportedin the collar neighbourhood { x < c } . We decompose the product uv as uv = (1 − χ ) uv + χ uv, and bound the H ,n/ κ,k M norm of both pieces.Since κ + k > n/ − χ has compact support, the standard algebra property for H κ + k ( M ◦ ), along with the equivalence of the H κ + k and H , ˜ (cid:96) ; κ,k M norms (any ˜ (cid:96) ∈ R ) on afixed compact subset of M ◦ , yields (cid:107) (1 − χ ) uv (cid:107) H ,n/ κ,k M ≤ C (cid:107) u (cid:107) H , κ,k M (cid:107) v (cid:107) H , κ,k M . On the other hand, u = χu, v = χv are supported in a collar neighbourhood of theboundary of ∂M , so can be viewed as belonging to C ∞ c ([0 , c ) × ∂M ). The functions˜ u ( t, y ) = u ( e t , y ) , ˜ v ( t, y ) = v ( e t , y ) , are then defined for ( t, y ) ∈ R × ∂M . Taking into account that xD x = D t if x = e t , we seethat(2.47) (cid:107) x n/ ( xD x ) j D βy w (cid:107) L ([0 ,c ) x × ∂M ; x − − n dxdµ ) = (cid:107) D jt D βy ˜ w (cid:107) L ( R t × ∂M ; dtdµ ) , for all w ∈ C ∞ c ([0 , c ) × ∂M ). Using Lemma 2.8 and (2.47), a short calculation shows (cid:107) u v (cid:107) H ,n/ κ,k M ≤ C (cid:107) u (cid:107) H , κ,k M (cid:107) v (cid:107) H , κ,k M , completing the proof of Lemma 2.7 for the case (cid:96) = (cid:96) (cid:48) = s = 0.One then proves the general case of (cid:96), (cid:96) (cid:48) ∈ R , s ∈ N using H , ˜ (cid:96) ; κ,k M = x ˜ (cid:96) H , κ,k M (any˜ (cid:96) ∈ R ) and the Leibniz rule. (cid:3) The following corollary follows from Lemma 2.7 and Lemma 2.3:
XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 19
Corollary 2.9.
Let s ∈ N and let γ , γ , . . . , γ p +1 be real parameters. Then provided κ ≥ and k ≥ ( n − / , pointwise multiplication of functions in C ∞ c ( M ) induces a boundedmultilinear map (2.48) H s,(cid:96) ; κ,k M γ · H s,(cid:96) ; κ,k M γ · · · H s,(cid:96) p ; κ,k M γp −→ H s,(cid:96) ; κ,k M γp +1 where (2.49) (cid:96) = (cid:96) + (cid:96) + · · · + (cid:96) p + ( p − n − κ. Proof.
When all γ j = 0, the result follows from applying Lemma 2.7 p − k on the right side of (2.49). In general, weuse Lemma 2.3 to write an element of H s,(cid:96) j ; κ,k M γj as e iγ j r times an element w j of H s,(cid:96) j ; κ,k M .The product is then the product of the w j , which lies in H s,(cid:96) + κ ; κ,k M , times the exponentialfactor e iγ (cid:48) r where γ (cid:48) = γ + · · · + γ p , which is in turn an element of H s,(cid:96) + κ ; κ,k M γp +1 times theexponential e i ( γ (cid:48) − γ p +1 ) r . Finally, multiplication by an exponential in r leads to a loss of κ in the spatial weight since we incur a factor of r each time the operator r ( D r − γ ) is appliedto the exponential factor. (cid:3) Remark . It is because of this loss of κ in the spatial weight that we work below withspaces where κ is as small as possible, namely κ = 1.3. Proof of Theorem 2.6
The organization of this section is as follows. We first prove invertibility of P actingbetween variable order spaces, as in (1.13). We then use this to prove invertibility on thespaces with extra module regularity, as in Theorem 2.6. The proof of invertibility is achievedby first proving that the map in question is Fredholm, and then establishing the trivialityof the kernel and cokernel. The Fredholm property is established by patching togethermicrolocal estimates of various sorts. In the elliptic region, we use a very standard ellipticestimate; on the characteristic variety, we use a standard positive commutator estimatewhere the Hamilton vector field is nonvanishing, and radial-point estimates originating withMelrose at the radial sets, where the Hamilton vector field vanishes. Theorem 3.1 ([32, Prop. 5.28]) . Let l ± ∈ S , ( sc T ∗ M ) (that is, let them be classicalscattering symbols of order (0 , on M , see Section 2) satisfy conditions (2.22) , (2.23) and (2.24) . Let s ∈ R . Then the map (1.13) is invertible. As just mentioned, the strategy is first to prove that the map (1.13) is Fredholm. Wewill prove
Lemma 3.2.
Suppose l ± ∈ S , ( sc T ∗ M ) satisfying the conditions in Theorem 3.1, let s ∈ R be arbitrary, and let M, N be such that
M < min { s, − s } and N < min { l + , l − } = 1 / − δ .Then there is a C > so that for all u ∈ X s, l ± , (3.1) (cid:107) u (cid:107) s, l ± ≤ C (cid:16) (cid:107) P u (cid:107) s − , l ± +1 + (cid:107) u (cid:107) M,N (cid:17) . Moreover, estimates (3.1) for both signs ± implies that the map (1.13) (for either sign) isFredholm.Remark . The estimate in Lemma 3.2 looks slightly different than similar estimates onefinds in the literature obtained using radial points estimates, e.g. that in [13, Eqn. 5.2], asit does not reflect the fact that near the higher decay region, e.g. near R − in the l + case,one obtains estimates where the right hand side has a term with above-threshold decay, see(3.8) below. This is often included in the global estimate, which can be written (cid:107) u (cid:107) s, l + ≤ C (cid:16) (cid:107) P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N + (cid:107) Gu (cid:107) M, l (cid:48) (cid:17) , − / < l (cid:48) < l +0 JESSE GELL-REDMAN, ANDREW HASSELL, JACOB SHAPIRO, AND JUNYONG ZHANG and WF (cid:48) ( G ) contained in a neighborhood of R − on which l + is constant. (See (3.12) below.)One can deduce the estimate in Lemma 3.2 from this estimate by an interpolation argument(see below) and we prefer (3.1) for its simplicity.To motivate the microlocal estimates below, we briefly explain why the estimates (3.1)imply the Fredholm statement. For definiteness, we consider only the + sign, so we areconsidering(3.2) P : X s, l + −→ Y s − , l + +1 . The estimate in (3.1) implies that the kernel (3.2) is finite dimensional by a standard ar-gument. Indeed, on the kernel of P we have (cid:107) u (cid:107) s, l + ≤ C (cid:107) u (cid:107) M,N , and the containment H s, l + ⊂ H M,N is compact, proving that the kernel must be finite dimensional.To show that the range is closed, consider a sequence u j in X s, l + in the subspace orthogo-nal to the kernel of P , for which P u j converges to some f ∈ H s − , l + +1+ . Then we apply (3.1)to u j , and observe that the (cid:107) u j (cid:107) M,N norms must be uniformly bounded; for if not, thenone can pass to a subsequence where (cid:107) u j (cid:107) M,N tends to infinity, rescale the u j to ˆ u j that (cid:107) ˆ u j (cid:107) M,N is fixed to be 1, and show, using the compact embedding H s, l + ⊂ H M,N again,that a subsequence of the ˆ u j converge to a limit v such that P v = 0. This implies that v = 0since the u j were chosen orthogonal to the kernel of P , and this is a contradiction since itwould imply convergence of the subsequence to zero also in the weaker norm H M,N , wherethe norm was fixed to be 1.Having thus observed that the (cid:107) u j (cid:107) M,N quantities are uniformly bounded, it followsfrom (3.1) that the (cid:107) u j (cid:107) s, l + quantities are uniformly bounded. Using the compactness ofthe inclusion H s, l + ⊂ H M,N once again we can extract a subsequence convergent in the H M,N norm, and applying (3.1) we obtain convergence in the H s, l + norm (since the P u j are converging in H s − , l + +1+ ). Thus we obtain a limit u for this subsequence, hence P u j converges to P u . So f = P u is in the range, proving closedness of the range.The last step is finite dimensionality of the cokernel. The cokernel can be identified withthose v ∈ ( H s − , l + +1 ) ∗ = H − s, − − l + with P v = 0, using the formal self-adjointness of theoperator P . Now recall from (2.24) that − − l + is precisely l − . So it suffices to prove thatthe kernel of P acting on H − s, l − is finite dimensional. But this follows from the estimate(3.1) for the opposite sign − , exactly as above. This completes the proof that the estimate(3.1), for both signs, implies the Fredholm property of the map (1.13) for the + sign (andthe argument for the − works in an exactly similar manner).3.1. Microlocal estimates.
In this section we review the specific microlocal estimates thatwe shall use to prove the estimate (3.1), which we shall refer to as the ‘Fredholm estimate’.The first type of estimate is an elliptic estimate, that applies on the elliptic set of P . Thisis very familiar from the theory of elliptic pseudodifferential operators. The only noveltyis that it applies here to the full elliptic set in the sense of the scattering calculus, thus,everywhere on the boundary of sc T ∗ M away from Σ( P ). Proposition 3.4 (microlocal elliptic regularity [32, Cor. 5.5] ) . Let u ∈ S (cid:48) and let Q , G ∈ Ψ , be such that WF (cid:48) ( Q ) ⊂ Ell( G ) ∩ Ell( P ) . Assume G P u ∈ H s − , l . Then Q u ∈ H s, l ,and for all M, N ∈ R , there is a constant C > such that if u ∈ H M,N , then (3.3) (cid:107) Q u (cid:107) s, l ≤ C ( (cid:107) G P u (cid:107) s − , l + (cid:107) u (cid:107) M,N ) . Thus, on the elliptic set, P , as an operator of order (2 , u is in H s, l microlocally on the elliptic set Ell( P ) if and onlyif P u is in H s − , l microlocally.On the characteristic set Σ( P ), wherever the Hamilton vector field H p is nonvanishing,we have propagation of singularities (as it is conventionally called — though it is moreaccurately called ‘propagation of regularity’). In the following proposition we specialize tothe variable order l + defined in Section 2.3, although only the condition (2.23) is necessaryfor the following result. Propagation of singularities goes back to H¨ormander’s paper [16], XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 21 and was first used at spatial infinity in the scattering calculus by Melrose [23], who viewedit as a microlocal version of the Mourre estimate [24]. The version we state is from [32,Thm. 5.4].
Proposition 3.5 (propagation of singularities/regularity estimate) . Let u ∈ S (cid:48) and let Q , Q (cid:48) , G ∈ Ψ , . Assume WF (cid:48) ( Q ) ⊆ Ell( G ) . Moreover, assume that (3.4) for every α ∈ WF (cid:48) ( Q ) ∩ Σ( P ) , there is a point α (cid:48) ∈ Ell( Q (cid:48) ) and a forward bicharacteristic segment γ from α (cid:48) to α such that γ ⊆ Ell( G ) .If Q (cid:48) u ∈ H s, l + and G P u ∈ H s − , l + +1 , then Q u ∈ H s, l + , and for all M, N there is
C > such that if u ∈ H M,N , then (3.5) (cid:107) Q u (cid:107) s, l + ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) s, l + + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N (cid:17) . That is, if u is in H s, l microlocally near a point α (cid:48) ∈ Σ( P ), if α is another point on thebicharacteristic γ through q , and if P u is sufficiently regular (namely, in H s − , l +1 ) along γ between α (cid:48) and α , then the regularity ‘propagates’ to α , in the sense that u is in H s, l at α , provided , in the case of a variable order l , that l is nonincreasing between α (cid:48) and α inthe direction of bicharacteristic flow. (If a variable weight is nondecreasing in the directionof bicharacteristic flow, as is the case with l − , then regularity propagates in the oppositedirection.)Neither of these estimates gives any information at the radial sets, which are the locationswithin Σ( P ) where the Hamilton vector field vanishes. At these sets, we have the followingradial point estimates, which come in two versions, one below and one above the spatialregularity level − / P u = 0. We only statethese for constant spatial weight, which suffices as we have assumed that l ± are constantin a neighbourhood of the radial sets. We also have stated this proposition with l + inmind, thus the below threshold result applies at the outgoing radial set R + , while the abovethreshold result applies at the incoming radial set R − ; for the other weight l − , the roles ofthe incoming and outgoing radial sets switch. It will help in understanding the statementsbelow to recall that R + is a sink, and R − a source, for the bicharacteristic flow, and allbicharacteristics inside Σ( P ) start at R − and end at R + . Again, these estimates were firstmade by Melrose in [23, Section 9]. Proposition 3.6 ([32, Prop. 5.27]) . (i) Below threshold regularity radial point estimate: Assume (cid:96) < − / . Let Q , Q (cid:48) , G ∈ Ψ , . Let U, U (cid:48) denote two open neighborhoods of R + with U (cid:98) U (cid:48) (cid:98) sc T ∗ ∂M M , andassume U ⊂ Ell( Q ) ⊂ WF (cid:48) ( Q ) ⊂ Ell( G ) ⊂ U (cid:48) . Assume that WF (cid:48) ( Q (cid:48) ) is con-tained in U (cid:48) \ U and that, (3.6) for every α ∈ WF (cid:48) ( Q ) ∩ (Σ( P ) \ R + ) , there is a point α (cid:48) ∈ Ell( Q (cid:48) ) and a forward bicharacteristic segment γ from α (cid:48) to α such that γ ⊆ Ell( G ) .If Q (cid:48) u ∈ H s,(cid:96) and G P u ∈ H s − ,(cid:96) +1 , then Q u ∈ H s,(cid:96) , and for all M, N there is
C > such that if u ∈ H M,N , then (3.7) (cid:107) Q u (cid:107) s,(cid:96) ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) s,(cid:96) + (cid:107) G P u (cid:107) s − ,(cid:96) +1 + (cid:107) u (cid:107) M,N (cid:17) . (ii) Above threshold regularity: Assume (cid:96), (cid:96) (cid:48) > − / and s, s (cid:48) ∈ R . Let U − (cid:98) sc T ∗ M bea sufficiently small neighborhood of R − . Then for all Q , G ∈ Ψ , such that R − ⊂ Ell( Q ) ⊂ WF (cid:48) ( Q ) ⊂ Ell( G ) ⊂ U − , if G P u ∈ H s − ,(cid:96) +1 and G u ∈ H s (cid:48) ,(cid:96) (cid:48) , then Q u ∈ H s,(cid:96) . Moreover, for all M, N ,there is
C > so that if u ∈ H M,N , then (3.8) (cid:107) Q u (cid:107) s,(cid:96) ≤ C (cid:16) (cid:107) G u (cid:107) s (cid:48) ,(cid:96) (cid:48) + (cid:107) G P u (cid:107) s − ,(cid:96) +1 + (cid:107) u (cid:107) M,N (cid:17) . Fredholm estimate.
In this subsection we explain how to piece together the microlo-cal estimates to produce a global estimate. See also Vasy, [32, Sect. 5.4.6] where this piecingtogether of estimates is discussed in terms of wavefront sets.We first note that, if U (cid:48) and U − are chosen small enough such that l + is constant, equalto − / − δ near U (cid:48) and − / δ near U − , and if we choose r to be equal to these respectivevalues in (3.7) and (3.8), then we can deduce the estimates with the variable weight l + , asfollows:(3.9) (cid:107) Q u (cid:107) s, l + ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) s, l + + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N (cid:17) and(3.10) (cid:107) Q u (cid:107) s, l + ≤ C (cid:16) (cid:107) G u (cid:107) s (cid:48) ,r (cid:48) + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N (cid:17) . This is because the ‘microlocal difference’ between, say, the norms (cid:107) Q u (cid:107) s, − / − δ and (cid:107) Q u (cid:107) s, l + is disjoint from the microlocal support of Q , so the difference can be controlledby (cid:107) u (cid:107) M,N for arbitrary M and N ; exactly the same argument applies to each of the otherterms in these two estimates.We then combine the estimates (3.3), (3.5), (3.9) and (3.10), noting that we may assumethat Q + Q + Q + Q = Id. In addition, we may assume that Q (cid:48) = Q and Q (cid:48) = Q ,as these satisfy the propagation conditions in (3.4) and (3.6). We add up these estimates,building in a large multiple of the second estimate and an even larger multiple of the fourth.Thus, for a constant C equal to the maximum constant in the four estimates (3.3), (3.5),(3.7) and (3.8), we have(3.11) (cid:107) u (cid:107) s, l + ≤ (cid:107) Q u (cid:107) s, l + + K (cid:107) Q u (cid:107) s, l + + (cid:107) Q u (cid:107) s, l + + K (cid:107) Q u (cid:107) s, l + ≤ C (cid:18) (cid:107) G P u (cid:107) s − , l + + (cid:107) u (cid:107) M,N + K (cid:16) (cid:107) Q u (cid:107) s, l + + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N (cid:17) + (cid:107) Q u (cid:107) s, l + + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N + K (cid:16) (cid:107) G u (cid:107) s (cid:48) ,(cid:96) (cid:48) + (cid:107) G P u (cid:107) s − , l + +1 + (cid:107) u (cid:107) M,N (cid:17)(cid:19)
Next, we estimate the Q and Q terms on the RHS as just done, by using (3.5), respec-tively (3.10). Notice that estimating the Q term produces an additional Q term, we againestimate using (3.10). So, finally, we obtain, for a new larger constant C (noting that s (cid:48) isarbitrary),(3.12) (cid:107) u (cid:107) s, l + ≤ C (cid:16) (cid:107) P u (cid:107) s − , r + +1 + (cid:107) u (cid:107) M,N + (cid:107) G u (cid:107) s (cid:48) ,r (cid:48) (cid:17) . Now, to handle the G term, choose (cid:96) (cid:48) = − / δ/ M < s (cid:48) < s . By Sobolev spaceinterpolation, for appropriate η ∈ (0 , (cid:107) G u (cid:107) s (cid:48) , − / δ/ ≤ (cid:107) G u (cid:107) − ηs, − / δ (cid:107) G u (cid:107) ηM,N . We can replace the norm (cid:107) G u (cid:107) s, − / δ with (cid:107) G u (cid:107) s (cid:48) , l + since l + = − / δ on the mi-crolocal support of G (which is contained in U − ). Then, by Young’s inequality, (cid:107) G u (cid:107) − ηs, l + (cid:107) G u (cid:107) ηM,N ≤ (cid:15) (cid:107) u (cid:107) s, l + + C ( (cid:15) ) (cid:107) u (cid:107) M,N for arbitrary (cid:15) > C ( (cid:15) ) is sufficiently large. The (cid:15) (cid:107) u (cid:107) s, l + term can be absorbedinto the RHS of (3.12) and the other term is a multiple of (cid:107) u (cid:107) M,N . This yields the Fredholmestimate (3.1).3.3.
Invertibility on variable order spaces.
We now prove Theorem 3.1 using the resultof Lemma 3.2. Given that the map (1.13) is Fredholm, it only remains to show that thekernel and cokernel are both trivial. In fact, due to the formal self-adjointness of the operator P , this amounts to showing that if P u = 0, and either u ∈ H s, l + or H s, l − , then u = 0. Asthe argument is essentially the same in both cases, we only consider the case that u ∈ H s, l + .So, assume that u ∈ H s, l + and P u = 0. Then, u is in H s, − / δ microlocally in aneighbourhood of R − , so we can apply Proposition 3.6, part (ii), and deduce that u is in H s,L XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 23 for arbitrarily large L microlocally near R − . The propagation theorem, Proposition 3.5,then shows that u is in H s,L for arbitrary L everywhere on Σ except, possibly, at R + . Theelliptic estimate implies that in fact, u is microlocally trivial except possibly at R + , in thesense that if A is such that WF (cid:48) ( A ) is disjoint from R + , then Au is smooth on M and allderivatives rapidly vanishing at the boundary (in particular, O ( r − N ) for every N ).We can thus apply [23, Prop. 12], which tells us that if u has wavefront set contained in R + , and P u ∈ S , then u has the form u = r − ( n − / e iλr ∞ (cid:88) j =0 r − j v j ( y ) , r → ∞ , where v j ∈ C ∞ ( ∂M ) . On the other hand, the “boundary pairing” lemma [23, Prop. 13] shows that the leadingcoefficient v in the expansion of u satisfies − iλ (cid:90) ∂M | v | = 2 Re (cid:90) M uP u. Since the right hand side is zero, v ≡ u ∈ H ∞ , − / δ ( M ) for δ small enough.Thus, u is above threshold decay at both radial sets, and again using Proposition 3.6, part(ii) (at the outgoing rather than incoming radial set) and P u = 0, it follows that u vanishesto infinity order along with its derivatives, i.e. u ∈ ˙ C ∞ ( M ). Finally, we can apply [17,Theorem 17.2.8], or alternatively [7], to deduce that u ≡
0. This completes the proof ofTheorem 3.1.
Remark . The fact that P − on these variable order spaces is equal to the action of theoutgoing resolvent is shown in [23, Sect. 11], or [29].3.4. Module regularity.
The next step is to adapt the argument above to the moduleregularity spaces H s,(cid:96) ; κ,k + instead of variable order spaces H s, l + . We shall prove each of themicrolocal estimates above in the module regularity setting. Proposition 3.8 (microlocal elliptic regularity — module version) . Let u ∈ S (cid:48) and let Q , G ∈ Ψ , be such that WF (cid:48) ( Q ) ⊂ Ell( G ) ∩ Ell( P ) . Assume G P u ∈ H s − ,(cid:96) ; κ,k + . Then Q u ∈ H s,(cid:96) ; κ,k + , and for all M, N ∈ R , there is a constant C > such that if u ∈ H M,N ,then (3.13) (cid:107) Q u (cid:107) s,(cid:96) ; κ,k ≤ C ( (cid:107) G P u (cid:107) s − ,(cid:96) ; κ,k + (cid:107) u (cid:107) M,N ) . Proof.
We prove this by induction on ( κ, k ). For ( κ, k ) = (0 ,
0) this is just Proposition 3.4.Now assume, for a given ( κ, k ) that the result is true for all ( κ (cid:48) , k (cid:48) ) < ( κ, k ) in the sense that κ (cid:48) ≤ κ , k (cid:48) ≤ k and ( κ (cid:48) , k (cid:48) ) (cid:54) = ( κ, k ). Then, generators A , . . . , A m of M + and B , . . . , B l of N , we have(3.14) A · · · A m B · · · B l Q u = Q A · · · A m B · · · B l u + m (cid:88) j =1 A · · · [ A j , Q ] · · · A m B · · · B l u + l (cid:88) j =1 A · · · A m B · · · [ B j , Q ] · · · B l u. We can shift the commutator factors to the left of the product modulo double commutatorfactors, shift the double commutator factors to the left modulo triple commutator factorsand so on. Notice that all of these multiple commutator factors are order (0 , (cid:48) ( Q ), hence contained in the elliptic set of G . Notefirst that G P A . . . A m B . . . B l u ∈ H s − ,(cid:96) . Indeed this can be seen by writing the operatorin terms of commutators as(3.15) G P A · · · A m B · · · B l = A · · · A k B · · · B l G P + m (cid:88) j =1 A · · · [ G P, A j ] · · · A m B · · · B l + l (cid:88) j =1 A · · · A m B · · · [ G P, B j ] · · · B l , so using that [ G P, A j ] , [ G P, B j ] ∈ Ψ , , we can therefore apply Proposition 3.4 to obtain(3.16) (cid:107) A · · · A m B · · · B l Q u (cid:107) s,(cid:96) (cid:46) (cid:107) G P A · · · A m B · · · B l u (cid:107) s − ,(cid:96) + commutator terms + (cid:107) u (cid:107) M,N . and we can perform a similar process as above, shifting the commutator factors to the leftmodulo double commutator factors, shifting those to the left modulo triple commutatorfactors, and so on. Each of these multiple commutator factors are order (2 , (cid:107) A · · · A m B · · · B l Q u (cid:107) s,(cid:96) (cid:46) (cid:107) A · · · A m B · · · B l G P u (cid:107) s − ,(cid:96) + (cid:88) (cid:107) ˜ C • (cid:89) A • (cid:89) B • u (cid:107) s,(cid:96) + (cid:88) (cid:107) C • (cid:89) A • (cid:89) B • u (cid:107) s − ,(cid:96) + (cid:107) u (cid:107) M,N , (3.17)where C • ∈ Ψ , and ˜ C • ∈ Ψ , are multi-commutators with wavefront set contained inWF (cid:48) ( Q ), and we have fewer than κ + k factors of the A • and the B • in total. The otherterms in (3.16) are estimated similarly. We thus obtain(3.18) (cid:107) A · · · A m B · · · B l Q u (cid:107) s,(cid:96) (cid:46) (cid:107) G P u (cid:107) s − ,(cid:96) ; κ,k + (cid:88) ( κ (cid:48) ,k (cid:48) ) < ( κ,k ) (cid:107) G (cid:48) u (cid:107) s,(cid:96) ; κ (cid:48) ,k (cid:48) + (cid:107) u (cid:107) M,N , where G (cid:48) is chosen so that WF (cid:48) ( Q ) ⊂ Ell( G (cid:48) ) ⊂ WF (cid:48) ( G (cid:48) ) ⊂ Ell( G ) and WF( I − G (cid:48) ) ∩ WF (cid:48) ( Q ) = ∅ . These conditions imply that G (cid:48) C = C modulo an operator of order( −∞ , −∞ ) which contributes to the (cid:107) u (cid:107) M,N term. We apply the inductive assumptionto the term G (cid:48) u , where G (cid:48) now plays the role of Q , and arrive at(3.19) (cid:107) A · · · A k B · · · B l Q u (cid:107) s,(cid:96) (cid:46) (cid:107) G P u (cid:107) s − ,(cid:96) ; κ,k + (cid:107) u (cid:107) M,N . After summing over all possible choices of the A . . . A m and the B . . . B l we obtain (3.13). (cid:3) Proposition 3.9 (propagation of regularity estimate — module version) . Let u ∈ S (cid:48) andlet Q , Q (cid:48) , G ∈ Ψ , . Assume WF (cid:48) ( Q ) ⊆ Ell( G ) \ ( R + ∪ R − ) . Moreover, assume that (3.4) holds. If Q (cid:48) u ∈ H s,(cid:96) ; κ,k + and G P u ∈ H s − ,(cid:96) +1; κ,k + , then Q u ∈ H s,(cid:96) ; κ,k + , and for all M, N there is
C > such that if u ∈ H M,N , then (3.20) (cid:107) Q u (cid:107) s,(cid:96) ; κ,k ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) s,(cid:96) ; κ,k + (cid:107) G P u (cid:107) s − ,(cid:96) +1; κ,k + (cid:107) u (cid:107) M,N (cid:17) . Remark . Here we included the extra assumption that WF (cid:48) ( Q ) is disjoint from theradial sets, so that the modules M + and N both become elliptic on WF (cid:48) ( Q ), in the sensethat at each point of WF (cid:48) ( Q ) there exists a module element of N (and hence also M + )that is elliptic. (It was not necessary to include this assumption in Proposition 3.5, but wecould have done so as the Proposition gives no information at the radial sets.) Proof.
When the modules are elliptic the proof becomes almost trivial. We note that themodule norm (cid:107)·(cid:107) s,(cid:96) ; κ,k is equivalent, microlocally on WF (cid:48) ( Q ), to the (cid:107)·(cid:107) s + κ + k,(cid:96) + κ + k norm.So the Proposition is actually equivalent to the previous one, with a shift in orders s and (cid:96) by κ + k . (cid:3) In the next proposition, all operators have microlocal support in a compact region of sc T ∗ M by assumption, thus disjoint from fibre-infinity. Hence the differential order is irrel-evant for both the operators and the spaces. We write it ∗ to emphasize this irrelevance. Proposition 3.11 (Radial point estimates — module version) . (i) Below threshold regularity radial point estimate: Assume (cid:96) < − / . Let Q , Q (cid:48) , G ∈ Ψ , . Let U, U (cid:48) denote two open neighborhoods of R + with U (cid:98) U (cid:48) (cid:98) sc T ∗ ∂M M , andassume that U ⊂ Ell( Q ) ⊂ WF (cid:48) ( Q ) ⊂ Ell( G ) ⊂ U (cid:48) . Assume that WF (cid:48) ( Q (cid:48) ) iscontained in U (cid:48) \ U and that (3.6) holds. If Q (cid:48) u ∈ H ∗ ,(cid:96) ; κ,k + and G P u ∈ H ∗ ,(cid:96) +1; κ,k + ,then Q u ∈ H ∗ ,(cid:96) ; κ,k + , and for all M, N there is
C > such that if u ∈ H M,N , then (3.21) (cid:107) Q u (cid:107) ∗ ,(cid:96) ; κ,k ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) ∗ ,(cid:96) ; κ,k + (cid:107) G P u (cid:107) ∗ ,(cid:96) +1; κ,k + (cid:107) u (cid:107) M,N (cid:17) . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 25 (ii)
Above threshold regularity: Assume (cid:96), (cid:96) (cid:48) > − / . Let U − (cid:98) sc T ∗ ∂M M be a sufficientlysmall neighborhood of R − . Then for all Q , G ∈ Ψ , such that R − ⊂ Ell( Q ) ⊂ WF (cid:48) ( Q ) ⊂ Ell( G ) ⊂ U − , if G P u ∈ H ∗ ,(cid:96) +1; κ,k + and G u ∈ H ∗ ,(cid:96) (cid:48) ; κ,k + , then Q u ∈ H ∗ ,(cid:96) ; κ,k + . Moreover, for all M, N , there is
C > so that if u ∈ H M,N , then (3.22) (cid:107) Q u (cid:107) ∗ ,(cid:96) ; κ,k ≤ C (cid:16) (cid:107) G u (cid:107) ∗ ,(cid:96) (cid:48) ; κ,k + (cid:107) G P u (cid:107) ∗ ,(cid:96) +1; κ,k + (cid:107) u (cid:107) M,N (cid:17) . Moreover, all of the above holds with H ∗ ,(cid:96) ; κ,k + , R ± replaced by H ∗ ,(cid:96) ; κ,k − , R ∓ .Proof. In this case, the argument is more elaborate than the previous two proofs, and relieson the construction of a positive commutator. The key fact we use is Lemma 2.5, that is,the P -positivity (in fact P -criticality) of module N . It is very similar to the argument from[12, Section 6], where test modules were introduced. To avoid a long exposition about testmodules and how positive commutator estimates are used to prove module regularity, wewill use Section 6 of [12] as a basis and only indicate the minor differences that arise in thepresent case.We first prove (3.21) for κ = 0, that is, when only the module N is involved. We fixa basis A = Id , . . . , A N of the module N , and use the notation A α , α = ( α , . . . , α N ) amulti-index, for the operator A α . . . A α N N and A α,(cid:96) for the operator x − (cid:96) A α . We note that the A α , as α ranges over all multi-indicesof length m , together with Id forms a basis for N m , the vector space of sums of m -foldproducts of elements of N , as a module over Ψ , .We prove the estimate by induction on k , the module order. For k = 0 the result isprecisely Proposition 3.6. We now assume inductively that result has been proved for all k (cid:48) < k . The positive commutator estimate arises from the following operator identity,which is equation (6.16) in [12]. In the following, Q is arbitrary, but we will choose it to bean operator which is microlocally equal to the identity on WF (cid:48) ( Q ), and with WF (cid:48) ( Q ) ⊂ Ell( G ). In the following identity, the C jk are defined by the commutators of P with basiselements A j , as in (2.38).(3.23) i [ A ∗ α,(cid:96) +1 / Q ∗ QA α,(cid:96) +1 / , P ] = A α,(cid:96) Q ∗ C + C ∗ + N (cid:88) j =1 α j ( C jj + C ∗ jj ) QA α,(cid:96) + (cid:88) | β | = k,β (cid:54) = α A α,(cid:96) Q ∗ C αβ QA β,(cid:96) + (cid:88) | β | = k,β (cid:54) = α A β,(cid:96) Q ∗ C ∗ αβ QA α,(cid:96) + A ∗ α,(cid:96) Q ∗ E α,(cid:96) + E ∗ α,(cid:96) QA α,(cid:96) + A ∗ α,(cid:96) +1 / i [ Q ∗ Q, P ] A α,(cid:96) +1 / where σ base , ( C ) | R + = − λ (2 (cid:96) + 1) σ base , ( C αβ ) | R + = 0 , C αβ ∈ Ψ , ( M ) ,E α,(cid:96) = x − (cid:96) E α , E α ∈ N k − . (3.24)The key point above is that the symbol of C , arising from the 2 νx∂ x component of minusthe Hamilton vector field from (2.20) hitting the x − (cid:96) factor, has a definite sign near R + —positive for (cid:96) less than the threshold exponent − /
2. Moreover, the P -criticality of N meansthat the diagonal operators C jj have symbols vanishing at R + — cf. (2.38). Similarly, theoff-diagonal terms C αβ vanish at R + due to (2.38). Now, we define a matrix C (cid:48) = ( C (cid:48) αβ ) ofoperators, as the indices α , β vary over multi-indices of length k , as follows: for α (cid:54) = β , C (cid:48) αβ = C αβ + C ∗ βα and on the diagonal, we define(3.25) C (cid:48) αα = C + C ∗ + N (cid:88) j =1 α j ( C jj + C ∗ jj ) . Thus, due to (3.24), the symbol of C (cid:48) at R + is diagonal with positive entries, and it istherefore positive as an matrix, provided that the microlocal support of Q is sufficientlyclose to R + . This means that we can write Q ∗ C (cid:48) Q = Q ∗ ( B ∗ B + G ) Q, where B is a matrix of operators of order ( ∗ ,
0) and G a matrix of operators of order ( ∗ , − u we also write Au for ( QA α,(cid:96) u ), regarded as a column vector indexed bymulti-indices α of length m . Thus, in this compact notation we can write the first two lineson the RHS of (3.23) as A ∗ ( B ∗ B + G ) A .Now we follow the argument of the proof of [12, Proposition 6.7]. We let u (cid:48) be an elementof H ∗ ,(cid:96) ; κ, . We have, in matrix notation,(3.26) (cid:88) | α | = k (cid:10) u (cid:48) , i [ A ∗ α,(cid:96) +1 / Q ∗ QA α,(cid:96) +1 / , P ] u (cid:48) (cid:11) = (cid:107) BAu (cid:48) (cid:107) + (cid:104) Au (cid:48) , GAu (cid:48) (cid:105) + (cid:88) | α | = k (cid:0) (cid:104) QA α,(cid:96) u (cid:48) , E α,(cid:96) u (cid:48) (cid:105) + (cid:104) E α,(cid:96) u (cid:48) , QA α,(cid:96) u (cid:48) (cid:105) (cid:1) + (cid:88) | α | = k (cid:104) A α,(cid:96) u (cid:48) , F A α,(cid:96) u (cid:48) (cid:105) . Here F = [ Q ∗ Q, P ] has operator wavefront set in U (cid:48) \ U , in particular, disjoint from R + ,and for elements which are understood to be vectors of distributions, the inner product isthe direct sum inner product.We may assume that there is a ˜ Q microlocally equal to the identity on WF (cid:48) ( Q ), and withWF (cid:48) ( ˜ Q ) ⊂ Ell( G ) ). We can therefore write (cid:104) A α,(cid:96) u (cid:48) , F A α,(cid:96) u (cid:48) (cid:105) = (cid:104) ˜ QA α,(cid:96) u (cid:48) , F A α,(cid:96) u (cid:48) (cid:105) + (cid:104) Eu (cid:48) , A α,(cid:96) u (cid:48) (cid:105) where E is order ( −∞ , −∞ ). Making this substitution, rearranging and applying theCauchy-Schwarz inequality, followed by the inequality ab ≤ (cid:15)a + (cid:15) − b , we obtain, for some C independent of u (cid:48) , and all norms understood to be L -norms unless otherwise stated,(3.27) (cid:107) BAu (cid:48) (cid:107) ≤ (cid:88) α (cid:12)(cid:12)(cid:12) (cid:104) QA α,(cid:96) +1 / u (cid:48) , QA α,(cid:96) +1 / P u (cid:48) (cid:105) (cid:12)(cid:12)(cid:12) + (cid:15)C (cid:16) (cid:107) Au (cid:48) (cid:107) + (cid:88) α (cid:107) ˜ QA α,(cid:96) u (cid:48) (cid:107) (cid:17) + (cid:15) − (cid:16) (cid:107) GAu (cid:48) (cid:107) + 2 (cid:88) α (cid:0) (cid:107) E α,(cid:96) u (cid:48) (cid:107) + (cid:107) F A α,(cid:96) u (cid:48) (cid:107) (cid:1)(cid:17) + C (cid:107) u (cid:48) (cid:107) M,N . We can treat the commutator term similarly (this is not done in [12], since there it wasassumed that
P u (cid:48) is Schwartz). Notice that Q − x − / Qx / is an operator of order (0 , − A α,(cid:96) +1 / this gives us an element of the ( k − M + ,which we shall write (abusing notation somewhat) as E α,(cid:96) +1 / . Then we have (cid:104) QA α,(cid:96) +1 / u (cid:48) , QA α,(cid:96) +1 / P u (cid:48) (cid:105) = (cid:104) QA α,(cid:96) u (cid:48) , x − / QA α,(cid:96) +1 / P u (cid:48) (cid:105) + (cid:104) E α,(cid:96) u (cid:48) , x − / QA α,(cid:96) +1 / P u (cid:48) (cid:105) and therefore (cid:12)(cid:12)(cid:12) (cid:104) QA α,(cid:96) +1 / u (cid:48) , QA α,(cid:96) +1 / P u (cid:48) (cid:105) (cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:107) QA α,(cid:96) u (cid:48) (cid:107) + (cid:107) E α,(cid:96) u (cid:48) (cid:107) +(1+ (cid:15) − ) (cid:107) x − / QA α,(cid:96) +1 / P u (cid:48) (cid:107) . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 27
Summing over α and combining this with (3.27) we have(3.28) (cid:107) BAu (cid:48) (cid:107) ≤ (cid:15) (cid:16) (cid:107) Au (cid:48) (cid:107) + (cid:88) α (cid:107) ˜ QA α,(cid:96) u (cid:48) (cid:107) (cid:17) + (cid:15) − (cid:16) (cid:107) GAu (cid:48) (cid:107) + 3 (cid:88) α (cid:0) (cid:107) E α,(cid:96) u (cid:48) (cid:107) + (cid:107) F A α,(cid:96) u (cid:48) (cid:107) + 2 (cid:107) x − / QA α,(cid:96) +1 / P u (cid:48) (cid:107) (cid:1)(cid:17) + C (cid:107) u (cid:48) (cid:107) M,N . The terms proportional to (cid:15) can be absorbed in the LHS, up to a term of the form C (cid:107) u (cid:48) (cid:107) M,N .In fact, on the microlocal support of Q , B has a microlocal inverse, that we will denote B − (despite not being an actual inverse of B ). So we have A = B − BA + E (cid:48) , where E (cid:48) hasorder ( −∞ , −∞ ). Then, estimating B − by its operator norm, we can absorb the (cid:107) Au (cid:48) (cid:107) terms provided (cid:15) is small compared to (cid:107) B − (cid:107) , while the E (cid:48) term only contributes a multipleof (cid:107) u (cid:48) (cid:107) M,N .We now notice that GA can be treated as being in the ( k − G has order ( ∗ , − E α,(cid:96) term, can be estimated using theinductive assumption. Similarly, we can commute the F factor to the right of the A α,(cid:96) (upto terms in the ( k − Q (cid:48) since it can bewritten F = F (cid:48) Q (cid:48) + E (cid:48)(cid:48) for some E (cid:48)(cid:48) of spatial order −∞ . In exactly the same way, we cancommute the Q to the right of the A α,(cid:96) +1 / factor and then replace it with G . Similarly,on the LHS of (3.28), Q can be moved to the right of the A α,(cid:96) , and then B can be removedjust as for the Au (cid:48) term above. Moreover, as Q is microlocally equal to the identity onWF (cid:48) ( Q ), we can replace it with Q on the LHS. After these manipulations, we obtain theestimate(3.29) (cid:107) Q u (cid:48) (cid:107) ∗ ,(cid:96) ;0 ,k ≤ C (cid:16) (cid:107) G P u (cid:48) (cid:107) ∗ ,(cid:96) +1;0 ,k + (cid:107) Q (cid:48) u (cid:48) (cid:107) ∗ ,(cid:96) ;0 ,k + (cid:107) u (cid:48) (cid:107) M,N (cid:17) . Now we let u (cid:48) = u (cid:48) ( η ) := (1 + ηr ) − u , u ∈ H ∗ ,(cid:96) ;0 ,k − , for η > u (cid:48) ∈ H ∗ ,(cid:96) ;0 ,k + for each η >
0, so the above computation is valid. Assuming that Q (cid:48) u ∈ H ∗ ,(cid:96) ;0 ,k + and G P u ∈ H ∗ ,(cid:96) +1;0 ,k + , then the RHS of (3.29) stays bounded as η →
0. Therefore, theLHS also stays bounded, and using the strong convergence of (1 + ηr ) − to the identity asin [12, Lemma 4.3], we see that we obtain estimate (3.29) also with u (cid:48) = u .Next, we shall show a slight strengthening of (3.21) for κ = 0: we shall show that(3.30) (cid:107) Q u (cid:107) ∗ ,(cid:96) ;1 ,k − ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) ∗ ,(cid:96) ;0 ,k + (cid:107) G P u (cid:107) ∗ ,(cid:96) +1;0 ,k + (cid:107) u (cid:107) M,N (cid:17) , k ≥ . To show this, notice that we have already shown that Q u is in H ∗ ,(cid:96) ;0 ,k + with the requiredestimate. So it only remains to prove an estimate for the additional element A + := r ( D r − λ )that is in M + but not in N . To do this, we write P u in the form
P u = ( D r + λ )( D r − λ ) u + i ( n − D r ur + r − Ω u, where Ω involves only tangential differentiation of order at most two, with coefficientssmooth on M . By assumption, this is in H ∗ ,(cid:96) +1;0 ,k + microlocally on WF (cid:48) ( Q ). We rear-range as ( D r + λ ) r − A + u = P u − i ( n − D r ur − r − Ω u, and apply Lemma 2.4. On the RHS, notice that P u by assumption is in H ∗ ,(cid:96) +1;0 ,k + microlo-cally on WF (cid:48) ( Q ); that D r u/r is in H ∗ ,(cid:96) +1;0 ,k + microlocally on WF (cid:48) ( Q ); and finally that r − Ω u is in H ∗ ,(cid:96) +1;0 ,k − microlocally on WF (cid:48) ( Q ) (by viewing one of the ∂ y i factors in Ωas being in the module N , and a second factor, r − ∂ y j , as in Ψ , , leaving an additionalvanishing factor r − ). We see then that( D r + λ ) r − A + u ∈ H ∗ ,(cid:96) +1;0 ,k − microlocally on WF (cid:48) ( Q ). Now using the ellipticity of D r + λ on this set, we find that r − A + u ∈ H ∗ ,(cid:96) +1;0 ,k − , which is equivalent to A + u ∈ H ∗ ,(cid:96) ;0 ,k − microlocally on WF (cid:48) ( Q ). Together with the fact that we have already Q u ∈ H ∗ ,(cid:96) ;0 ,k + shows that Q u ∈ H ∗ ,(cid:96) ;1 ,k − ,with the required estimate.Now we show, by induction on κ , that we have the following estimate for all ( κ, k ) providedthat k ≥ (cid:107) Q u (cid:107) ∗ ,(cid:96) ; κ +1 ,k − ≤ C (cid:16) (cid:107) Q (cid:48) u (cid:107) ∗ ,(cid:96) ; κ,k + (cid:107) G P u (cid:107) ∗ ,(cid:96) +1; κ,k + (cid:107) u (cid:107) M,N (cid:17) . We have already shown this for κ = 0. So given κ >
0, assume that (3.31) has alreadybeen proved for all ( κ (cid:48) , k ) with κ (cid:48) < κ . The only thing left to prove is to show that A + u ∈ H ∗ ,(cid:96) ; κ,k − microlocally on WF (cid:48) ( Q ), with the corresponding estimate. By induction,this follows if we show that A + u is in H ∗ ,(cid:96) ; κ − ,k + microlocally on WF (cid:48) ( Q (cid:48) ) and P A + u is in H ∗ ,(cid:96) +1; κ − ,k + microlocally on WF (cid:48) ( G ). The first statement is immediate from the inductionhypothesis. For the second, we commute P and A + , obtaining, for certain constants a, b , P A + u = A + P u − D r r − A + u + r − ( aD r u + bu ) − r − Ω u, where Ω involves only tangential differentiation of order at most two, with coefficientssmooth on M . Since by assumption, P u ∈ H ∗ ,(cid:96) +1; κ,k + , microlocally on WF (cid:48) ( Q ) (an as-sumption in force throughout this paragraph, but which we shall omit repeating), we have A + P u ∈ H ∗ ,(cid:96) +1; κ − ,k + . Since we already know that u ∈ H ∗ ,(cid:96) ; κ,k + , we see that the term r − D r A + u is in H ∗ ,(cid:96) +1; κ − ,k + . Similarly the term r − ( aD r u + bu ) is in H ∗ ,(cid:96) +1; κ,k + . Finally,we split the two tangential derivatives of Ω as above to see that r − Ω u ∈ H ∗ ,(cid:96) +1; κ − ,k + . Weconclude that P A + u is in H ∗ ,(cid:96) +1; κ − ,k + and therefore A + u is in H ∗ ,(cid:96) ; κ,k − , which combinedwith u ∈ H ∗ ,(cid:96) ; κ,k + shows that u ∈ H ∗ ,(cid:96) ; κ +1 ,k − , with the corresponding estimate. The proofof (3.31) is complete, and immediately implies (3.21).We next turn to the proof of (3.22). This works quite differently in relation to the twomodules. At the incoming radial set R − , the module M + is elliptic, while N is characteris-tic. The effect of the κ th power of the module M + is thus just to increase the spatial order (cid:96) by κ . So, without loss of generality, we may assume that κ = 0.We then employ a very similar argument to the one above. Notice that, instead of having σ base , ( C ) = − λ (2 (cid:96) + 1) at the radial set, as above, we now have σ base , ( C ) = λ (2 (cid:96) + 1).On the other hand, now (cid:96) > − /
2, so the 2 (cid:96) + 1 factor has also switched sign, so this symbolremains positive at the radial set (now R − ). Using the fact that the module N is P -critical,we find that the matrix C (cid:48) in this case is again positive definite at the incoming radial set.Then we run the same argument as above, with the following twist: In this case, the F term arising from [ Q ∗ Q, P ] has the same sign as C (cid:48) , namely it is positive, as it arises fromminus the (rescaled) Hamilton vector field H p hitting σ ( Q ) . Taking σ ( Q ) to be a functiononly of | µ | h at x = 0 we see that from (2.20) that − H p ( σ ( Q ) ) is nonnegative. Since thishas the same sign as that of C (cid:48) , to leading order, we can discard this term up to a lowerorder term. This lower order term accounts for the presence of (cid:107) G u (cid:107) ∗ ,(cid:96) (cid:48) ,k in the estimate;at first sight it appears that we could take (cid:96) (cid:48) = (cid:96) − / (cid:96) (cid:48) asmuch as we like), but the regularization required to make the estimate hold requires that (cid:96) (cid:48) is greater than the threshold value of − /
2. See the proof of [32, Proposition 5.26], between(5.61) and (5.62), for the details of the regularization step.This proves the proposition for H ∗ ,(cid:96) ; κ,k + . For the space H ∗ ,(cid:96) ; κ,k − , where the index κ nowindicates module regularity with respect to M − , a similar argument applies. Here, it isimportant that the module N is P -critical and not just P -positive. In this case, the abovethreshold estimate is localized near R + . For M − regularity we proceed as in the proof of M + for H ∗ ,(cid:96) ; κ,k + , i.e. we use the ellipticity of M − elements at R + . For N regularity, in theexpression (3.25) we have that σ base , ( C ) = − λ (2 (cid:96) + 1) where now (cid:96) > − / C is actually negative in this case. For C (cid:48) αα to have a sign, we must then knowthat the other terms defining it do are not too positive. The fact that N is P -critical meansthey vanish, and hence the matrix C (cid:48) is negative definite, rather than positive definite, near R + . Then all proceeds as above. (cid:3) XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 29
Corollary 3.12.
Suppose − / < (cid:96) (cid:48) < (cid:96) < − / and κ ≥ . Then, under the sameassumptions as in part (ii) of Proposition 3.11, we have (3.22) .Proof. This follows since the module M + is elliptic at R − . So when κ ≥
1, the estimate(3.22) is equivalent to the same estimate with (cid:96), (cid:96) (cid:48) increased by κ , and κ set to zero. Underthe assumption that − / < (cid:96) (cid:48) < (cid:96) < − / κ ≥ (cid:3) Invertibility on module regularity spaces.
Our final piece of preparation for theproof of Theorem 2.6 is the following result relating our module regularity spaces to variableorder spaces.
Lemma 3.13.
Assume l + ∈ C ∞ ( sc T ∗ M ) satisfies (2.22) and (2.23) . Then for (cid:15) < δ and (cid:96) = − / − (cid:15) , (3.32) κ ≥ ⇒ H s,(cid:96) ; κ,k + ⊂ H s, l + . Proof.
Since H s,(cid:96) ; κ,k + ⊂ H s,(cid:96) ;1 , , to show (3.32), it suffices to assume κ = 1 and k = 0. Let U be a small neighborhood of R + near which l + = − / − δ , and let V (cid:98) U be a smallerneighborhood of R + . For each q ∈ sc T ∗ M \ V , there is an element A q of M + , ellipticon a neighborhood U q of q . Form a partition of unity subordinate to the cover of sc T ∗ M ,consisting of U and finitely many of the U q , say, U q , . . . U q m . We take { Q q j } mj =1 , Q ∈ Ψ , tobe the corresponding left quantizations of the microlocal cutoffs that comprise the partitionof unity. Clearly, Qu ∈ H s,(cid:96) , since u ∈ H s,(cid:96) . Thus Qu ∈ H s, l + since, on WF (cid:48) ( Q ), we have l + = − / − δ < − / − (cid:15) = (cid:96) .On the other hand, by the assumption of module regularity, each A q j u ∈ H s,(cid:96) . Because A q j is order (1 ,
1) and WF (cid:48) ( Q q j ) ⊂ Ell( A q j ), microlocal elliptic regularity (Proposition 3.4)asserts that Q q j u ∈ H s +1 ,(cid:96) +1 . Then we note that H s +1 ,(cid:96) +1 ⊂ H s, l + since (cid:96) + 1 = 1 / − (cid:15) ≥− / δ = max l + for δ sufficiently small. (cid:3) Remark . The point of this Lemma is that the different behaviour at the incoming andoutgoing radial set is enforced by the module regularity instead of by a variable weightfunction, due to the fact that the module M + is elliptic at the incoming radial set R − butcharacteristic at the ougoing radial set R + .We are now in a position to prove Theorem 2.6. Proof of Theorem 2.6.
Let s ∈ R , (cid:96) ∈ ( − / , − / κ ≥ k be given. We first combinethe estimates (3.13), (3.20), (3.21) and (3.22) (in the latter case, for (cid:96) ∈ ( − / , − /
2) asallowed by Corollary 3.12). This is done exactly as in Section 3.2, so we do not repeat theargument. We obtain, for u ∈ H M,N such that
P u ∈ H s,(cid:96) ; κ,k + , that u ∈ H s,(cid:96) ; κ,k + and(3.33) (cid:107) u (cid:107) s,(cid:96) ; κ,k ≤ C (cid:16) (cid:107) P u (cid:107) s − ,(cid:96) ; κ,k + (cid:107) u (cid:107) M,N (cid:17) . We next use Lemma 3.13 to assert that H s,(cid:96) ; κ,k + ⊂ H s, l + , provided (cid:96) is sufficiently close to − /
2. The proof Lemma 3.13 may be trivially modified to also show H s − ,(cid:96) ; κ,k + ⊂ H s − , l + +1 .Thus we have the following diagram:(3.34) X s, l + P (cid:47) (cid:47) Y s − , l + +1 X s,(cid:96) ; κ,k + (cid:63)(cid:31) (cid:79) (cid:79) Y s − ,(cid:96) +1; κ,k + (cid:63)(cid:31) (cid:79) (cid:79) . Our goal is to show that the restriction of P to X s,(cid:96) ; κ,k yields a bijection X s,(cid:96) ; κ,k →Y s − ,(cid:96) +1; κ,k + . Injectivity follows immediately since the top row is an injective map. To showsurjectivity, we suppose f ∈ Y s − ,(cid:96) +1; κ,k + = H s − ,(cid:96) +1; κ,k + . In, particular f ∈ H s − , l + +1 . So,by surjectivity of the top row there is u ∈ X s, l + with P u = f . Then, thanks to (3.33), we see that, provided ( M, N ) are sufficiently small, u ∈ H s,(cid:96) ; κ,k + . As a bounded bijection, themap X s,(cid:96) ; κ,k + → Y s − ,(cid:96) +1; κ,k + is automatically a Hilbert space isomorphism. (cid:3) Proof of the Main Theorem
In this section, we find a nonlinear eigenfunction with prescribed incoming data by findinga fixed point of the map in (1.16), which we shall show is a contraction map on the space X ,(cid:96) ;1 ,k + . Here and below we fix s = 2, and let k be any integer strictly larger than ( n − / (cid:96) = − / − δ , for some fixed δ with 0 < δ ≤ (4 p ) − ≤ / Linear eigenfunction.
Fix f ∈ H k +2 ( ∂M ). Let u be the unique solution to the freeequation P ( λ ) u = 0 , subject to the condition that the coefficient on its incoming part at infinity is f . By [23], if f is C ∞ , we have a decomposition(4.1) u = r − ( n − / ( e − iλr g − + e iλr g + ) , where the g ± ∈ C ∞ ( M ), and g − | ∂M = f . For f of finite regularity, this expansion onlymakes sense in a distributional sense. In our case, we only use the ‘leading part’ of thisexpansion to express the linear eigenfunction u as an element in H ,(cid:96) ;1 ,k − ⊕ H ,(cid:96) ;1 ,k + . Thus,let u − ( r, y ) = χ ( r ) r − ( n − / e − iλr f ( y ) , u + := u − u − . Here χ is a cutoff function, supported in r > R and identically equal to 1 for r ≥ R . Byinspection, we see that u − ∈ H ,(cid:96) ;1 ,k − ∩ H ,(cid:96) ;1 ,k +2 − . Moreover, it is clear that if (cid:107) f (cid:107) H k +2 ( ∂M ) is sufficiently small, then u − is small in the norms of both these spaces.Moreover, by direct calculation we have( P u − )( r, y ) = ˜ χ ( r ) r − ( n +3) / e − iλr g ( r, y ) , x = r − where ˜ χ is a similar cutoff function, equal to 1 on supp χ and supported in r > R , and g ( r, y ) is a smooth function of r − with values in H k ( ∂M ). The key point is the gain of twopowers of r − as r → ∞ . It follows that P u − ∈ H ,(cid:96) +2;1 ,k − . Now we want to view this as an element of H ,(cid:96) +1;1 ,k + ; to accommodate the M + -moduleregularity of order k = 1, we lose one order of vanishing. Thus P u − ∈ H ,(cid:96) +1;1 ,k + = Y ,(cid:96) +1;1 ,k + . Then we claim that u + is equal to − R ( λ + i P u − ). Indeed, v := u − − R ( λ + i (cid:0) P u − (cid:1) solves P v = 0and according to Theorem 2.6, R ( λ + i (cid:0) P u − (cid:1) is in X ,(cid:96) ;1 ,k + ; in particular, it has no incomingdata at order r − ( n − / due to the module regularity at the incoming radial set ( M + iselliptic at this set). Thus, v is the linear eigenfunction with incoming data equal to f , so itcoincides with u by definition. Hence u − u − = v − u − = − R ( λ + i P u − ).4.2. Contraction mapping on X ,(cid:96) ;1 ,k + . We return to the discussion of Section 1.2. There,it was explained how finding a nonlinear eigenfunction amounts to finding a fixed point ofthe map Φ given by(4.2) Φ( w ) = u + + R ( λ + i (cid:0) N [ u − + w ] (cid:1) . Let us check that a fixed point w provides us with a nonlinear eigenfunction u := u − + w .Adding u − to both sides of (4.2), we obtain(4.3) Φ( w )+ u − = w + u − = u + + u − + R ( λ + i (cid:0) N [ u − + w ] (cid:1) = u + R ( λ + i (cid:0) N [ u − + w ] (cid:1) . Thus,(4.4) u = u + R ( λ + i (cid:0) N [ u ] (cid:1) . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 31
Now we apply P to both sides. This annihilates u and we find that P u = N [ u ] , as claimed.We now show that Φ is a contraction mapping on X ,(cid:96) ;1 ,k + , provided that (cid:107) f (cid:107) H k +2 ( ∂M ) issufficiently small (and hence (cid:107) u − (cid:107) H ,(cid:96) ;1 ,k + is small), and provided that w is small. The firstthing to check is that Φ is a mapping on this space.We have already seen that u + lies in this space, since u + is in H ,(cid:96) ;1 ,k + and P u + = − P u − is in the space H ,(cid:96) +1;1 ,k + from the discussion above.Next, recall that the nonlinear term N [ v ] is a product of ˜ p ≥ p factors of the form Qv or Qv , where Q is a scattering differential operator of order (2 ,
0) (in the case of R n itjust means a combination of the usual coordinate partial derivatives multiplied by C ∞ ( M )functions; see Remark 1.5). This is, therefore, a product of ˜ p factors, each of which lies in H ,(cid:96) ;1 ,k + . It follows that N [ u − + w ] is a finite sum of products of such factors. We havealready seen that u − lies in H ,(cid:96) ;1 ,k − , and w by assumption lies in H ,(cid:96) ;1 ,k + . Also, we noticethat complex conjugation is an involution between H ,(cid:96) ;1 ,k − and H ,(cid:96) ;1 ,k + . So N [ u − + w ] is asum of products of factors, each of which lies in H ,(cid:96) ;1 ,k − or H ,(cid:96) ;1 ,k + . Applying Corollary 2.9,we find that the product lies in H ,(cid:96) (cid:48) ;1 ,k + , provided that for all ˜ p ≥ p ,(4.5) (cid:96) (cid:48) ≤ ˜ p(cid:96) + (˜ p − n − . We would like to know when this product is in H ,(cid:96) +1;1 ,k . This is the case provided that(4.6) (cid:96) + 1 ≤ ˜ p(cid:96) + (˜ p − n − ⇐⇒ ≤ (˜ p − (cid:96) + (˜ p − n . Since (cid:96) ≥ − /
8, the RHS is increasing in ˜ p . So it is only necessary to demand (4.6) for˜ p = p . Since (cid:96) < − /
2, a necessary condition is that(4.7) 2 < ( p − n − , which is precisely condition (1.7). When this holds, we automatically have(4.8) 5 / ≤ ( p − n − n and p are integers. It is straightforward to check that provided 0 < δ ≤ (4 p ) − ,given (4.8), we have (4.6), and in fact, in anticipation of Proposition 4.1, we note that wecan take (cid:96) (cid:48) = 3 / X ,(cid:96) ;1 ,k + , provided that theprescribed incoming data f is small in H k +2 ( ∂M ). We have(4.9) Φ( w ) − Φ( w ) = R ( λ + i (cid:0) N [ u − + w ] − N [ u − + w ] (cid:1) . Since N is a monomial of degree p , the RHS is a sum of terms the form Q ( w − w ) orits complex conjugate, times a monomial of degree p − Q (cid:48) u − , Q (cid:48)(cid:48) w or Q (cid:48)(cid:48)(cid:48) w or their complex conjugates, where the Q , Q (cid:48) , etc., are scattering differential operators oforder (2 , η > f (cid:55)→ u − , u − = χ ( r ) r − ( n − / e − iλr f ( y ) , is a bounded map from H k +2 ( ∂M ) to H ,(cid:96) ;1 ,k − , so we may assume that u − is sufficientlysmall in this norm, say ≤ η . Supposing that w and both w are both less than η in thenorm H ,(cid:96) ;1 ,k + , then we find that N [ u − + w ] − N [ u − + w ] is a finite number, say c ( p ), ofterms each of which is in H ,(cid:96) +1;1 ,k + by Corollary 2.9 with norm in this space bounded by C (cid:107) w − w (cid:107) H ,(cid:96) ;1 ,k + (cid:16) (cid:107) u − (cid:107) H ,(cid:96) ;1 ,k − + (cid:107) w (cid:107) H ,(cid:96) ;1 ,k + + (cid:107) w (cid:107) H ,(cid:96) ;1 ,k + (cid:17) p − . Applying R ( λ + i P acting between Y ,(cid:96) +1;1 ,k + and X ,(cid:96) ;1 ,k + , the norm ofΦ( w ) − Φ( w ) in X ,(cid:96) ;1 ,k + is at most (cid:107) w − w (cid:107) H ,(cid:96) ;1 ,k + times c ( p ) C (cid:107) R ( λ + i (cid:107) (3 η ) p − . It follows that provided η is chosen small enough so that c ( p ) C (cid:107) R ( λ + i (cid:107) (3 η ) p − isstrictly less than 1, the map Φ is a contraction on the ball of radius η centred at theorigin in X ,(cid:96) ;1 ,k + . By the contraction mapping theorem, we deduce the existence of a fixedpoint w ∈ X ,(cid:96) ;1 ,k + . In view of the previous discussion this furnishes us with a nonlineareigenfunction u − + w .4.3. Outgoing boundary data.
Continuing the proof of Theorem 1.4, we show that w , thefixed point of Φ given incoming data f , has zero incoming boundary data and well-definedoutgoing boundary data. Proposition 4.1.
Let u = u − + w be the nonlinear eigenfunction constructed above given f ∈ H k +2 ( ∂M ) . Then u has an asymptotic expansion at infinity of the form (4.11) u = r − ( n − / (cid:16) e − iλr f ( y ) + e iλr b ( y ) + O ( r − (cid:15) (cid:48) ) (cid:17) , r → ∞ , for some (cid:15) (cid:48) > , where b ∈ H k ( ∂M ) .Proof. We know that
P u = N [ u ], and, in view of the discussion below (4.7), that the RHSis in H , / ,k + .The proof is therefore completed by the following lemma. (cid:3) Lemma 4.2.
Suppose (cid:96) ∈ ( − / , − / , k > ( n − / , and that w ∈ H ,(cid:96) ;1 ,k +2+ satisfiesthe equation (4.12) P w = F, F ∈ H , / (cid:15) ;1 ,k + ( M ) for some (cid:15) > . Then lim r →∞ r ( n − / e − iλr w ( r, · ) exists in H k ( ∂M ) . Letting b ∈ H k ( ∂M ) denote the limit, we have (4.13) r ( n − / e − iλr w ( r, · ) − b = O ( r − (cid:15) (cid:48) ) in H k ( ∂M ) , r → ∞ for any (cid:15) (cid:48) < (cid:15) .Remark . Since H k ( ∂M ) embeds into C ( ∂M ), due to the assumption k > ( n − /
2, thisalso shows that we have the asymptotic (4.13) pointwise in y ∈ ∂M . Proof.
It suffices to decompose w = w + + w − , where r ( n − / e − iλr w + ( r, · ) − b = O ( r − (cid:15) (cid:48) ) in H k ( ∂M ) , r → ∞ and r ( n − / e + iλr w − ( r, · ) = O ( r − (cid:15) (cid:48) ) in H k ( ∂M ) , r → ∞ . We do this by choosing a pseudodifferential cutoff B ∈ Ψ , such that B is microlocally equalto the identity near a neighbourhood U of the outgoing radial set R + , and microlocally equalto zero outside some slightly larger neighbourhood V , i.e., for some open neighborhoods U ⊂ V of R + we have WF (cid:48) ( I − B ) ∩ U = ∅ , WF (cid:48) ( B ) ⊂ V . Then we set(4.14) w + = Bw, w − = (Id − B ) w. From (4.12) we get(4.15)
P w + = P ( Bw ) = BF + [ P, B ] w. We claim that the RHS is in H , / (cid:15) ;0 ,k + . Certainly this is true for the term BF since F isin this space and B ∈ Ψ , . For the term [ P, B ] w , we claim that(4.16) [ P, B ] = r − A where A ∈ M + . Since w ∈ H ,(cid:96) ;1 ,k +2+ , this would imply Aw ∈ H ,(cid:96) ;0 ,k +2+ , and so [ P, B ] w = r − Aw ∈ H ,(cid:96) +2;0 ,k + ⊂ H , / (cid:15) ;0 ,k + for (cid:15) sufficiently small. But (4.16) follows immediately for V sufficiently small since r [ P, B ] has order (1 , V \ U . Thereforeit is characteristic at R + , which is a sufficient condition for an operator of order (1 ,
1) tobelong to M + . XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 33
Write ˜ w + = χ ( r ) r ( n − / e − iλr w + , where χ is supported in r > R and identically 1 near r ≥ R , and we assume R is sufficiently large that the support of χ is contained in a collarneighbourhood of the boundary. Our first goal is to show that ˜ w + ( r, y ) has a limit b ( y ) as r → ∞ , and that ˜ w + ( r, y ) − b ( y ) = O ( r − (cid:15) (cid:48) ). To do this, we write the operator P in theform (2.39):(4.17) P = D r − λ − i ( n − r D r + r − Q + r − ˜ Q, where Q is a second order differential operator involving only tangential D y j derivatives,and ˜ Q is a scattering differential operator of order (1 , (cid:16) D r + λ (cid:17)(cid:16) D r − λ − i ( n − r (cid:17) w + = BF + [ P, B ] w + i ( n − r ( r ( D r − λ )) w + + (cid:16) r − Q + r − ˜ Q (cid:17) w + . Notice that the RHS is in H , / (cid:15) ;0 ,k + , using assumption (4.12) for F , the M + moduleregularity of order 1 for the term involving r ( D r − λ ) ∈ M + , and the N module regularity oforder ≥ D r + λ is elliptic everywhereexcept at the set { ν = − λ } ; in particular, it is elliptic on WF (cid:48) ( B ), provided that V is takensufficiently small. Thus we may invert this operator microlocally, obtaining(4.19) (cid:16) D r − λ − i ( n − r (cid:17) w + ∈ H , / (cid:15) ;0 ,k + . Now, observing that D r ˜ w + = r ( n − / e − iλr (cid:16) D r − λ − i ( n − r (cid:17) w + + ( D r χ ) r ( n − / e − iλr w + , we find that D r ˜ w + ∈ H , / (cid:15) − ( n − / ,k + ⇐⇒ D r ˜ w + ∈ r n/ − − (cid:15) L (cid:0) (([ R, ∞ ) , r n − dr ); H k ( ∂M ) (cid:1) where we used the support property of D r χ for the inclusion in H ,n/ − (cid:15) ;0 ,k + of the secondterm. We can express this with respect to the measure dr as follows: D r ˜ w + ∈ r − / − (cid:15) L (cid:0) ([ R, ∞ ) , dr ); H k ( ∂M ) (cid:1) ⊂ r − (cid:15) (cid:48) L (cid:0) ([ R, ∞ ) , dr ); H k ( ∂M ) (cid:17) . Notice that, by assumption, w is locally H in r with values in H k ( ∂M ), so it is thereforein H k ( ∂M ) for each fixed r . We can therefore integrate to infinity and find that b ( y ) = ˜ w + ( R, y ) + (cid:90) ∞ R ∂ r ˜ w + ( r (cid:48) , y ) dr (cid:48) is well-defined as an element of H k ( ∂M ). Moreover,˜ w + ( r, y ) − b ( y ) = − (cid:90) ∞ r ddr (cid:48) ˜ w + ( r (cid:48) , y ) dr (cid:48) = O H k ( ∂M ) ( r − (cid:15) (cid:48) ) . A very similar argument can be applied to the w − term. We define ˜ w − = r ( n − / e iλr w − and compute as above. However, we switch the sign of λ in (4.18) to obtain(4.20) (cid:16) D r − λ (cid:17)(cid:16) D r + λ − i ( n − r (cid:17) w − = (Id − B ) F − [ P, B ] w + i ( n − r ( D r + λ ) w − + (cid:16) − r − ∆ ∂M + r − Q (cid:48) (cid:17) w − . where Q (cid:48) = Q (cid:48) + r − Q (cid:48) , Q (cid:48) i ∈ Ψ , , i = 1 , Q (cid:48) a scattering differential operatorinvolving only tangential derivatives. On the microlocal support of Id − B , the module M + is elliptic, hence w − is actually in H ,(cid:96) +1;0 ,k +2 in this region. So the third term on the RHS is in H ,(cid:96) +2;0 ,k +2+ . We see that the RHS is in H , / (cid:15) ;0 ,k + as before. We may assume that D r − λ is elliptic on the microsupport of Id − B , so we may invert it microlocally to obtain(4.21) (cid:16) D r + λ − i ( n − r (cid:17) w − ∈ H , / (cid:15) ;0 ,k + . Now, observing that D r ˜ w − = r ( n − / e iλr (cid:16) D r + λ − i ( n − r (cid:17) w + + ( D r χ ) r ( n − / e iλr w + , we find that D r ˜ w − ∈ r − / − (cid:15) L (cid:0) (([ R, ∞ ) , dr ); H k ( ∂M ) (cid:1) ⊂ r − (cid:15) (cid:48) L (cid:0) (([ R, ∞ ) , dr ); H k ( ∂M ) . The rest of the argument can be followed to obtain a limit b − ( y ) = lim r →∞ ˜ w − ( r, y )in H k ( ∂M ), with ˜ w − ( r, y ) − b − ( y ) = O H k ( ∂M ) ( r − (cid:15) (cid:48) ) . This is only possible if b − vanishes identically. Indeed, otherwise w − would fail to be in H s, − / ; however, since WF (cid:48) (Id − B ) ∩ R + = ∅ and the module M + is elliptic off R + ,(Id − B ) w ∈ H s,(cid:96) +1 . This completes the proof of the lemma. (cid:3) Uniqueness.
To complete the proof of Theorem 1.4, we show that the solution u obtained above is unique in the following sense: Proposition 4.4.
Suppose that (1.7) is satisfied, that N satisfies the conditions of The-orem 1.4, and that f is sufficiently small in H k +2 ( ∂M ) , so that the proof above of theexistence of a nonlinear eigenfunction u with incoming data f is valid. Let u − be given interms of f by (4.10) .Then the solution u is unique in the following sense. Let u , u satisfy P u i = N [ u i ] andassume u i − u − = w i both lie in H ,(cid:96) ;1 ,k + . Then there exists η > such that (cid:107) f (cid:107) H k +2 ( ∂M ) , (cid:107) w (cid:107) H ,(cid:96) ;1 ,k + , (cid:107) w (cid:107) H ,(cid:96) ;1 ,k + < η = ⇒ u = u . Proof.
Let u and u be nonlinear eigenfunctions as in the proposition. It suffices to showthat the corresponding w i are both fixed points of the map Φ, since a contraction map hasonly one fixed point.We first note that the w i are in X ,(cid:96) ;1 ,k + . First note that u − ∈ H ,(cid:96) ;1 ,k − and w i is byassumption in H ,(cid:96) ;1 ,k + , so, as we saw above, this means that N [ u i ] = N [ u − + w i ] ∈ H ,(cid:96) +1;1 ,k + .Since P w i = − P u − + N [ u i ] ∈ H ,(cid:96) +1;1 ,k + , this confirms that w i ∈ X ,(cid:96) ;1 ,k + .Next, from P ( u − + w i ) = N [ u i ] , we apply R ( λ + i
0) and note that R ( λ + i P w i = w i since w i ∈ X ,(cid:96) ;1 ,k + , while, as we haveseen, R ( λ + i P u − = − u + . Therefore, − u + + w i = R ( λ + i N [ u − + w i ]and this rearranges to Φ( w i ) = w i for each i . The proof is complete. (cid:3) References [1] H. Berestycki and P.-L. Lions. Nonlinear scalar field equations. I. Existence of a ground state.
Arch.Rational Mech. Anal. , 82(4):313–345, 1983. 6[2] H. Berestycki and P.-L. Lions. Nonlinear scalar field equations. II. Existence of infinitely many solutions.
Arch. Rational Mech. Anal. , 82(4):347–375, 1983. 6[3] A. Hassell C. Guillarmou and A. Sikora. Restriction and spectral multiplier theorems on asymptoticallyconic manifolds.
Analysis and PDE , 6:893–950, 2013. 6[4] H. Christianson and J. Marzuola. Existence and stability of solitons for the nonlinear schr¨odingerequation on hyperbolic space.
Nonlinearity , 23:89–106, 2010. 6[5] Alberto Enciso and Daniel Peralta-Salas. Bounded solutions to the Allen-Cahn equation with level setsof any compact topology.
Anal. PDE , 9(6):1433–1446, 2016. 6
XISTENCE AND ASYMPTOTICS OF NONLINEAR HELMHOLTZ EIGENFUNCTIONS 35 [6] Gilles Evequoz and Tobias Weth. Dual variational methods and nonvanishing for the nonlinearHelmholtz equation.
Adv. Math. , 280:690–728, 2015. 6[7] Richard Froese, Ira Herbst, Maria Hoffmann-Ostenhof, and Thomas Hoffmann-Ostenhof. On the ab-sence of positive eigenvalues for one-body Schr¨odinger operators.
J. Analyse Math. , 41:272–284, 1982.23[8] Jesse Gell-Redman, Nick Haber, and Andr´as Vasy. The Feynman propagator on perturbations ofMinkowski space.
Comm. Math. Phys. , 342(1):333–384, 2016. 5, 6[9] C. Guillarmou and A. Hassell. Uniform sobolev estimates for non-trapping metrics.
Journal of Inst.Math. Jussieu , 13:599–632, 2014. 6[10] Susana Guti´errez. Non trivial L q solutions to the Ginzburg-Landau equation. Math. Ann. , 328(1-2):1–25, 2004. 1, 4, 6[11] N. Haber and A. Vasy. Propagation of singularities around a Lagrangian submanifold of radial points.
Bulletin de la SMF, arXiv:1110.1419 , To appear. 15[12] Andrew Hassell, Richard Melrose, and Andr´as Vasy. Spectral and scattering theory for symbolic po-tentials of order zero.
Adv. Math. , 181(1):1–87, 2004. 5, 14, 25, 26, 27[13] Peter Hintz and Andr´as Vasy. Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitterand Minkowski spacetimes.
Anal. PDE , 8(8):1807–1890, 2015. 6, 18, 19[14] Peter Hintz and Andr´as Vasy. Global analysis of quasilinear wave equations on asymptotically Kerr–deSitter spaces.
Int. Math. Res. Not. IMRN , (17):5355–5426, 2016. 7[15] Peter Hintz and Andr´as Vasy. The global non-linear stability of the Kerr–de Sitter family of black holes.
Acta Math. , 220(1):1–206, 2018. 7[16] Lars H¨ormander. On the existence and the regularity of solutions of linear pseudo-differential equations.
Enseignement Math. (2) , 17:99–163, 1971. 20[17] Lars H¨ormander.
The analysis of linear partial differential operators III . Springer, Berlin, 1985. 7, 23[18] Dietrich Hfner, Peter Hintz, and Andrs Vasy. Linear stability of slowly rotating Kerr black holes.
Preprint, arXiv:1906.00860 , 2019. 7[19] C. E. Kenig, A. Ruiz, and C. D. Sogge. Uniform Sobolev inequalities and unique continuation for secondorder constant coefficient differential operators.
Duke Math. J. , 55(2):329–347, 1987. 4[20] Yong Liu and Juncheng Wei. On the Helmholtz equation and Dancer’s-type entire solutions for nonlinearelliptic equations.
Proc. Amer. Math. Soc. , 147(3):1135–1148, 2019. 6[21] Rainer Mandel, Eugenio Montefusco, and Benedetta Pellacci. Oscillating solutions for nonlinearHelmholtz equations.
Z. Angew. Math. Phys. , 68(6):Art. 121, 19, 2017. 6[22] J. Marzuola and E.M. Taylor. Higher dimensional vortex standing waves for nonlinear schr¨odingerequations,.
Commu. PDE , 41:398–446, 2016. 6[23] Richard B. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces.In
Spectral and scattering theory (Sanda, 1992) , volume 161 of
Lecture Notes in Pure and Appl. Math. ,pages 85–130. Dekker, New York, 1994. 5, 6, 7, 12, 21, 23, 30[24] D Mourre. Absence of singular spectrum for certain self-adjoint operators.
Comm. Math. Phys. ,78(1):391–400, 1981. 21[25] Zeev Nehari. On a nonlinear differential equation arising in nuclear physics.
Proc. Roy. Irish Acad.Sect. A , 62:117–135 (1963), 1963. 6[26] Cesare Parenti. Operatori pseudo-differenziali in R n e applicazioni. Ann. Mat. Pura Appl. (4) , 93:359–389, 1972. 6[27] Peter A. Tomas. A restriction theorem for the Fourier transform.
Bull. Amer. Math. Soc. , 81:477–478,1975. 4[28] A. Vasy. Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a La-grangian approach.
Preprint, arXiv:1905.12809 , 2019. 7[29] Andras Vasy. Limiting absorption principle on riemannian scattering (asymptotically conic) spaces, alagrangian approach.
Preprint, arXiv:1905.12587 . 7, 23[30] Andras Vasy. Resolvent near zero energy on riemannian scattering (asymptotically conic) spaces.
Preprint, arXiv:1808.06123 . 7[31] Andr´as Vasy. Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an ap-pendix by Semyon Dyatlov).
Invent. Math. , 194(2):381–513, 2013. 4, 6[32] Andr´as Vasy. A minicourse on microlocal analysis for wave propagation. In
Asymptotic analysis ingeneral relativity , volume 443 of
London Math. Soc. Lecture Note Ser. , pages 219–374. CambridgeUniv. Press, Cambridge, 2018. 1, 5, 6, 7, 19, 20, 21, 22, 28
School of Mathematics and Statistics, University of Melbourne
E-mail address : [email protected] Mathematical Sciences Institute, Australian National University
E-mail address : [email protected]
Mathematical Sciences Institute, Australian National University
E-mail address : [email protected]
Department of Mathematics, Beijing Institute of Technology and, Cardiff University, UK
E-mail address ::