Limiting absorption principle on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach
LLIMITING ABSORPTION PRINCIPLE ON RIEMANNIANSCATTERING (ASYMPTOTICALLY CONIC) SPACES, ALAGRANGIAN APPROACH
ANDR ´AS VASY
Abstract.
We use a Lagrangian perspective to show the limiting absorptionprinciple on Riemannian scattering, i.e. asymptotically conic, spaces, and theirgeneralizations. More precisely we show that, for non-zero spectral parameter,the ‘on spectrum’, as well as the ‘off-spectrum’, spectral family is Fredholm infunction spaces which encode the Lagrangian regularity of generalizations of‘outgoing spherical waves’ of scattering theory, and indeed this persists in the‘physical half plane’. Introduction and outline
The purpose of this paper is to prove the limiting absorption principle, concern-ing the limit of the resolvent at the spectrum on appropriate function spaces, forLaplace-like operators on Riemannian scattering (asymptotically conic at infinity)spaces (
X, g ) using a description that focuses on the outgoing radial set, which inphase space corresponds to the well-known outgoing spherical waves in Euclideanscattering theory. Thus, the result is a precise description of the limiting resolventin terms of mapping properties on spaces of (finite regularity) Lagrangian distri-butions, where now the Lagrangian is conic in the base manifold, rather than thefibers of the cotangent bundle as familiar from standard microlocal analysis. Sucha result is well suited for the analysis of waves, especially at the ‘radiation face’, or‘scri’, see [5], though we do not pursue this aspect here. We explain more of the his-toric context of Lagrangian analysis in scattering theory below, but already remarkthat recently such a Lagrangian analysis proved very effective in the description ofinternal waves in fluids by Dyatlov and Zworski [2].The basic setting is Melrose’s scattering pseudodifferential algebra Ψ ∗ , ∗ sc ( X ), see[11], which for X the radial compactification of R n (to a ball) goes back to Parentiand Shubin [14, 15], and which corresponds to any standard quantization of symbolswith the property | D αz D βζ a ( z, ζ ) | ≤ C αβ (cid:104) z (cid:105) r −| α | (cid:104) ζ (cid:105) s −| β | , with r the decay order and s the differential order. The key property of thisalgebra is that the principal symbol is taken modulo (cid:104) z (cid:105) − (cid:104) ζ (cid:105) − better terms, thusalso captures decay at infinity; see Section 2 for more detail. Date : July 16, 2019. Original version: May 29, 2019.2000
Mathematics Subject Classification.
Primary 35P25; Secondary 58J50, 58J40, 35P25,35L05, 58J47.The author gratefully acknowledges partial support from the NSF under grant numbers DMS-1361432 and DMS-1664683 and from a Simons Fellowship. a r X i v : . [ m a t h . A P ] J u l ANDRAS VASY
With this in mind, recall first that for σ (cid:54) = 0 real, elements of the spectral family∆ g − σ are not elliptic in this algebra due to the part of the principal symbolcapturing decay (essentially as | ζ | − σ can vanish), rather have a non-degeneratereal principal symbol with a source-to-sink Hamilton flow within their characteristicset (the zero set of the principal symbol). One obtains a Fredholm problem usingvariable decay order weighted scattering Sobolev spaces (which are the standardSobolev space on R n , albeit of a microlocally variable order), where the order onlymatters on the characteristic set, needs to be monotone along the Hamilton flow,and be greater than a threshold value ( − / i − i σ , thus Im σ = 2 i Re σ Im σ shows that Im σ ≥ i σ >
0, and the − i σ <
0) correspond to propagatingestimates forward along the Hamilton flow, i.e. having high decay order at thesource, vs. propagating estimates backwards, i.e. having high decay order at thesink. This can then be extended uniformly to zero energy, see [22], using secondmicrolocal methods discussed below.A different way of arranging a Fredholm setup is by considering a fixed decayorder Sobolev space which is lower than the threshold order, but adding to it extraLagrangian regularity relative to elements of Ψ , characteristic on the outgoingradial set (referred to as ‘module regularity’, see [6, 7, 4], see also [2]). (Since theLagrangian is at finite points in the fibers of the scattering cotangent bundle, i.e.where ζ is finite in the Euclidean picture, the differential order is immaterial; onlythe decay order matters.) For instance, the background decay order can be taken1 / / x − ( x D x + σ ) , x − ( xD y j ) = D y j , with x the boundary defining function, y j local coordinates on the boundary, andthe metric is to leading order warped product type relative to these. (So in theasymptotically Euclidean setting, one could have x = r − , and y local coordinateson the sphere with respect to the standard spherical coordinate decomposition.)Thus, the domain space is the modified version of { u ∈ x − L : x − ( x D x + σ ) u, D y j u ∈ x − L } , with the modification just so that the operator maps it to the target which simplyhas 1 additional order of decay(1.1) { u ∈ x L : x − ( x D x + σ ) u, D y j u ∈ x L } . Using the variable order Fredholm theory it is straightforward to show (using avariable order that is in ( − / ,
0) at the incoming radial set, and is < − IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 3 the additional regularity domain. However, it is harder to directly run Fredholmarguments since these involve duality and inversion, and the additional moduleregularity gives a dual space for which it is harder to prove estimates since the dualof, for instance, the space (1.1) is x L + x − ( x D x + σ ) x L + (cid:88) j D y j x L , see [12, Appendix A].A way around this difficulty with dualization, which we pursue in this paper,is to use even stronger, second microlocal, spaces, see [22, Section 5] in this scat-tering context, and see [1, 16, 19] in different contexts. Recall that these secondmicrolocal techniques play a role in precise analysis at a Lagrangian, or more gen-erally coisotropic, submanifold. These second microlocal techniques were employedin [22] due to the degeneration of the principal symbol at zero energy, correspond-ing to the quadratic vanishing of any dual metric function at the zero section; thechosen Lagrangian is thus the zero section, really understood as the zero sectionat infinity. In a somewhat simpler way than in other cases, this second microlocal-ization at the zero section is accomplished by simply using the b-pseudodifferentialoperator algebra of Melrose [13]. In an informal way, this arises by blowing up thezero section of the scattering cotangent bundle at the boundary, though a moreprecise description (in that it makes sense even at the level of quantization, thespaces themselves are naturally diffeomorphic) is the reverse: blowing up the cor-ner (fiber infinity over the boundary) of the b-cotangent bundle: see Section 2 formore detail and additional references. (But the basic point is that the scatteringvector fields x D x , xD y j are replaced by totally characteristic, or b-, vector fields xD x , D y j .) In [22] this was used to show a uniform version of the resolvent es-timates down to zero energy using variable differential order b-pseudodifferentialoperators. Indeed, the differential order of these, cf. the aforementioned blow-up ofthe corner, corresponds to the scattering decay order away from the zero section,thus this allows the uniform analysis of the problem to zero energy. However, herethe decay order (of the b-ps.d.o.) is also crucial, for it corresponds to the spaces onwhich the exact zero energy operator (i.e. with σ = 0) is Fredholm, which, with H b denoting weighted b-Sobolev spaces relative to the scattering (metric) L -density,are H ˜ r,l b → H ˜ r − ,l +2b with | l + 1 | < n − , where ˜ r is the variable order (which isirrelevant at zero energy since the operator is elliptic in the b-pseudodifferentialalgebra then). (The more refined, fully 2-microlocal, spaces H s,r,l sc , b , see Section 2,corresponding to the blow-up of the corner, have three orders: sc-differential s ,sc-decay/b-differential r and b-decay l ; using all of these is convenient, as the oper-ators are sc-differential-elliptic, so one can use easily that this order, s , is essentiallyirrelevant; this modification is not crucial.)Now, for σ (cid:54) = 0 real, one can work in a second microlocal space by simplyconjugating the spectral family P ( σ ) by e iσ/x (this being the multiplier from theright), with the point being that this conjugation acts as a canonical transformationof the scattering cotangent bundle, moving the outgoing radial set to the zerosection, see Sections 2-3. Then the general second microlocal analysis becomesb-analysis. Indeed, note that this conjugation moves x − ( x D x + σ ) , resp. x − ( xD y j ) , ANDRAS VASY to x − ( x D x ) = xD x , resp. x − ( xD y j ) = D y j , so the Lagrangian regularity becomes b-differential-regularity indeed. Notice thatthe conjugate of the simplest model operator P ( σ ) = ( x D x ) + i ( n − x ( x D x ) + x ∆ h − σ ∈ Diff ( X ) ⊂ Diff ( X ) , which is the Laplacian of the conic metric g = x − dx + x − h ( y, dy ) (considerednear the ‘large end’, x = 0), is thenˆ P ( σ ) = e − iσ/x P ( σ ) e iσ/x = ( x D x − σ ) + i ( n − x ( x D x − σ ) + x ∆ y − σ = ( x D x ) − σ ( x D x ) + i ( n − x ( x D x ) − i ( n − xσ + x ∆ y ∈ x Diff ( X ) , which has one additional order of vanishing in this b-sense (the factor of x on theright). (This is basically the effect of the zero section of the sc-cotangent bundlebeing now in the characteristic set.) Moreover, to leading order in terms of theb-decay sense, i.e. modulo x Diff ( X ), this is the simple first order operator − σx (cid:16) xD x + i n − (cid:17) . (In general, decay is controlled by the normal operator of a b-differential operator,which arises by setting x = 0 in its coefficients after factoring out an overall weight,and where one thinks of it as acting on functions on [0 , ∞ ) x × ∂X , of which [0 , δ ) x × ∂X is identified with a neighborhood of ∂X in X .) This is non-degenerate for σ (cid:54) = 0in that, on suitable spaces, it has an invertible normal operator; of course, this isnot an elliptic operator, so some care is required. Notice that terms like ( x D x ) and σx D x have the same scattering decay order, i.e. on the front face of the blownup b-corner they are equally important. Thus, we use real principal type plusradial points estimates at finite points in the scattering cotangent bundle, togetherwith a radial point type analysis of the zero section, but now interpreted in thesecond microlocal setting. This gives, for the general class of operators discussedin Section 3, which includes the spectral family of the Laplacian of Riemannianscattering metrics, with Im α ± ( σ ) = 0 in the case of the operator discussed above: Theorem 1.1.
Suppose that P ( σ ) satisfies the hypotheses of Section 3 and let α + ( σ ) , α − ( σ ) be as given there, see (3.10) and (3.11) ; thus, Im α ± ( σ ) = 0 if P ( σ ) is formally self-adjoint, and ∓ σxα ± ( σ ) is the subprincipal symbol at ∓ σ dxx .Suppose also that ˜ r + (cid:96) + 1 / − Im α − ( σ ) > , (cid:96) + 1 / − Im α + ( σ ) < , and K a compact subset of { σ ∈ C : Im σ ≥ , σ (cid:54) = 0 } . For σ ∈ K , let ˆ P ( σ ) = e − iσ/x P ( σ ) e iσ/x . Then ˆ P ( σ ) : { u ∈ H ˜ r,(cid:96) b : ˆ P ( σ ) u ∈ H ˜ r,(cid:96) +1b } → H ˜ r,(cid:96) +1b is Fredholm, and if P ( σ ) = P ( σ ) ∗ for σ ∈ R \ { } then it is invertible, with thisinverse being the ± i resolvent limit (in the sense of σ ± i ) of P ( σ ) correspondingto ± Re σ > , and the norm of ˆ P ( σ ) − as an element of L ( H ˜ r,(cid:96) +1b , H ˜ r,(cid:96) b ) is uni-formly bounded for σ ∈ K . Furthermore, invertibility is preserved under suitablysmall perturbations of P ( σ ) . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 5
These statements also hold if both inequalities on the orders are reversed: ˜ r + (cid:96) + 1 / − Im α − ( σ ) < , (cid:96) + 1 / − Im α + ( σ ) > , provided one also reverses the sign of Im σ to Im σ ≤ , and thus takes { σ ∈ C :Im σ ≤ , σ (cid:54) = 0 } above.Furthermore, the statements hold on second microlocal spaces, recalled in Sec-tion 2, ˆ P ( σ ) : { u ∈ H s,r,(cid:96) sc , b : ˆ P ( σ ) u ∈ H s − ,r +1 ,(cid:96) +1sc , b } → H s − ,r +1 ,(cid:96) +1sc , b with (cid:96) + 1 / − Im α + ( σ ) < , r + 1 / − Im α − ( σ ) > , as well as with (cid:96) + 1 / − Im α + ( σ ) > , r + 1 / − Im α − ( σ ) < (again reversing the sign of Im σ ).Remark . Note that H ˜ r,(cid:96) b = H ˜ r, ˜ r + (cid:96),(cid:96) sc , b , so ˜ r + (cid:96) is the scattering decay orderaway from the zero section. Thus the statements on H b and H sc , b spaces in thetheorem are very similar, including in terms of the restrictions on the orders, withthe main advantage of the H sc , b statements being the ability to use ellipticity inthe sc-differential sense, making the order s arbitrary. Remark . Here α ± ( σ ) are functions on ∂X , and the stated inequalities, such as (cid:96) + 1 / − Im α + ( σ ) <
0, are assumed to hold at every point on ∂X .In the case of the vector valued version, i.e. if P ( σ ) acts on sections of a vectorbundle equipped with a fiber inner product, such as on scattering one-forms orsymmetric scattering 2-cotensors, the statement and the proof are completely par-allel, with the only change that now α ± ( σ ) are valued in endomorphisms, and theinequalities involving α ± are understood in the sense of bounds for endomorphisms(such as positive definiteness). Remark . We in fact show regularity statements below of the kind that if u ∈ H s (cid:48) ,r (cid:48) ,(cid:96) sc , b with r (cid:48) satisfying an inequality like r , and if ˆ P ( σ ) u ∈ H s − ,r +1 ,(cid:96) +1sc , b , then u ∈ H s,r,(cid:96) sc , b , and the estimate for u in terms of ˆ P ( σ ) u (and a relatively compactterm) implied by the Fredholm property holds. See for instance Proposition 4.16.One can also improve the b-decay order (cid:96) ; see Remark 4.15.Notice that, in terms of the limiting absorption principle, there are two waysto implement this conjugation: one can conjugate either by e iσ/x , where σ is nowcomplex, or by e i Re σ/x . The former, which we pursue, gives much stronger spaceswhen σ is not real with Im σ > e iσ/x entails an exponentially decaying weight e − Im σ/x , so if the original operatoris applied to u , the conjugated operator is applied to e Im σ/x u times an oscillatoryfactor.We also note that under non-trapping assumptions, mutatis mutandis, all thearguments extend to the large σ (with Im σ bounded) setting via a semiclassicalversion of the argument presented below, as we show in Section 5, namely one has Theorem 1.5.
With ˆ P ( σ ) as above, ˜ r + (cid:96) + 1 / − Im α − ( σ ) > , (cid:96) + 1 / − Im α + ( σ ) < , r + 1 / − Im α − ( σ ) > , ANDRAS VASY and under the additional assumption that the bicharacteristic flow is non-trapping,for and σ > there is C > such that high energy estimates hold on the semi-classical spaces, h = | σ | − , Im σ ≥ : (cid:107) u (cid:107) H ˜ r,l b , (cid:126) ≤ C | σ | − (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r,l +1b , (cid:126) and (cid:107) u (cid:107) H s,r,l sc , b , (cid:126) ≤ C | σ | − (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b , (cid:126) uniformly in | σ | > σ .The analogous conclusion also holds with ˜ r + (cid:96) + 1 / − Im α − ( σ ) < , (cid:96) + 1 / − Im α + ( σ ) > , r + 1 / − Im α − ( σ ) < and Im σ ≤ .Remark . Note that the estimates in Theorem 1.5 have a loss of | σ | − relative toelliptic large-parameter estimates that hold for P ( σ ) when σ is in a cone boundedaway from the real axis: the latter correspond to P ( σ ) : H s,r sc , (cid:126) → H s − ,r sc , (cid:126) . This isdue to the fact that in the more precise function spaces used in this statement ˆ P ( σ )is not elliptic.The structure of this paper is the following. In Section 2 we recall the necessarybackground for pseudodifferential operator algebras. In Section 3 we discuss indetail the assumptions on P ( σ ), and the form of the conjugate ˆ P ( σ ), as well aselliptic estimates. In Section 4 we then provide the positive commutator estimatesthat prove Theorem 1.1. Finally in Section 5 we prove the high energy version,Theorem 1.5.I am very grateful for numerous discussions with Peter Hintz, various projectswith whom have formed the basic motivation for this work. I also thank DietrichH¨afner and Jared Wunsch for their interest in this work which helped to push ittowards completion, and Jesse Gell-Redman for comments improving it.2. Pseudodifferential operator algebras
Three operator algebras play a key role in this paper on the manifold with bound-ary X . Below we use x as a boundary defining function, and y j , j = 1 , . . . , n − ∂X , extended to a collar neighborhood of the boundary.We also use the convention that vector fields and differential operators, of variousclasses discussed below, have smooth, i.e. C ∞ ( X ), coefficients unless otherwise in-dicated. The notation for symbolic coefficients of order l is S l Diff( X ), where Diffobtains subscripts according to the algebra being studied. Here recall that sym-bols, or conormal functions, of order l , are C ∞ ( X ◦ ) functions which are bounded by C x − l , and for which iterated application of vector fields tangent to the boundary ∂X , i.e. elements of V b ( X ), results in a similar (with different constants) bound. Inlocal coordinates, elements of V b ( X ) are linear combinations of x∂ x and ∂ y j , so thecontrast between C ∞ and S coefficients is regularity with respect to ∂ x vs. x∂ x .Classical symbols are those with a one-step polyhomogeneous asymptotic expansionat ∂X ; thus, classical elements of S are exactly elements of C ∞ .The first algebra that plays a role is Melrose’s scattering algebra, [11], Ψ s,r sc ( X );the spectral family of the Laplacian of a scattering metric lies in Ψ , ( X ). Thisalgebra is based on the Lie algebra of scattering vector fields V sc ( X ) = x V b ( X ),where we recall that V b ( X ) is the Lie algebra of b-vector fields, i.e. vector fields IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 7 tangent to ∂X , and the corresponding algebra Diff sc ( X ) consisting of finite sumsof finite products of scattering vector fields and elements of C ∞ ( X ). In local coor-dinates as above, elements of V sc ( X ) are linear combinations of x ∂ x , x∂ y j . Thesevector fields are all smooth sections of the vector bundle sc T X , with local basis x ∂ x , x∂ y j , and thus their principal symbols are exactly smooth (in the base point)fiber-linear functions on the dual bundle sc T ∗ X (with local basis x − dx, x − dy j ,the coefficients, which give fiber coordinates, are denoted by τ and µ j , i.e. a covec-tor is of the form τ ( x − dx ) + (cid:80) j µ j ( x − dy j )); the differential operators have thusprincipal symbols which are fiber-polynomials. In order to familiarize ourselveswith this, we note that if X is the radial compactification R n of R nz , i.e. a sphereat infinity is added, so that the result is a closed ball, with x = r − , y j being localcoordinates near a point on the boundary, where r is the Euclidean radius functionand y j are local coordinates on the sphere, then V sc ( X ) is exactly the collection ofvector fields of the form (cid:80) a j ∂ z j , where a j are smooth on X . Correspondingly, inthis case, sc T ∗ X is naturally identified with R nz × ( R n ) ∗ ζ , i.e. (a partially compact-ified version of) the most familiar phase space in microlocal analysis. The class ofpseudodifferential operators Ψ s,r sc ( X ) in this case, going back to Parenti and Shubin[14, 15], is standard quantizations of symbols a ∈ S s,r ( sc T ∗ X ) on sc T ∗ X of orders( s, r ), where s is the differential and r is the decay order: | D αz D βζ a ( z, ζ ) | ≤ C αβ (cid:104) z (cid:105) r −| α | (cid:104) ζ (cid:105) s −| β | . The phase space in general for Ψ s,r sc ( X ) is thus sc T ∗ X , quantization maps can berealized by using a partition of unity within coordinate charts each of which iseither disjoint from the boundary or is of the form as above, i.e. a coordinate charton the sphere times [0 , (cid:15) ) x , which in turn can be identified with an asymptoticallyconic region at infinity in Euclidean space, so the R n -quantization can be used.(One also adds general Schwartz kernels which are Schwartz on X × X , i.e. are in˙ C ∞ ( X × X ).) The principal symbols in this algebra are taken modulo lower orderterms in terms of both orders, i.e. in S s,r ( sc T ∗ X ) /S s − ,r − ( sc T ∗ X ) = S s,r ( sc T ∗ X ) /S s − ,r − ( sc T ∗ X ) , where sc T ∗ X denotes the fiber radial compactification of sc T ∗ X . In particular, van-ishing of this principal symbol captures relative compactness on L -based Sobolevspaces; here L is the L -space with respect to any Riemannian sc-metric (i.e. asmooth positive definite inner product on sc T X ), which is the standard L spaceon R n in case X = R n .The second algebra is Melrose’s b-algebra [13], whose Lie algebra of vector fields, V b ( X ), has already been discussed. In local coordinates, elements of the latter arelinear combinations of x∂ x and ∂ y j , so again are all smooth sections of a vectorbundle, b T X , with local basis x∂ x and ∂ y j , and thus their principal symbols aresmooth fiber-linear functions on the dual bundle b T ∗ X , with local basis x − dx and dy j (with coefficients denoted by τ b and ( µ b ) j , so covectors are written as τ b ( x − dx ) + (cid:80) j ( µ b ) j dy j ). The corresponding pseudodifferential algebra Ψ ˜ r,l b ( X ),with ˜ r the differential, l the decay, order, which Melrose defined via describingtheir Schwartz kernels on a resolved space, called the b-double space, is closelyrelated to H¨ormander’s uniform algebra Ψ ˜ r ∞ ( R n ) [9, Chapter 18.1]. Namely, using t = − log x we are working in a cylinder [ T, ∞ ) t × U , U a coordinate chart on ∂X ,and for instance Schwartz kernels of elements of x − l Ψ ˜ r ∞ ( R n ) = e lt Ψ ˜ r ∞ ( R n ) which ANDRAS VASY have support (with prime denoting the right, unprime the left, factor on the productspace) in | t − t (cid:48) | < R are elements of Ψ ˜ r,l b ( X ) and indeed capture (locally) Ψ ˜ r,l b ( X )modulo smoothing operators, Ψ −∞ ,l b ( X ). In general one adds smooth Schwartzkernels which are superexponentially decaying in | t − t (cid:48) | , as well as Schwartz kernelsrelating to disjoint coordinate charts on ∂X with similar decay, see [21, Section 6]for a more thorough description from this perspective. In this algebra the principalsymbol map captures only the behavior at fiber infinity, i.e. in the differential ordersense, and takes values in S ˜ r,l ( b T ∗ X ) /S ˜ r − ,l ( b T ∗ X ) = S ˜ r,l ( b T ∗ X ) /S ˜ r − ,l ( b T ∗ X ) . This principal symbol is a ∗ -algebra homomorphism, so σ ˜ r +˜ r (cid:48) ,l + l (cid:48) ( AA (cid:48) ) = σ ˜ r,l ( A ) σ ˜ r (cid:48) ,l (cid:48) ( A (cid:48) ) , A ∈ Ψ ˜ r,l b ( X ) , A (cid:48) ∈ Ψ ˜ r (cid:48) ,l (cid:48) b ( X ) , so the algebra is commutative to leading order in the differential sense, i.e.[ A, A (cid:48) ] ∈ Ψ ˜ r +˜ r (cid:48) − ,l + l (cid:48) b ( X ) , but there is no gain in decay. The principal symbol of the commutator as an elementof Ψ ˜ r +˜ r (cid:48) − ,l + l (cid:48) b ( X ) is given by the usual Hamilton vector field expression: σ m + m (cid:48) − ,l + l (cid:48) ([ A, A (cid:48) ]) = 1 i H a a (cid:48) , a = σ m ( A ) , a (cid:48) = σ m (cid:48) ( A (cid:48) ) . For l = 0, H a is a b-vector field on b T ∗ X , i.e. is tangent to b T ∗ ∂X X (and in generalit simply has an extra weight factor); indeed in local coordinates it takes the form(2.1) ( ∂ τ b a )( x∂ x ) − ( x∂ x a ) ∂ τ b + (cid:88) j (cid:0) ( ∂ ( µ b ) j a ) ∂ y j − ( ∂ y j a ) ∂ ( µ b ) j (cid:1) = ( − ∂ τ b a ) ∂ t − ∂ t a ( − ∂ τ b ) + (cid:88) j (cid:0) ( ∂ ( µ b ) j a ) ∂ y j − ( ∂ y j a ) ∂ ( µ b ) j (cid:1) , where the − signs in the ∂ t -version correspond to τ b dxx = − τ b dt ; notice that thesecond line is the standard form of the Hamilton vector field taking into accountthat τ b is the negative of the canonical dual coordinate of t .Principal symbol based constructions and considerations (ellipticity, propagationof singularities, etc.) do not give rise to relatively compact errors on L -basedSobolev spaces; here L is the L -space with respect to any Riemannian b-metric(i.e. a smooth positive definite inner product on b T X ), which in the cylindricalpicture above is simply the standard L space on the cylinder. However, in additionthere is a normal operator, which captures the behavior of an element of Ψ ˜ r,l b ( X )at X . For differential operators, P ∈ S Diff ˜ r,l b ( X ), which is at least to leadingorder at the boundary is smooth (which in the cylindrical picture means that thecoefficients have a limit as t → + ∞ , with exponential convergence to the limit),this amounts to restricting the coefficients of x l times the operator to the boundaryand obtaining a model operator on [0 , ∞ ) x × ∂X which is dilation invariant in x (which amounts to translation invariance in t on R t × ∂X ); there is an analogousstatement for pseudodifferential operators. If an operator is elliptic in the principalsymbol sense, and its normal operator is invertible on a weighted Sobolev space,then the original operator is Fredholm between correspondingly weighted b-Sobolevspaces (shifted by the decay order l we factored out).There is a common resolution of these two algebras in the form of the thirdrelevant algebra, which is the second microlocalized, at the zero section, scattering IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 9 algebra, Ψ s,r,l sc , b ( X ), which is described in more detail in this context in [22, Section 5].Here the symbol space S s,r,l can be arrived at in two different ways. From the secondmicrolocalization perspective, one takes sc T ∗ X , and blows up the zero section overthe boundary o ∂X . The new front face is naturally identified with b T ∗ ∂X X . Inthis perspective, one is looking at scattering pseudodifferential operators which aresingular at the zero section. Now the three orders of Ψ s,r,l sc , b ( X ), and correspondinglyof S s,r,l , are the sc-differential order s , the sc-decay order r and the b-decay order l respectively, i.e. they are the symbolic orders of amplitudes used for the quantizationat the three hypersurfaces: sc-fiber infinity, the lift of sc T ∗ ∂X X , and the new frontface. Thus, we adopt a second microlocalization-centric approach in the orderconvention, see Figure 1. . .................................................................................................................................................................................................................................................................................................................................................................................................................. .......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................................................................................................................................................................................... R n ×{ }{ }× R n ∂ R n × R n R n × ∂ R n ! !! ! ΣΣ ΣΣ . .................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................................................................... ......................................................................................................
Char( ˆ P )Char( ˆ P ) . .................................................................................................................................................................................................................................................................................................................................................................................................................. .......................................................................................................................................................................................................................................................... ........... ............. ................ ................... .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................... ................ ............. ........... ......................................................................................................................................................................................................................................................... . .............................................................................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................................................................................................................................................................................... R n ×{ }{ }× R n R n × ∂ R n b T ∗ ∂ R n R n !! !!! ! Σ Σ Σ . .................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................................................................... ......................................................................................................
Char( ˆ P )Char( ˆ P ) Figure 1.
Second microlocalized Euclidean space R n . The lefthand side is the fiber-compactified sc-cotangent bundle, sc T ∗ R n = R n × ( R n ) ∗ , the right hand side is its blow-up at the boundary ofthe zero section. The (interior of the) front face of the blow-up,shown by the curved arcs, can be identified with b T ∗ ∂ R n R n . Thecharacteristic set of ˆ P ( σ ), σ (cid:54) = 0, discussed in Section 3, is alsoshown, both from the compactified perspective, as Σ, which is asubset of the boundary, and from the conic perspective, here conicin the base (i.e. the dilations are in the R nz , spatial, factor), asChar( ˆ P ). On the second microlocal figure on the right, the char-acteristic set within the boundary lies at the lift of the fibers of thesc-cotangent bundle over the boundary; from the b-perspective, itthus corresponds to symbolic behavior, and lies at fiber infinity.The fiber of cotangent bundle over the origin, i.e. { } × ( R n ) ∗ , isalso indicated; this is only special from the conic (dilation) per-spective, in which it is the analogue of the zero section in standardmicrolocal analysis.On the other hand, from an analytically better behaved, but geometrically equiv-alent, perspective, one takes b T ∗ X , and blows up the corner, namely fiber infinityat ∂X . The new front face is then sc T ∗ ∂X X , blown up at the zero section, see Fig-ure 2. These two resolved spaces are naturally the same, see [22, Section 5], in thesense that the identity map in the interior (as both are identified with T ∗ X ◦ there) extends smoothly to the boundary; this can be checked easily by noting that τ ( x − dx ) + (cid:88) j µ j ( x − dy j ) = τ b ( x − dx ) + (cid:88) j ( µ b ) j dy j shows τ = xτ b , µ = xµ b . The advantage of the b-perspective is that the b-quantization, etc., procedures workwithout a change, since the space of conormal functions, i.e. symbols, is unchangedunder blowing up a corner. Moreover, it allows to capture global phenomena at theLagrangian, and thus compactness properties, unlike the usual second microlocalperspective in which Lagrangianizing errors are treated as residual. In particular,we have Ψ ˜ r, ˜ r + l,l sc , b ( X ) = Ψ ˜ r,l b ( X ) . The algebra Ψ sc , b ( X ) combines the features of the previous two algebras, thusthe principal symbol is in S s,r,l /S s − ,r − ,l , does not capture relative compactness,and there is a normal operator, which when combined with the principal symbol,does capture relative compactness and thus Fredholm properties. Because of theaforementioned identification, one can consider the sc-decay part of the principalsymbol to be described by a function on [ sc T ∗ ∂X X ; o ∂X ], up to overall weight factors,at least if the pseudodifferential operator is to leading order (in sc-decay) classical. . ............................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ . ............................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................ o bb T ∗ X b T ∗ ∂X X . ........................................................................................................................................................................ ...................................... ................. ............... ............... ................. .................. ................... .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ................... .................. ................. ............... ............... ................. ..................................... . ............................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................ o bb T ∗ X b T ∗ ∂X X ff= [ sc T ∗ ∂X X ; o ∂X ] Figure 2.
The second microlocal space, on the right, obtained byblowing up the corner of b T ∗ X , shown on the left.In all these cases one has corresponding Sobolev spaces, namely H s,r sc , H ˜ r,l b , H s,r,l sc , b ,all of which are subspaces of tempered distributions u , i.e. the dual space of ˙ C ∞ ( X ),i.e. C ∞ functions vanishing to infinite order at ∂X . Of these, H s,r sc is locally, in thesense of asymptotically conic regions discussed earlier in this section, the standardweighted Sobolev space on R n , (cid:104) z (cid:105) − r H s ( R n ). Alternatively, for s ≥ A of Ψ s, ( X ), elliptic in the sense of the sc-differentialorder (i.e. as | ζ | → ∞ in the local model), and define the space as u ∈ x r L , L = L ( X ), for which Au ∈ x r L . Here the choice of the elliptic element isirrelevant, and all such elliptic elements A give rise to equivalent squared norms:(2.2) (cid:107) Au (cid:107) L + (cid:107) x − r u (cid:107) L . Similarly, in the cylindrical identification discussed earlier, H ˜ r,l b is locally the weightedSobolev space e ( l − n/ t H ˜ r ( R n ), where the distribution should be supported in IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 11 ( T, ∞ ) t × U , the type of region discussed before. Here the exponent − nt/ l = 0 space is L ( X ) = L ( X ): the density is apositive non-degenerate multiple of x − n − | dx dy | = e nt | dt dy | . This is a shift by e − nt/ = x n/ relative to the usual convention for b-Sobolevspaces, see e.g. [13] or [21, Section 5.6] , and is made, as in [22] , so that the basespaces, corresponding to all orders being , are the same L -space for all Sobolevscales we consider. For ˜ r ≥ A of Ψ ˜ r, ( X ), elliptic in the symbolic sense (i.e. in the usual sense for H¨ormander’suniform algebra in the local model discussed earlier), mapping the distribution u to x l L , with norm(2.3) (cid:107) Au (cid:107) L + (cid:107) x − l u (cid:107) L . The second microlocal spaces are refinements of H ˜ r,l b ; choosing any ˜ r ≤ min( s, r − l ), H s,r,l sc , b ( X ) is the subspace of H ˜ r,l b for which there exists an elliptic element A ofΨ s,r, , b ( X ), elliptic in the standard symbolic sense corresponding to the first twoorders, for which Au ∈ x l L with the squared norm(2.4) (cid:107) x − l Au (cid:107) L + (cid:107) u (cid:107) H ˜ r,l b , and the second term can be replaced by (cid:107) u (cid:107) H ,l b = (cid:107) x − l u (cid:107) L if ˜ r can be taken tobe ≥
0. We reiterate that we use the scattering L space as the base space for ournormalization of orders in all cases, so when all their indices are 0, all these spacesare simply L = L ( X ).In the high energy setting we also need the semiclassical version of all thesealgebras. In these one adds a parameter h ∈ (0 , h > h has no significant effect, so the main point is the uniform behavior as h → sc , (cid:126) ( X ), resp. Diff b , (cid:126) ( X ), over C ∞ ([0 , h × X ), are h times the standard vector fields, i.e. h V sc ( X ), resp. h V b ( X ). Thus, for instance,semiclassical scattering differential operators are built from hx D x and hxD y j inlocal coordinates. There are then semiclassical pseudodifferential algebras in bothof these cases. Much as Ψ sc ( X ) is one of the standard pseudodifferential algebraswhen X = R n , Ψ sc , (cid:126) ( X ) is one of the standard semiclassical pseudodifferentialalgebras in this case; elements are semiclassical quantizations( A h u )( z, h ) = (2 πh ) − n (cid:90) e iζ · ( z − z (cid:48) ) /h a ( z, ζ, h ) u ( z (cid:48) ) dz (cid:48) dζ of symbols a ∈ C ∞ ([0 , S s,r ( sc T ∗ X )) on sc T ∗ X of orders ( s, r ), where s is thedifferential and r is the decay order: | D jh D αz D βζ a ( z, ζ, h ) | ≤ C αβj (cid:104) z (cid:105) r −| α | (cid:104) ζ (cid:105) s −| β | ;here one can simply demand boundedness in h (not in its derivatives) instead. Thephase space is then sc T ∗ X × [0 , h , and the principal symbol is understood moduloadditional decay in h , i.e. in S s,r ( sc T ∗ X × [0 , h ) /hS s − ,r − ( sc T ∗ X × [0 , h ) , so there is a new, semiclassical, principal symbol, given by the restriction of a to h = 0. Since the localization becomes stronger as h →
0, one can transfer this algebra to manifolds with boundary just as we did for Ψ sc ( X ). We refer to [20] formore details, and [25] for a general discussion of semiclassical microlocal analysis.The b-version is completely similar, locally (using the logarithmic identificationabove) based on the semiclassical quantization of H¨ormander’s uniform algebra, i.e.symbols in C ∞ ([0 , h ; S ˜ r ∞ ( b T ∗ X )): | D jh D α ˜ z D β ˜ ζ a (˜ z, ˜ ζ, h ) | ≤ C αβj (cid:104) ˜ ζ (cid:105) ˜ r −| β | , where ˜ z = ( t, y ) = ( − log x, y ) locally. In particular, principal symbols are in S ˜ r,l ( b T ∗ X × [0 , h ) /hS ˜ r − ,l ( b T ∗ X × [0 , h ) , and again there is a normal operator. We refer to [8, Appendix A.3] for moredetails.Finally the second microlocalized at the zero section algebra arises, as before, byblowing up the zero section at ∂X × [0 , h in sc T ∗ X × [0 , h , though it is better toconsider it from the b-perspective, blowing up the corner ∂ T ∗ X of b T ∗ X , times[0 , h , in b T ∗ X × [0 , h . Here [0 , h is a parameter for both perspectives, namelyit is a factor both in the space within which the blow-up is taking place and in thesubmanifold being blown up, so the resulting space is[ b T ∗ X ; ∂ T ∗ X ] × [0 , h = [ sc T ∗ X ; o ∂X ] × [0 , h , i.e. the symbols are smooth functions of h with values in the non-semiclassicalsecond microlocal space. Since this is a blow up of the codimension 2 corner of b T ∗ X × [0 , h in the first factor, much as in the non-semiclassical setting, one canuse the usual (now semiclassical) b-pseudodifferential algebra for quantizations,properties, etc.The semiclassical Sobolev spaces are the standard Sobolev spaces, but with an h -dependent norm. Thus, on R n , these are defined using the semiclassical Fouriertransform ( F h u )( ζ, h ) = (2 πh ) − n (cid:90) R n e − iz · ζ/h u ( z, h ) dz ;so that (cid:107) u (cid:107) H s (cid:126) ( R n ) = (cid:107)(cid:104) ζ (cid:105) s F h u (cid:107) L ( R n ) , while (cid:107) u (cid:107) H s,r (cid:126) = (cid:107)(cid:104) z (cid:105) r u (cid:107) H s (cid:126) , and then the definition of H s,r sc , (cid:126) ( X ) locally reduces to this. The Sobolev spaces forthe other operator algebras are analogous. Thus, (2.2), (2.3), (2.4) are replaced byan equation of the same form but with A ∈ Ψ s, , (cid:126) ( X ), resp. A ∈ Ψ ˜ r, , (cid:126) ( X ), resp. A ∈ Ψ s,r, , b , (cid:126) ( X ), elliptic in the relevant symbolic senses.3. The operator
We first define the class of operator families we consider, drawing comparisonswith [22], where the σ → g be a scattering metric, g − g ∈ S − ( X ; sc T ∗ X ⊗ s sc T ∗ X ) , g = x − dx + x − g ∂X , IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 13 g ∂X a metric on ∂X , so g is asymptotic to a conic metric g on (0 , ∞ ) × ∂X , andwe indeed make the assumption that g − g even has a leading term, i.e. for some δ > g − g ∈ x C ∞ ( X ; sc T ∗ X ⊗ s sc T ∗ X ) + S − − δ ( X ; sc T ∗ X ⊗ s sc T ∗ X )These are stronger requirements than in [22], where g − g ∈ S − δ ( X ; sc T ∗ X ⊗ s sc T ∗ X ) was allowed, δ >
0, but this is due to our desire to obtain a more preciseconclusion, albeit in a non-zero energy regime. In fact, these requirements can berelaxed, as only the normal-normal component of g − g actually needs to have suchan asymptotic behavior, but we do not comment on this further.In [22], due to the near zero energy regime being considered, we worked in ab-framework from the beginning. For non-zero energies weaker (in terms of theoperator algebra), scattering, assumptions are natural, though we impose strongerasymptotic requirements on these. Then we consider P ( σ ) = P (0) + σQ − σ , P (0) ∈ S Diff ( X ) , Q ∈ S − Diff ( X ) ,P (0) elliptic, P (0) − ∆ g ∈ S − Diff ( X ) , Notice that this means that for real σ , P ( σ ) − P ( σ ) ∗ ∈ S − Diff ( X ).We in fact make the stronger assumption that P (0) − ∆ g , Q have leading terms:(3.1) P (0) − ∆ g ∈ x Diff ( X ) + S − − δ Diff ( X ) ,Q ∈ x Diff ( X ) + S − − δ Diff ( X );thus σ ∈ R ⇒ P ( σ ) − P ( σ ) ∗ ∈ x Diff ( X ) + S − − δ Diff ( X ) . Here in fact Q can have arbitrary smooth dependence on σ if one stays sufficientlyclose to real values of σ , in particular for real σ . Note that for fixed real σ , Q canbe incorporated into P (0). While the restrictions we imposed can be relaxed, inthat leading terms are only required in some particular components, essentiallyamounting to the radial set that is moved to the zero section, in this paper we keepthe assumption of this form. Note that x Diff ( X ) + S − − δ Diff ( X ) ⊂ x Diff ( X ) + S − − δ Diff ( X )and x Diff ( X ) + S − − δ Diff ( X ) ⊂ x Diff ( X ) + S − − δ Diff ( X );the expressions on the left hand side correspond to the ‘category’ of operatorsused in [22], in so far as b-spaces are used, although here we have stronger decayassumptions.We mention that(3.2) ∆ g = ∆ sc = x n +1 D x x − n − x D x + x ∆ ∂X is the model scattering Laplacian at infinity.From the Lagrangian perspective we consider a conjugated version of P ( σ ).Thus, let ˆ P ( σ ) = e − iσ/x P ( σ ) e iσ/x . Since conjugation by e iσ/x is well-behaved in the scattering, but not in the b-sense,it is actually advantageous to first perform the conjugation in the scattering setting,and then convert the result to a b-form. At the scattering principal level, the effect of the conjugation is to replace τ by τ − σ and leave µ unchanged, correspondingto e − iσ/x ( x D x ) e iσ/x = x D x − σ, e − iσ/x ( xD y j ) e iσ/x = xD y j . Since the principal symbol of P ( σ ) in the scattering decay sense, so at x = 0, is p ( σ ) = τ + µ − σ , the principal symbol ˆ p ( σ ) of ˆ P ( σ ) isˆ p ( σ ) = x ( τ + µ ) − σxτ b = τ + µ − στ. Moreover, for σ real, if P ( σ ) is formally self-adjoint, so is ˆ P ( σ ); in general σ ∈ R ⇒ ˆ P ( σ ) − ˆ P ( σ ) ∗ ∈ x Diff ( X ) + S − − δ Diff ( X ) . In order to have a bit more precise description, it is helpful to compute ˆ P ( σ )somewhat more explicitly. Proposition 3.1.
We have (3.3) ˆ P ( σ ) = ˆ P (0) + σ ˆ Q − σ (cid:16) x D x + i n − x + x ˜ α + ( σ ) (cid:17) with ˆ P (0) ∈ x Diff ( X ) + S − − δ Diff ( X ) , ˆ Q ∈ x Diff ( X ) + S − − δ Diff ( X ) , ˜ α + ( σ ) ∈ C ∞ ( X ) + S − δ ( X ) . Remark . A simple computation shows that if we regard ˆ P ( σ ) as an operatoron half-densities, using the metric density to identify functions and half-densities,then the subprincipal symbol of ˆ P ( σ ) at the sc-zero section (i.e. regarding ˆ P ( σ ) asan element of Diff ( X )) is − σx ˜ α + ( σ ) modulo S − + S − τ + S − · µ , with the S − terms corresponding to the statement holding at the zero section. Proof.
To start with, in local coordinates, we have(3.4) P (0) =(1 + xa )( x D x ) + (cid:88) j xa j (( x D x )( xD y j ) + ( xD y j )( x D x ))+ (cid:88) i,j a ij ( xD y i )( xD y j )+ ( i ( n −
1) + a ) x ( x D x ) + (cid:88) j xa j ( xD y j ) + xa (cid:48) , and(3.5) Q = b x ( x D x ) + (cid:88) j xb j ( xD y j ) + b (cid:48) x, with a , a j , a , a j , a (cid:48) , b , b j , b (cid:48) ∈ C ∞ ( X ) + S − δ ( X ), a ij ∈ C ∞ ( X ) + S − − δ ( X ), andwith b , b j , b (cid:48) smoothly depending on σ . Here i ( n −
1) is taken out of the x ( x D x )term of P (0) because this way for a formally selfadjoint operator a | ∂X is real, cf.(3.2); similarly for a formally selfadjoint operator a j , a (cid:48) , b , b j , b (cid:48) have real restric-tions to ∂X . (Note that a ij is real by standard principal symbol considerations, asthat of P ( σ ) is the dual metric function G .) IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 15
This gives e − iσ/x P (0) e iσ/x =(1 + xa )( x D x − σ ) + (cid:88) j xa j (( x D x − σ )( xD y j ) + ( xD y j )( x D x − σ ))+ (cid:88) i,j a ij ( xD y i )( xD y j ) + ( i ( n −
1) + a ) x ( x D x − σ ) + (cid:88) j xa j ( xD y j ) + xa (cid:48) , and e − iσ/x Qe iσ/x = b x ( x D x − σ ) + (cid:88) j xb j ( xD y j ) + b (cid:48) x, Combining the terms gives(3.6) ˆ P ( σ ) = ˆ P (0)+ σ ˆ Q − σ (cid:16) x D x + i n − x + 12 x (cid:0) − a σ + a + b σ − σ − a (cid:48) − b (cid:48) (cid:1)(cid:17) with(3.7) ˆ P (0) = P (0) − xa (cid:48) ∈ x Diff ( X ) + S − − δ Diff ( X ) , ˆ Q = Q − b (cid:48) x − xa ( x D x ) − (cid:88) j xa j ( xD y j ) ∈ x Diff ( X ) + S − − δ Diff ( X ) . (cid:3) Notice that if the coefficients were smooth, rather than merely symbolic, ˆ P ( σ )would be in x Diff ( X ); with the actual assumptions in generalˆ P ( σ ) ∈ x Diff ( X ) + S − − δ Diff ( X ) + S − − δ Diff ( X ) , with the only term of (3.3) that is not in a faster decaying space being the lastone; this is unlike P ( σ ) which is merely in Diff ( X ) + S − − δ Diff ( X ) due to the σ term; this one order decay improvement plays a key role below. Note also the σ − in front of a (cid:48) in the last parenthetical expression of (3.6); this corresponds tothe Laplacian with the Coulomb potential having significantly different low energybehavior than with a short range potential (or no potential). On the other hand,long range terms in the higher order terms make no difference even in that case;indeed, the a contribution even decays as σ → P (0) vanishes quadratically at thescattering zero section, τ = 0, µ = 0, x = 0, hence the subprincipal symbol makessense directly there (without taking into account contributions from the principalsymbol, working with half-densities, etc.), and this in turn vanishes. (The same isnot true for P (0) due to the xa (cid:48) term.) Since it will be helpful when considering non-real σ below, we note positivity properties of ˆ P (0) and related structural propertiesof ˆ Q . Lemma 3.3.
The operator ˆ P (0) is non-negative modulo terms that are either sub-sub-principal or subprincipal but with vanishing contribution at the scattering zerosection, in the sense that it has the form (3.8) ˆ P (0) = (cid:88) j T ∗ j T j + (cid:88) j T ∗ j T (cid:48) j + (cid:88) j T † j T j + T (cid:48)(cid:48) where T j ∈ x Diff ( X ) + S − − δ Diff ( X ) , T (cid:48) j , T † j ∈ x C ∞ ( X ) + S − − δ ( X ) , T (cid:48)(cid:48) ∈ x C ∞ ( X ) + S − − δ ( X ) . Moreover, (3.9) ˆ Q = (cid:88) j T ∗ j ˜ T (cid:48) j + (cid:88) j ˜ T † j T j + ˜ T (cid:48)(cid:48) with ˜ T (cid:48) j , ˜ T † j ∈ x C ∞ ( X ) + S − − δ ( X ) , ˜ T (cid:48)(cid:48) ∈ x C ∞ ( X ) + S − − δ ( X ) .Remark . Technically it would be slightly better to replace T j by a one-formvalued differential operator as that would remove the need of discussing coordinatecharts, and then the form of ˆ P (0) would be immediate from the definition of theLaplacian, with T j replaced by the exterior differential or the covariant derivative. Proof.
We work in local coordinates, to which we can reduce by taking T j to becutoff versions of what we presently state, with a union taken over charts. Thenwe can take the T j to be x D x and xD y j ; then the adjoints differ from x D x , resp. xD y j , by elements of x C ∞ ( X ) + S − − δ ( X ), thus the difference can be absorbedinto T (cid:48) j , T † j . The statements then follow from the coordinate form obtained in theproof of Proposition 3.1. Note that the removal of the terms xa (cid:48) from P (0) and xb (cid:48) from Q (they being shifted into ˜ α + ) is important in making the membershipstatements hold. (cid:3) In terms of the local coordinate description of P ( σ ) and ˆ P ( σ ), see (3.4) and(3.5), the normal operator of ˆ P ( σ ) in x Diff ( X ), which arises by considering theoperator x − ˆ P ( σ ) and freezing the coefficients at the boundary, is(3.10) N ( ˆ P ( σ )) = − σ (cid:16) x D x + i n − x + α + x (cid:17) ,α + = α + ( σ ) = ˜ α + ( σ ) | ∂X = 12 (cid:0) − a σ + a + b σ − σ − a (cid:48) − b (cid:48) (cid:1) | ∂X ;notice that (for a (cid:48) = 0) this degenerates at σ = 0. Invariantly, see Remark 3.2, − σxα + ( σ ) is the subprincipal symbol at the sc-zero section, where the quotient isbeing taken with S − − δ in place of S − . Note that the normal operator is x timesthe normal vector field to the boundary plus a smooth function, which, for σ (cid:54) = 0,corresponds to the asymptotic behavior of the solutions of ˆ P ( σ ) v ∈ ˙ C ∞ ( X ) being x ( n − − iα + ) / C ∞ ( ∂X ) , modulo faster decaying terms. Here for formally self-adjoint P ( σ ) when σ is real,the α + term changes the asymptotics in an oscillatory way (as α + is real then) butnot the decay rate, but for complex σ the decay rate may also be affected. Thiscorresponds to the asymptotics e iσ/x x ( n − − iα + ) / C ∞ ( ∂X )for solutions of P ( σ ) u ∈ ˙ C ∞ ( X ) for σ (cid:54) = 0. This indicates that we can remove thecontribution of α + to leading decay order by conjugating the operator by x iα + , aswell as have analogous achievements for the x ( n − / part of the asymptotics, butactually that factor is useful for book-keeping when using the L , rather than the L inner product, and we do not remove this here. IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 17
Note also that if we instead conjugated by e − iσ/x , moving the other radial pointto the zero section, we would obtain the normal operator(3.11) 2 σ (cid:16) x D x + i n − x + α − x (cid:17) ,α − = α − ( σ ) = 12 (cid:0) a σ + a + b σ + σ − a (cid:48) + b (cid:48) (cid:1) | ∂X . Again invariantly, cf. Remark 3.2, 2 σxα − ( σ ) is the subprincipal symbol at thesc-zero section, where the quotient is being taken with S − − δ in place of S − .Next, we consider the principal symbol behavior at, as well as near, ∂X . While(ignoring the irrelevant S − − δ Diff ( X )+ S − − δ Diff ( X ) terms, which are irrelevantthat they do not affect ellipticity near ∂X ) ˆ P ( σ ) ∈ x Diff ( X ), it is degenerate atthe principal symbol level since it is actually in x Diff ( X ) + σx Diff ( X ) ⊂ Ψ , − ( X ) + Ψ , − ( X ) . Correspondingly, we consider ˆ P ( σ ) as an element of the second microlocalized scat-tering pseudodifferential operators, concretelyˆ P ( σ ) ∈ Ψ , , − , b ( X ) . Recall that this space of operators is formally arrived at by blowing up the zerosection of the scattering cotangent bundle at the boundary, but more usefully(in that quantizations, etc., make sense still) by blowing up the corner of thefiber-compactified b-cotangent bundle); in this sense the summands Ψ , − ( X ) andΨ , − ( X ) have the same (sc-decay) order since on the front face x and | ( τ b , µ b ) | − are comparable.Note that in Ψ , , − , b ( X ) the operator ˆ P ( σ ) is elliptic in the sc-differential sense,with principal symbol given by the dual metric function G ; it is also elliptic in thesc-decay sense in a neighborhood of the corner corresponding to sc-fiber-infinity atthe boundary, with now the principal symbol being G − στ , τ = xτ b , the sc-fibercoordinate. Correspondingly, one has elliptic estimates in this region:(3.12) (cid:107) B u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) B ˆ P ( σ ) u (cid:107) H s − ,r,l +1sc , b + (cid:107) u (cid:107) H − N, − N, − N sc , b ) , with the third, b-decay, order actually irrelevant, where B , B ∈ Ψ , , , b ( X ) =Ψ , ( X ), B microlocalizes to the aforementioned region (i.e. has wave front setthere, understood in the strong sense that we can consider B as a scatteringps.d.o., so this imposes triviality at the b-front face), as does B , but B is ellipticon the wave front set of B . Correspondingly, below we always work microlocal-ized away from sc-fiber infinity , in which region H s,r,l sc , b is the same as H r − l,l b , andΨ s,r,l sc , b ( X ) is the same as Ψ r − l,l b ( X ). Thus, if one so wishes, one can use purely theb-pseudodifferential and Sobolev space notation.On the other hand, the principal symbol of ˆ P ( σ ) in the scattering decay sense isˆ p ( σ ) = x ( τ + µ ) − σxτ b = τ + µ − στ, so there is a non-trivial characteristic set, namely where ˆ p ( σ ) vanishes. Notice thatif one wants to consider ˆ P ( σ ) ∈ Ψ , , − , b ( X ) and its principal symbol as a function on[ sc T ∗ ∂X X ; o ∂X ], one should factor out (corresponding to the order − τ + µ ) / ;we mostly do not do this explicitly here. (There is an analogous phenomenon at fiber infinity, but as we discussed already, the operator is elliptic there, so this isnot a region of great interest.) Concretely we have(3.13) Re ˆ p ( σ ) = x ( τ + µ ) − σ ) xτ b = τ + µ − σ ) τ, Im ˆ p ( σ ) = − σ ) xτ b = − σ ) τ. For real (non-zero) σ thus the characteristic set is the translated sphere bundle(with a sphere over each base point in ∂X ),(3.14) 0 = Re ˆ p ( σ ) = ( τ − Re σ ) + µ − (Re σ ) . For non-real complex σ this set is intersected with τ = 0, and thus becomes almost trivial: one concludes that points in the characteristic set have µ = 0, so points ofnon-ellipticity are necessarily at the front face [ sc T ∗ ∂X X ; o ∂X ]. However, to see thebehavior there one actually does need to rescale by ( τ + µ ) / to obtain( τ + µ ) / − σ τ ( τ + µ ) / , which does vanish within the front face, ( τ + µ ) / = 0, namely at τ ( τ + µ ) / = 0,but notice that this vanishing is simple .For real σ , considering sc T ∗ X (rather than its blow-up), the conjugation of theoperator is simply pullback by a symplectomorphism at the phase space level, andthe Hamilton flow has exactly the same structure as in the unconjugated case,except translated by the symplectomorphism. Thus, there are two submanifolds ofradial points, one of which is now the zero section, the other is { τ = 2 Re σ, µ = 0 } , τ is monotone decreasing along the flow, so for Re σ >
0, the non-zero section radialset is a source, for Re σ < without apropagation term (‘estimate for free’), and below which one can propagate estimatesinto the radial points from a punctured neighborhood; see [18, Section 2.4] in thestandard microlocal context, and [21, Section 5.4.7] for a more general discussionthat explicitly includes the scattering setting. The relevant quantities are x − H ˆ p ( σ ) x = ∓ β x, with β > − is for sink, the top line here and thereafter,+ for source),(3.15) σ sc , ∗ , − (cid:16) ˆ P ( σ ) − ˆ P ( σ ) ∗ i (cid:17) = ± β ˜ β ± x at the radial set (where ∗ in the subscript denotes the irrelevant sc-differentialorder), and then the threshold value is r ± = − − ˜ β ± . In our case, β = 2 | Re σ | , while (3.15) at the radial point moved to the zero section is − σ (Im α + ( σ )) x ,giving for Re σ >
0, when this is a sink, ˜ β + = − Im α + ( σ ), while for Re σ <
0, when
IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 19 this is a source, ˜ β − = − Im α + ( σ ) again, hence in either case we have a thresholdregularity r = −
12 + Im α + ( σ ) . Similarly, for the radial point not moved to the zero section, the conjugation corre-sponding to the reversed sign of σ would move it there, and this conjugation gives(3.11) as the normal operator, so we have r (cid:54) =0 = −
12 + Im α − ( σ ) . The resulting estimate, combining propagation estimates from the radial pointoutside the sc-zero section and standard propagation estimates, is(3.16) (cid:107) B u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) B ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N, − N, − N sc , b ) , with the third, b-decay, order actually irrelevant, r > r (cid:54) =0 , and where B , B ∈ Ψ , , , b ( X ) = Ψ , ( X ), B microlocalizes away from the zero section (again in thestrong sense that we can consider B as a scattering ps.d.o., so this imposes trivialityat the b-front face), as does B , but B is elliptic on the wave front set of B ,and also on all bicharacteristics in the characteristic set of ˆ P ( σ ) emanating frompoints in WF (cid:48) sc ( B ) towards the non-zero section radial point, including at theseradial points. On the other hand, the estimate that propagates estimates from aneighborhood of the sc-zero section to the other radial set is(3.17) (cid:107) B u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) B u (cid:107) H s,r,l sc , b + (cid:107) B ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N, − N, − N sc , b ) , with the third, b-decay, order again irrelevant, r < r (cid:54) =0 , and where B , B , B ∈ Ψ , , , b ( X ) = Ψ , ( X ), B microlocalizes away from the zero section in the samesense as above, B is elliptic on an annular region surrounding the zero section, B elliptic on the wave front set of B , and also on all bicharacteristics in thecharacteristic set of ˆ P ( σ ) emanating from points in WF (cid:48) sc ( B ) towards WF (cid:48) sc ( B ),including at the radial points outside the zero section.Moreover, (3.14) shows that τ has the same sign as Re σ along the characteristicset, in the extended sense that it is allowed to become zero. In view of (3.13)thus Im ˆ p ( σ ) thus has − Im σ times the sign of Re σ . Correspondingly, the standardcomplex absorption estimates, see [21] in the present context, allow propagation ofestimates forward along the Hamilton flow when Im σ ≥ σ → σ ≤
0, which means for both signs of Re σ that wecan propagate estimates towards the zero section when Im σ ≥
0, i.e. (3.16) holdsthen, and away from the zero section when Im σ ≤
0, i.e. (3.17) holds then; thesestatements (by standard scattering results) are valid as long as one stays away fromthe zero section itself (where we are using the second microlocal pseudodifferentialalgebra). 4.
Commutator estimates
Since from the standard conjugated scattering picture we already know that thezero section has radial points, the only operator that can give positivity microlocallyin a symbolic commutator computation is the weight. Here, in the second microlocalsetting at the zero section, this means two different kinds of weights, corresponding to the sc-decay (thus microlocally b-differential) and the b-decay orders. Recallthat the actual positive commutator estimates utilize the computation of(4.1) i ( ˆ P ( σ ) ∗ A − A ˆ P ( σ )) = i ( ˆ P ( σ ) ∗ − ˆ P ( σ )) A + i [ ˆ P ( σ ) , A ]with A = A ∗ , so for non-formally-self-adjoint ˆ P ( σ ) there is a contribution from theskew-adjoint part Im ˆ P ( σ ) = 12 i ( ˆ P ( σ ) − ˆ P ( σ ) ∗ )of ˆ P ( σ ), most relevant for us when σ is not real; here the notation ‘Im ˆ P ( σ )’ ismotivated by the fact that its principal symbol is actually Im ˆ p ( σ ), with ˆ p ( σ ) beingthe principal symbol of ˆ P ( σ ). It is actually a bit better to rewrite this, withRe ˆ P ( σ ) = 12 ( ˆ P ( σ ) + ˆ P ( σ ) ∗ )denoting the self-adjoint part of ˆ P ( σ ), as(4.2) i ( ˆ P ( σ ) ∗ A − A ˆ P ( σ )) = (Im ˆ P ( σ ) A + A Im ˆ P ( σ )) + i [Re ˆ P ( σ ) , A ] . If A ∈ Ψ r − , l +1b , ˆ P ( σ ) ∈ Ψ , − ( X ) implies that the second term is a priori inΨ r, l b . However, it is actually in a smaller space since ˆ P ( σ ) ∈ Ψ , − ( X )+Ψ , − ( X ),which in terms of the second microlocal algebra means that ˆ P ( σ ) ∈ Ψ , , − , b ( X ).Thus, taking A ∈ Ψ r − , l +1b = Ψ r − , r + l ) , l +1sc , b ( X ), the commutator is in fact inΨ r, r + l ) − , l sc , b ( X ), so the scattering decay order is 2(˜ r + l ) −
1, and microlocallynear the scattering zero section (where it will be of interest) it is in Ψ r − , l b ( X ).Via the usual quadratic form argument this thus estimates u in H ˜ r − / ,l b in termsof ˆ P ( σ ) u in H ˜ r − / ,l +1b , assuming non-degeneracy. On the other hand, in the firstterm we only have Im ˆ P ( σ ) ∈ Ψ , − ( X ) = Ψ , , − , b ( X ) when σ / ∈ R , so the first termis in Ψ r, r + l ) , l sc , b ( X ), so is the same order, 2 l , in the b-decay sense, but is actuallybigger, order 2(˜ r + l ), in the sc-decay sense, which is the usual situation when oneruns positive commutator arguments with non-real principal symbols, as we will doin the sc-decay sense.Now, going back to the issue of the zero section consisting of radial points,we compute the principal symbol of the second term of (4.2) (which is the onlyterm when σ is real and P ( σ ) is formally self-adjoint) when A ∈ Ψ r − , l +1b ismicrolocally the weight (as mentioned above, only this can give positivity), i.e.(4.3) a = x − l − ( τ + µ ) ˜ r − / , is the principal symbol, so in the second microlocal algebra, A ∈ Ψ r − , r + l ) , l +1sc , b ,i.e. the scattering decay order is 2(˜ r + l ). Lemma 4.1.
The principal symbol H Re ˆ p ( σ ) a of i [Re ˆ P ( σ ) , A ] in Ψ r, r + l ) − , l sc , b ( X ) is (4.4) x − l ( τ + µ ) ˜ r − / (cid:16) σ ) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) = x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) σ ) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − l + ˜ r ) τ ( τ + µ ) (cid:17) . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 21
Proof.
It is a bit simpler (and more standard) to compute Poisson brackets using b T ∗ X rather than sc T ∗ X , cf. (2.1), so we proceed this way, and then we re-expressthe result in sc T ∗ X afterwards. Since the principal symbol of Re ˆ P ( σ ) isRe ˆ p ( σ ) = x ( τ + µ ) − x Re στ b , we compute(4.5) { x ( τ + µ ) − x Re στ b , x − l − ( τ + µ ) ˜ r − / } = (2 x τ b − x Re σ )( − l − x − l − ( τ + µ ) ˜ r − / − (2 x ( τ + µ ) − x Re στ b ) x − l − r − / τ b ( τ + µ ) ˜ r − / . Expanding and rearranging, we have(4.6)= 4(Re σ )( l + 1 / x − l ( τ + µ ) ˜ r − / + 4(Re σ )(˜ r − / x − l τ ( τ + µ ) ˜ r − / − l + 1 / x − l +1 τ b ( τ + µ ) ˜ r − / − r − / x − l +1 τ b ( τ + µ ) ˜ r − / = x − l ( τ + µ ) ˜ r − / (cid:16) σ ) (cid:0) ( l + 1 / τ + µ ) + (˜ r − / τ (cid:1) − x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) = x − l ( τ + µ ) ˜ r − / (cid:16) σ ) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) , giving the left hand side of (4.4) as desired. Finally, substituting τ = xτ b , µ = xµ b yield the right hand side. (cid:3) Remark . For future reference, we record the impact of having an additionalregularizer factor, namely replacing a by a ( (cid:15) ) = af (cid:15) . The role of this is very much standard in positive commutator estimates, see [21,Section 5.4] for a discussion in a similar form, though is slightly delicate in radialpoints estimates as radial points limit the regularizability, see [21, Section 5.4.7],[18, Proof of Proposition 2.3], as well as earlier work going back to [11] and including[4, Theorem 1.4]. However, in our second microlocal setting in fact there is no suchlimitation as it is the b-decay order that is microlocally limited (near the scatteringzero section), and we are not regularizing in that.One can take the regularizer of the form f (cid:15) ( τ + µ ) , f (cid:15) ( s ) = (1 + (cid:15)s ) − K/ , where K > (cid:15) ∈ [0 , (cid:15) → f (cid:15) ( τ + µ ) is a symbol of order − K for (cid:15) >
0, but is only uniformlybounded in symbols of order 0, converging to 1 in symbols of positive order. Then sf (cid:48) (cid:15) ( s ) = − K (cid:15)s (cid:15)s f (cid:15) ( s ) , and 0 ≤ (cid:15)s (cid:15)s ≤
1, so in particular sf (cid:48) (cid:15) ( s ) /f (cid:15) ( s ) is bounded. The effect of this isto add an overall factor of f (cid:15) ( τ + µ ) to (4.4) and (4.5)-(4.6), and replace every occurrence of ˜ r , except those in the exponent, by(4.7) ˜ r + ( τ + µ ) f (cid:48) (cid:15) ( τ + µ ) f (cid:15) ( τ + µ ) = ˜ r − K (cid:15) ( τ + µ )1 + (cid:15) ( τ + µ ) . Corollary 4.3.
The principal symbol of i [Re ˆ P ( σ ) , A ] in Ψ r, r + l ) − , l sc , b ( X ) is apositive elliptic multiple of Re σ in S r, r + l ) − , l on the characteristic set near theimage of the scattering zero section, i.e. the b-face, if l + 1 / > , and it is anegative elliptic multiple there if l + 1 / < .Remark . The sign restrictions on l +1 / P ( σ ), as discussed at the end of Section 3 (cf. r there).Note also that adding a regularizer factor as in Remark 4.2 leaves the conclusionvalid with Re σ replaced by (Re σ ) f (cid:15) ( τ + µ ), and the ellipticity uniform in (cid:15) ∈ [0 , r in (4.4) (apart from those in theexponent), that is thus replaced by (4.7), comes with an additional vanishing factorat the zero section (via τ or τ b ) and thus is lower order in the b-decay sense. Proof.
On the characteristic set of ˆ P ( σ ), where thus Re ˆ p ( σ ) = 0, we have0 = Re ˆ p ( σ ) = ( τ − Re σ ) + µ − (Re σ ) , so | τ − Re σ | ≤ | Re σ | , and thus τ has the same sign as Re σ , but only in an indefinitesense (thus it may vanish). Restricted to Re ˆ p ( σ ) = 0, µ has a simple zero at thezero section while τ vanishes quadratically since at τ = 0 , µ = 0, dp is − σ ) dτ ,i.e. τ is equal to Re ˆ p ( σ ) up to quadratic errors (while dµ is linearly independent ofthis). Correspondingly, on the right hand side of (4.4), not only is the second termof the big parentheses smaller than the first near the zero section on account ofthe extra τ vanishing factor, but even the τ term is negligible compared to the µ term, provided that the latter has a non-degenerate coefficient, i.e. provided l + 1 / l + 1 / (cid:3) While one could simply (and most naturally) use a microlocalizer to a neighbor-hood of the characteristic set in Ψ , , , b ( X ) via using a cutoff on the second microlocalspace, [ sc T ∗ ∂X X ; o ∂X ], to obtain a positive commutator, see the discussion below inthe non-real spectral parameter setting after Lemma 4.18, one can in fact modifythe commutator (in a somewhat ad hoc manner) by adding an additional term thatgives the correct sign everywhere near the image of the scattering zero section, i.e.the b-face, and we do so here. Lemma 4.5.
Let ˜ s = (˜ r − / τ b ( τ + µ ) − = (˜ r − / xτ ( τ + µ ) − . Then H Re ˆ p ( σ ) a + 2˜ sa Re ˆ p ( σ ) is a positive elliptic multiple of Re σ in S r − , r + l ) − , l near the image of thescattering zero section, i.e. the b-face, if l + 1 / > , and it is a negative ellipticmultiple there if l + 1 / < . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 23
In fact, (4.8) H Re ˆ p ( σ ) a + 2˜ sa Re ˆ p ( σ )= x − l +˜ r )+1 ( τ + µ ) ˜ r − / (4(Re σ )( l + 1 / − l + ˜ r + 1 / τ ) . Remark . Due to localization near the scattering zero section added explicitly inthe discussion after the proof, the first, sc-differentiability, order in S r − , r + l ) − , l is actually irrelevant.Moreover, the analogue of the conclusion remains valid with a regularizer as inRemark 4.2, i.e. a replaced by a ( (cid:15) ) , provided in the definition of ˜ s as well as in theconclusion, ˜ r is replaced by (4.7) (except in the exponent), and in the conclusionan overall factor of f (cid:15) ( τ + µ ) is added. Remark . As the proof below shows, replacing ˜ s byˆ s = 2( l + ˜ r )( τ + µ ) − τ b = 2( l + ˜ r ) x ( τ + µ ) − τ replaces the right hand side of (4.8) by x − l +˜ r )+1 ( τ + µ ) ˜ r − / σ ) (cid:0) − ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) . This is manifestly definite with the same sign as Re σ if l + 1 / > l + ˜ r <
0, andwith the opposite sign if l + 1 / < l + ˜ r >
0, and the sign requirements for l + ˜ r turn out to be natural for the global problem, namely these give the signs requiredto obtain microlocal estimates at the other radial set. However, the terms fromIm ˆ P ( σ ), relevant due to (4.2), give rise to terms like − l + ˜ r + 1 / τ above in(4.8) (including with µ j in place of τ ), unless stronger assumptions are imposed onIm ˆ P ( σ ), so in the generality of the present paper this alternative approach is notparticularly fruitful. Nonetheless, the alternative approach becomes very useful inthe companion paper [23], where the zero energy limit is studied and where strongerassumptions are imposed on Im ˆ P ( σ ); it is this perspective that enables us to obtainuniform estimates as σ → Proof.
Adding to (4.4)2˜ sa = 2(˜ r − / xτ ( τ + µ ) − a = 2(˜ r − / τ b ( τ + µ ) − a = 2 x − l ( τ + µ ) ˜ r − / x − (˜ r − / τ b = 2 x − l +˜ r )+1 ( τ + µ ) ˜ r − / (˜ r − / τ times Re ˆ p ( σ ), namely x τ − σ ) xτ b + x µ = τ + µ − σ ) τ, we obtain(4.9) x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) σ ) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − l + ˜ r ) τ ( τ + µ )+ 2(˜ r − / τ ( τ + µ ) − σ )(˜ r − / τ (cid:17) = x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) σ )( l + 1 / τ + µ ) − l + ˜ r + 1 / τ ( τ + µ ) (cid:17) = x − l +˜ r )+1 ( τ + µ ) ˜ r − / (4(Re σ )( l + 1 / − l + ˜ r + 1 / τ ) . As already mentioned, the factor τ is small near the scattering zero section, so4(Re σ )( l +1 / − l +˜ r +1 / τ has the same (definite) behavior as 4(Re σ )( l +1 / − Re σ if l + 1 / < σ if l + 1 / > (cid:3) Using an additional cutoff factor χ which is identically 1 in a neighborhood of thezero section (where the above computation already gave the correct sign), one cancombine this with standard scattering estimates by making this factor microlocalizenear the zero section, so depending on the sign of the Hamilton derivative, theremay be an error arising from the support of its differential, but this is controlledfrom the incoming radial set, see the discussion around (3.16). (An alternativeis instead making the factor monotone along the Hamilton flow with a strict signoutside a small neighborhood of the radial sets.) Recall that as the scattering decayorder of this operator is 2 l + 2˜ r −
1, the requirement for the incoming radial pointestimate (away from the sc-zero section), for formally self-adjoint operators (thusignoring the Im ˆ P ( σ ) terms), is 2 l + 2˜ r − > − /
2) = −
1, i.e. ˜ r + l >
0. Thismeans that using such a cutoff we have microlocal control on the support of dχ if˜ r + l > l + 1 / < . ............................................................................................................................................................................................................................................................. ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................. ................................... ..................................... ....................................... ....................... .................... ................. ................ ................ ............... ✲ x . ............... ................ ................ ................. .................... ....................... ....................................... ..................................... ................................... .................................. .......................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................. .................. ......... ........ ........ ......... ................... ................. .............. ............. ............................................................................................................. ............................................................................................................................... ✻ τ ................................................................ ✛ µ ........................................................................................................................... .................. .................. ............ .............. ............... ................. ............... ............... .............. .............. ................ ............... ...................................................................................................... ................................................................................................................................................................................................................................................................................................................. .............................................................................................................................. ................................................................................................................................................................................. χ Figure 3.
The support of χ on the second microlocal space, indi-cated by the rectangular box. The characteristic set is the circularcurve tangent to the µ axis at the b-face, given by the sc-zerosection.On the other hand, if l + 1 / >
0, then it is not hard to see that the additionalterm caused by the commutator with χ contributes a term with the same sign asthe weight term, i.e. a sign that agrees with that of Re σ . Indeed, we can take χ = χ ( µ ) χ ( τ ) , with χ , χ identically 1 near 0 of compact support sufficiently close to 0 and with χ having relatively large support so that supp χ ( . ) ∩ supp dχ ( . ) is disjoint fromthe zero set of Re ˆ p ( σ ). See Figure 3. Thus, elliptic scattering estimates control the IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 25 dχ term. On the other hand, doing the computation in the b-notation, { x ( τ + µ ) − x Re στ b , χ ( x µ ) } = 2(2 x τ b − x Re σ ) x µ χ (cid:48) ( x µ )= − x (Re σ − τ ) µ χ (cid:48) ( µ ) , so if χ is arranged to have sufficiently small support, say in [ − (Re σ ) / , (Re σ ) / σ . Arranging that − χ (cid:48) is a square,this simply adds another term of the correct sign to our symbolic computation.As non-real σ complicates the arguments, we first consider real σ . We have: Lemma 4.8.
Suppose σ is real. Let S = S ∗ ∈ Ψ − , − , , b ( X ) have principal symbol ˜ s , A = A ∗ ∈ Ψ −∞ , r + l ) , l +1sc , b ( X ) have principal symbol χa , and consider (4.10) i ( ˆ P ( σ ) ∗ A − A ˆ P ( σ )) + AS ˆ P ( σ ) + ˆ P ( σ ) ∗ SA = Im ˆ P ( σ ) A + A Im ˆ P ( σ ) + i [Re ˆ P ( σ ) , A ] + AS Re ˆ P ( σ ) + Re ˆ P ( σ ) SA ∈ Ψ −∞ , r + l ) − , l sc , b ( X ) . With the notation of (3.4) and (3.5) , the principal symbol of (4.10) is (4.11) x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) σ ( l + 1 / − Im α + ( σ )) − l + ˜ r + 1 / − Im a − σ Im b ) τ + 2 (cid:88) j (Im a j + σ Im b j ) µ j (cid:17) χ, modulo terms involving derivatives of χ .Remark . Equation (4.11) shows that the threshold value − / l is shifted to − / α + ( σ ). Note that due to the support condition on χ , the τ and µ j termsin the parentheses can be absorbed into 4 σ ( l + 1 / − Im α + ( σ )) when the latter hasa definite sign, as discussed in the proofs of Corollary 4.3 and Lemma 4.5.Moreover, the analogue of the conclusion remains valid with a regularizer as inRemark 4.2, i.e. a replaced by a ( (cid:15) ) , and correspondingly A by A ( (cid:15) ) , provided in thedefinition of ˜ s as well as in the conclusion, ˜ r is replaced by (4.7) (except in theexponent), and in the conclusion an overall factor of f (cid:15) ( τ + µ ) is added. Proof.
We observe that (4.10) has principal symbol given by (4.8) times χ , plus2 χa times the principal symbol of Im ˆ P ( σ ) ∈ Ψ , − , − , b ( X ), modulo the term arisingfrom the cutoff. Now, the principal symbol of Im ˆ P ( σ ) isIm (cid:16) x ( a + σb ) τ b + (cid:88) j x ( a j + σb j )( µ b ) j − xσα + ( σ ) (cid:17) , as follows from (3.6), (3.7) and (3.1). Thus, the principal symbol of (4.10) is (4.11)modulo terms involving derivatives of χ , as desired. (cid:3) The below-threshold regularity statement (so the sc-zero section is the outgoingradial set, corresponding to low decay) is:
Proposition 4.10.
Suppose l + 1 / − Im α + ( σ ) < , ˜ r + l − Im α − ( σ ) > and σ is real. Then (4.12) (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l b ) . This estimate holds in the strong sense that if u ∈ H ˜ r (cid:48) − / ,l b for some ˜ r (cid:48) satisfyingthe inequality above in place of ˜ r and if ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b then u ∈ H ˜ r − / ,l b andthe estimate holds.Similarly, if l + 1 / − Im α + ( σ ) < , r = ˜ r + l − / > − / α − ( σ ) , and σ is real, then (4.13) (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N, − N,l sc , b ) . Again this holds in the analogous sense that if ˆ P ( σ ) u is in the space on the righthand side and u ∈ H s (cid:48) ,r (cid:48) ,l sc , b for some s (cid:48) , r (cid:48) satisfying the inequality above with r (cid:48) inplace of r , then u is a member of the space on the left hand side, and the estimateholds.Proof. Consider σ > u so that all computations directly makesense. Concretely, everything below works directly if u ∈ H ˜ r,l b , resp. H s,r +1 / ,l sc , b ,with the loss of 1 / P ( σ ) ∗ A separately from thecommutator. However, a very simple regularization argument (even simpler thanthe one discussed below, i.e. it has even less impact on the argument), see [21, Proofof Proposition 5.26] as well as [4, Lemma 3.4], removes this restriction and allows u ∈ H ˜ r − / ,l b , resp. H s,r,l sc , b (though the regularization discussed below completelyremoves the need for this, just as in the low regularity case of the aforementionedreferences).The principal symbol (4.11) of (4.10) can be written as − b + e by Remark 4.9,with e arising from the Poisson bracket with the cutoff χ , thus supported on supp dχ ,so we obtain(4.14) i ( ˆ P ( σ ) ∗ A − A ˆ P ( σ )) + AS ˆ P ( σ ) + ˆ P ( σ ) ∗ SA = − B ∗ B + E + F, where B ∈ Ψ ∗ ,l +˜ r − / ,l sc , b has principal symbol b , E ∈ Ψ ∗ , l +˜ r ) − ,l sc , b has principalsymbol e , and F ∈ Ψ ∗ , l +˜ r ) − ,l sc , b is lower order in the sc-decay sense. Applying to u and pairing with u gives(4.15) (cid:107) Bu (cid:107) ≤ |(cid:104) P ( σ ) u, Au (cid:105)| + |(cid:104) Eu, u (cid:105)| + |(cid:104) F u, u (cid:105)| . Here the E term is controlled by the incoming radial point and propagation esti-mates (as well as the elliptic estimates, including near sc-fiber infinity!), see (3.17).It is helpful to write A = A , A = A ∗ , as arranged by taking χ , χ to be squaresand letting the principal symbol of a to be the square root of that of A . Then b isan elliptic multiple of x / a , so (cid:107) x / A u (cid:107) is controlled by (cid:107) Bu (cid:107) modulo termsthat can be absorbed into |(cid:104) F u, u (cid:105)| . Thus, modulo terms absorbed into the F term, (cid:104) P ( σ ) u, Au (cid:105) = (cid:104) x − / A P ( σ ) u, x / A u (cid:105) is controlled by (cid:107) Bu (cid:107)(cid:107) x − / A P ( σ ) u (cid:107) ≤ (cid:15) (cid:107) Bu (cid:107) + (cid:15) − (cid:107) x − / A P ( σ ) u (cid:107) , and now the first term can be absorbed into the left hand side of (4.15). This gives,using the controlled E terms, with elliptic estimates for the scattering differentia-bility order, with r = ˜ r + l − / (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N,r − / ,l sc , b ) . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 27
Since (cid:107) u (cid:107) H − N,r − / ,l sc , b can be bounded by a small multiple of (cid:107) u (cid:107) H − N,r,l sc , b plus a largemultiple of (cid:107) u (cid:107) H − N, − N,l sc , b , with the former being absorbable into the left hand side,this proves (4.13), and thus (4.12) as a special case, under the additional assumptionof membership of u in the space on the left hand side.In fact the standard regularization argument, using the second part of Re-mark 4.9, shows that the estimate (4.16) holds in the stronger sense that if theright hand side is finite, so is the left hand side, and iterating the estimate gives(4.13) and (4.12). (cid:3) On the other hand, for l + 1 / − Im α + ( σ ) > Proposition 4.11.
Suppose l + 1 / − Im α + ( σ ) > , ˜ r + l − Im α − ( σ ) < and σ is real. Then (4.17) (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l b ) . Similarly, if l + 1 / − Im α + ( σ ) > , r = ˜ r + l − / < − / α − ( σ ) , and σ is real, then (4.18) (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N, − N,l sc , b ) . These estimates hold in the sense analogous to Proposition 4.10, except thereis no need for ˜ r (cid:48) , resp. r (cid:48) to satisfy any inequalities, since for sufficiently negative ˜ r (cid:48) , r (cid:48) , as one may always assume in this context, the inequalities involving these areautomatically satisfied.Remark . Note that in this proposition there is no limit to background decay,as represented by ˜ r (cid:48) , r (cid:48) , thus technically to regularizability, unlike what happens instandard radial point estimates, see [11, 18, 4, 21]. The reason is that the analogueof the limitation of regularizability in the standard setting is the b-decay order,which we here fix, thus we do not improve it over a priori expectations. Proof.
In this case the cutoff term H Re ˆ p ( σ ) χ is also principally positive (for σ >
0, otherwise there is an overall sign switch, though that has no impact on theargument), so the principal symbol of (4.10) is b + b , and we obtain that (4.14)is replaced by i ( ˆ P ( σ ) ∗ A − A ˆ P ( σ )) + AS ˆ P ( σ ) + ˆ P ( σ ) ∗ SA = B ∗ B + B ∗ B + F, where B, B ∈ Ψ ∗ ,l +˜ r − / ,l sc , b have principal symbol b, b , and F ∈ Ψ ∗ , l +˜ r ) − ,l sc , b islower order in the sc-decay sense. Combining this with the outgoing radial pointand propagation estimates (as well as the elliptic estimates) as in (3.16), we canproceed as in the proof of Proposition 4.10 to conclude (4.17) as well as (4.18). (cid:3) Now, the last term of (4.12) and of (4.17) can be estimated using the normaloperator N ( ˆ P ( σ )) = − σ (cid:16) x D x + i n − x + α + ( σ ) x (cid:17) , noting that x − N ( ˆ P ( σ )) = − σ (cid:16) xD x + i n −
12 + α + ( σ ) (cid:17) should be considered as an operator from H − N,l b to H − N − ,l b .Concretely we have: Lemma 4.13.
For l < − / α + ( σ ) , we have (cid:107) v (cid:107) H − N,l b ≤ C (cid:107) N ( ˆ P ( σ )) v (cid:107) H − N,l +1b whenever v ∈ H − N,l b and N ( ˆ P ( σ )) v ∈ H − N,l +1b .The same estimate also holds for l > − / α + ( σ ) .Remark . Here Im α + ( σ ) is a function on ∂X , and the inequalities l < − / α + ( σ ), resp. l > − / α + ( σ ), need to hold at each point of ∂X . Remark . It is straightforward to formalize and prove, via a contour shiftingargument on the Mellin transform side, a version of this lemma that assumes that v is supported in x ≤
1, say (as relevant below in the setting of Proposition 4.16),and that v ∈ H − N,l (cid:48) b only for some l (cid:48) < l , with l (cid:48) satisfying the same inequality as l ,and concludes that v ∈ H − N,l b . However, we do not need this in the present paper,and in any case one can run such an argument as an a posteriori ‘regularity’ (heremeaning b-decay) argument. Proof.
For our immediate purposes it is more convenient to work with L , sowe set ˜ H b to be the b-Sobolev space relative to L , here this really is of in-terest in [0 , ∞ ) × ∂X , with density dxx dg ∂X . Since the quadratic form on L is (cid:104) x n · , ·(cid:105) g , L = x − n/ L , so x − N ( ˆ P ( σ )) mapping from H − N,l b to H − N − ,l b amounts to x − n/ x − N ( ˆ P ( σ )) x n/ being considered from ˜ H − N,l b to ˜ H − N − ,l b , or x − n/ − l x − N ( ˆ P ( σ )) x n/ l from ˜ H − N, to ˜ H − N − , . But this is − σ (cid:16) xD x − i ( l + 1 /
2) + α + ( σ ) (cid:17) , which on the Mellin transform side is multiplication by(4.19) − σ (cid:16) τ b − i ( l + 1 /
2) + α + ( σ ) (cid:17) , which is invertible for real α + if l + 1 / (cid:54) = 0, and in general if l + 1 / (cid:54) = Im α + ( σ ).The differential order is not an issue: the Mellin transform, with image restricted tothe real line, is an isomorphism from the Sobolev spaces ˜ H s (cid:48) , on [0 , ∞ ) × ∂X andthe large parameter, in τ b , i.e. semiclassical in the reciprocal (cid:104) τ b (cid:105) − , Sobolev spaces H s (cid:48) (cid:104) τ b (cid:105) − , see [13] around equation (5.41) and [18] around equation (3.8). Since themultiplication operator (4.19), which is multiplication by a constant for each fixed τ b , has a bounded inverse on these spaces when l + 1 / (cid:54) = Im α + ( σ ), the conclusionfollows. (cid:3) Proposition 4.16.
Suppose l + 1 / − Im α + ( σ ) < , ˜ r + l − Im α − ( σ ) > and σ (cid:54) = 0 is real. Then (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l − δ b ) . This estimate holds in the sense that if u ∈ H ˜ r (cid:48) − / ,l b for some ˜ r (cid:48) satisfying theinequality above in place of ˜ r and if ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b then u ∈ H ˜ r − / ,l b and theestimate holds.Similarly, if l + 1 / − Im α + ( σ ) < , r = ˜ r + l − / > − / α − ( σ ) , and σ is real, then (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N, − N,l − δ sc , b ) . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 29
Again this holds in the analogous sense that if ˆ P ( σ ) u is in the space on the righthand side and u ∈ H s (cid:48) ,r (cid:48) ,l sc , b for some s (cid:48) , r (cid:48) satisfying the inequality above with r (cid:48) inplace of r , then u is a member of the space on the left hand side, and the estimateholds.The analogous conclusions also hold if l +1 / − Im α + ( σ ) > , ˜ r + l − Im α − ( σ ) < , r = ˜ r + l − / < − / α − ( σ ) , except that ˜ r (cid:48) , r (cid:48) do not need to satisfy anyinequalities, cf. Proposition 4.11.Proof. Applying Lemma 4.13 with v = ψu , ψ supported near x = 0, identically 1in a smaller neighborhood, using ˆ P ( σ ) − N ( ˆ P ( σ )) ∈ Ψ , − − δ b ( X ) (note the δ > (cid:107) u (cid:107) H − N,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H − N,l +1b + (cid:107) u (cid:107) H − N +2 ,l − δ b ) . In combination with (4.12) we have (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N +2 ,l − δ b ) , where − N + 2 may simply be replaced by − N in the notation, giving the firststatement of the proposition.Using the second microlocal estimate (4.13) instead of (4.12) gives, completelyanalogously, the second statement of the proposition.The reversed inequality version on the orders is completely analogous. (cid:3) Proof of Theorem 1.1 for real σ . We start by showing a slight improvement of thestatement of Proposition 4.16. Namely, as soon as the nullspace of P ( σ ) is trivial,the usual argument allows the last relatively compact term in Proposition 4.16 tobe dropped, so that (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b , and this is uniform for σ in compact sets in R \ { } . Indeed, if this is not true, thereare sequences σ j in the fixed compact set, u j ∈ H ˜ r − / ,l b , which we may normalizeto (cid:107) u j (cid:107) H ˜ r − / ,l b = 1, with ˆ P ( σ j ) u j ∈ H ˜ r − / ,l +1b such that1 = (cid:107) u j (cid:107) H ˜ r − / ,l b > j (cid:107) ˆ P ( σ j ) u j (cid:107) H ˜ r − / ,l +1b , so ˆ P ( σ j ) u j → H ˜ r − / ,l +1b . But by the weak compactness of the unit ball,there is a weakly convergent subsequence, which we do not indicate in notation,converging to some u ∈ H ˜ r − / ,l b and one may also assume that σ j also converges(by passing to another subsequence). In particular, due to the compactness of theinclusion H ˜ r − / ,l b → H − N,l − δ b , u j converges to u in H − N,l − δ b strongly, so by thefirst estimate of Proposition 4.16, using that the first term on the right hand sidegoes to 0, we conclude that 1 ≤ C (cid:107) u (cid:107) H − N,l − δ b , so in particular u (cid:54) = 0. On the other hand, ˆ P ( σ j ) u j → ˆ P ( σ ) u in H ˜ r − / − (cid:15),l +1 − (cid:15) b , (cid:15) >
0, so we conclude that ˆ P ( σ ) u = 0, which contradicts the triviality of thenullspace.If P ( σ ) = P ( σ ) ∗ , the triviality of the nullspace, on the other hand, follows fromthe standard results involving the absence of embedded eigenvalues: the resultsthus far, as ˜ r + l > fact in H ∞ ,l b , i.e. is conormal. Then a generalized and extended version of theboundary pairing formula of [11], using the approach of Isozaki [10], as given in[24, Proposition 7] (the Feynman and anti-Feynman function spaces correspondto the incoming and outgoing resolvents), shows that in fact it is in ˙ C ∞ ( X ) andthen unique continuation arguments at infinity conclude the proof. Note that if P ( σ ) (cid:54) = P ( σ ) ∗ , the uniformity of our estimates still implies that for P ( σ ) − P ( σ ) ∗ suitably small, the triviality of the nullspace holds.Notice that for ˆ P ( σ ) ∗ has the same properties as ˆ P ( σ ) except that we need toreplace Im α ± ( σ ) by their negatives, so actually we have proved two estimates(4.20) (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b ,u ∈ H ˜ r − / ,l b , ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b , (4.21) (cid:107) v (cid:107) H ˜ r (cid:48)− / ,l (cid:48) b ≤ C (cid:107) ˆ P ( σ ) ∗ v (cid:107) H ˜ r (cid:48)− / ,l (cid:48) +1b ,v ∈ H ˜ r (cid:48) − / ,l (cid:48) b , ˆ P ( σ ) ∗ v ∈ H ˜ r (cid:48) − / ,l (cid:48) +1b , where we may take l < − / α + ( σ ) , ˜ r + l − Im α − ( σ ) > , ˜ r (cid:48) − / − (˜ r − / , l (cid:48) = − l − , for then ˜ r (cid:48) + l (cid:48) + Im α − ( σ ) = − ˜ r + 1 − l − α − ( σ ) < ,l (cid:48) = − / − ( l + 1 / > − / − Im α + ( σ ) , and now the spaces on the left, resp. right, hand side of (4.20) and right, resp. left,hand side of (4.21) are duals of each other.There is a slight subtlety in that we only have(4.22) ˆ P ( σ ) : X σ = { u ∈ H ˜ r − / ,l b : ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b } → Y = H ˜ r − / ,l +1b , rather than H ˜ r − / ,l b → H ˜ r − / ,l +1b , but the treatment of this is standard, as in[18, Section 2.6] and [17, Section 4.3]. Indeed, certainly injectivity is immediatefrom (4.20). For surjectivity note that (4.21) implies that given f in the dual of H ˜ r (cid:48) − / ,l (cid:48) b , which is H ˜ r − / ,l +1b , there exists u in the dual of H ˜ r (cid:48) − / ,l (cid:48) +1b , whichis H ˜ r − / ,l b such that ˆ P ( σ ) u = f . To see this claim, one considers the conjugatelinear functional v (cid:55)→ (cid:104) f, v (cid:105) , defined for v ∈ H ˜ r (cid:48) +3 / ,l (cid:48) b (so ˆ P ( σ ) ∈ H ˜ r (cid:48) − / ,l (cid:48) +1b automatically) which by (4.21) satisfies |(cid:104) f, v (cid:105)| ≤ C (cid:107) ˆ P ( σ ) ∗ v (cid:107) H ˜ r (cid:48)− / ,l (cid:48) +1b ; thus wecan consider the conjugate linear functional from the range of ˆ P ( σ ) ∗ on H ˜ r (cid:48) +3 / ,l (cid:48) b to C given by ˆ P ( σ ) ∗ v (cid:55)→ (cid:104) f, v (cid:105) which is therefore continuous when the range isequipped with the H ˜ r (cid:48) − / ,l (cid:48) +1b norm. By the Hahn-Banach theorem, it can beextended to H ˜ r (cid:48) − / ,l (cid:48) +1b , i.e. there exists an element u of the dual space H ˜ r − / ,l b such that (cid:104) u, ˆ P ( σ ) ∗ v (cid:105) = (cid:104) f, v (cid:105) for all v ∈ H ˜ r (cid:48) +3 / ,l (cid:48) b , in particular for all Schwartz v , which is to say ˆ P ( σ ) u = f . But then ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b , so u ∈ X σ , showingsurjectivity. This establishes the invertibility of ˆ P ( σ ) as stated.The case of second microlocal spaces is completely analogous, and gives theinvertibility of ˆ P ( σ ) as a map(4.23) ˆ P ( σ ) : X σ = { u ∈ H s,r,l sc , b : ˆ P ( σ ) u ∈ H s − ,r +1 ,l +1sc , b } → Y = H s − ,r +1 ,l +1sc , b . IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 31 (cid:3)
Remark . We remark here that X σ given by (4.23) is easily seen to have theproperty that H s,r +1 ,l sc , b (which is a subspace of it) is dense in it. Indeed, one simplyneeds to show regularizability in the second, sc-decay, order. This is accomplishedby taking a family Λ (cid:15) ∈ Ψ , − , , b ( X ) uniformly bounded in Ψ , , , b ( X ), converging toId in Ψ ,(cid:15), , b ( X ), (cid:15) >
0. Now, for u ∈ X σ we have Λ (cid:15) u → u in H s,r,l sc , b (which followsfrom Λ (cid:15) → Id strongly on H s,r,l sc , b ), and similarly Λ (cid:15) ˆ P ( σ ) u → ˆ P ( σ ) u in H s − ,r +1 ,l +1sc , b .However, regarding ˆ P ( σ ) u , what we must actually show is that ˆ P ( σ )Λ (cid:15) u → ˆ P ( σ ) u in H s − ,r +1 ,l +1sc , b . But ˆ P ( σ )Λ (cid:15) u = Λ (cid:15) ˆ P ( σ ) u + [ ˆ P ( σ ) , Λ (cid:15) ] u , with [ ˆ P ( σ ) , Λ (cid:15) ] uniformlybounded in a space with one additional order of sc-decay (and differential order!)relative to the products, namely Ψ , − , − , b ( X ), converging to 0 in Ψ , − (cid:15), − , b ( X ), so[ ˆ P ( σ ) , Λ (cid:15) ] u → H s − ,r +1 ,l +1sc , b , and thus in H s − ,r +1 ,l +1sc , b . This shows Λ (cid:15) u → u in X σ , so H s,r +1 ,l sc , b is dense in X σ . Since the inclusion map H s,r +1 ,l sc , b → X σ is continuous,and since C ∞ ( X ) is dense in H s,r +1 ,l sc , b , we conclude that ˙ C ∞ ( X ) is also dense in it.As for X σ in (4.22), one can show the density statement by noting that if u ∈ H ˜ r − / ,l b with ˆ P ( σ ) u ∈ H ˜ r − / ,l +1b then u ∈ H ˜ r − / , ˜ r + l − / ,l sc , b with ˆ P ( σ ) u ∈ H ˜ r − / , ˜ r + l +1 / ,l +1sc , b . This is almost a special case of the above discussion taking s = ˜ r + 3 / r = ˜ r + l − /
2, with the only issue being that u ∈ H s − ,r,l sc , b (andˆ P ( σ ) u ∈ H s − ,r +1 ,l +1sc , b ) not u ∈ H s,r,l sc , b . But this is easily overcome: ˆ P ( σ ) u ∈ H s − ,r +1 ,l +1sc , b and ellipticity of ˆ P ( σ ) in the first order shows that u ∈ H s,r,l sc , b . Thus,the argument of the previous paragraph is applicable, and shows that ˙ C ∞ ( X ) isdense in X σ . It also shows that even though elements of X σ only have a prioridifferential regularity ˜ r − /
2, in fact, in the scattering sense, they have differentialregularity ˜ r + 3 / σ . We remark that the regular-ization issues and the ways of dealing with them are completely analogous to thereal σ case, and we will not comment on these explicitly. As we have already seen,near the zero section the term − στ is the most important part of the principalsymbol since the other terms vanish quadratically at the zero section, so it is usefulto consider ˜ P ( σ ) = σ − ˆ P ( σ ) , so ˜ p ( σ ) = σ − ˆ p ( σ ) = − τ + σ | σ | − ( τ + µ ) , hence Re ˜ p ( σ ) = − τ + (Re σ ) | σ | − ( τ + µ ) , Im ˜ p ( σ ) = − (Im σ ) | σ | − ( τ + µ ) . Thus, Im ˜ p ( σ ) ≤ σ ≥
0, which means one can propagate estimates forwardsalong the Hamilton flow of Re ˜ p ( σ ); similarly, if Im σ ≤
0, one can propagateestimates backwards along the Hamilton flow of Re ˜ p ( σ ). As we have seen, forIm σ (cid:54) = 0, the operator is actually only characteristic at the front face. The principalsymbol computation replacing Lemma 4.1 is: Lemma 4.18.
Let A ∈ Ψ r − , r + l ) , l +1sc , b have principal symbol a given by (4.3) .The principal symbol H Re ˜ p ( σ ) a of i [Re ˜ P ( σ ) , A ] in Ψ r, r + l ) − , l sc , b ( X ) is (4.24) x − l ( τ + µ ) ˜ r − / (cid:16) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − σ | σ | x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) = x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − σ | σ | ( l + ˜ r ) τ ( τ + µ ) (cid:17) . Proof.
We have (cid:110) Re σ | σ | x ( τ + µ ) − xτ b , x − l − ( τ + µ ) ˜ r − / (cid:111) = (2 Re σ | σ | x τ b − x )( − l − x − l − ( τ + µ ) ˜ r − / − (2 Re σ | σ | x ( τ + µ ) − xτ b ) x − l − r − / τ b ( τ + µ ) ˜ r − / . Expanding and rearranging,= 4( l + 1 / x − l ( τ + µ ) ˜ r − / + 4(˜ r − / x − l τ ( τ + µ ) ˜ r − / − σ | σ | ( l + 1 / x − l +1 τ b ( τ + µ ) ˜ r − / − σ | σ | (˜ r − / x − l +1 τ b ( τ + µ ) ˜ r − / = x − l ( τ + µ ) ˜ r − / (cid:16) (cid:0) ( l + 1 / τ + µ ) + (˜ r − / τ (cid:1) − σ | σ | x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) = x − l ( τ + µ ) ˜ r − / (cid:16) (cid:0) ( l + ˜ r ) τ + ( l + 1 / µ (cid:1) − σ | σ | x ( l + ˜ r ) τ b ( τ + µ ) (cid:17) . Rewriting from the second microlocal perspective, substituting τ = xτ b , µ = xµ b ,completes the proof. (cid:3) For a moment, let us ignore the contributions to Im ˜ P ( σ ) from subprincipalterms. Again, the ( l + 1 / µ term is the dominant one in the expression on theright hand side of (4.24), so the commutator has a sign that agrees with that of l + 1 /
2. Since the imaginary part has the same (indefinite) sign as − Im σ , thismeans that for Im σ > l + 1 / <
0, the twosigns agree, and one has the desired estimate; a similar conclusion holds if Im σ < l + 1 / >
0. We can ensure the negativity/positivity of the parenthetical term(in terms of a multiple of τ + µ which, or whose negative, is bounded belowby a positive constant) by using a cutoff on the blown up space, in τ τ + µ (or τ /µ ), which is identically 1 near 0 and has small support and whose differential issupported in the elliptic set: χ ( τ τ + µ ). We do need to add a cutoff to localize nearthe scattering zero section, but as the operator is elliptic outside the zero section,on the differential of such a cutoff we have elliptic estimates, so these terms arecontrolled. See Figure 4. IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 33
In more detail, the full computation then involves(4.25) i ( ˜ P ( σ ) ∗ A − A ˜ P ( σ )) = (Im ˜ P ( σ ) A + A Im ˜ P ( σ )) + i [Re ˜ P ( σ ) , A ]where A ∈ Ψ r − , r + l ) , l +1sc , b ( X ) as before, namely has principal symbol χa with χ = χ ( µ ) χ ( τ ) χ (cid:16) τ τ + µ (cid:17) . Now, the first term of (4.25) has the correct sign at the principal symbol level asalready discussed (when l < − / σ >
0, as well as when l > − / σ <
0, and when the subprincipal terms of (4.25) are ignored). However, as ithas one order less sc-decay than the main term (but it degenerates as Im σ → . ............................................................................................................................................................................................................................................................. ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................. ................................... ..................................... ....................................... ....................... .................... ................. ................ ................ ............... ✲ x . ............... ................ ................ ................. .................... ....................... ....................................... ..................................... ................................... .................................. .......................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................. .................. ......... ........ ........ ......... ................... ................. .............. ............. ............................................................................................................. ............................................................................................................................... ✻ τ ................................................................ ✛ µ ........................................................................................................................... .................. .................. ............ .............. ............... ................. ............... ............... .............. .............. ................ ............... ...................................................................................................... ............................................................................................................... ................................ χ Figure 4.
The support of χ on the second microlocal space, indi-cated by the rectangular box. The characteristic set is the circularcurve tangent to the µ axis at the b-face, given by the sc-zerosection. Lemma 4.19.
We have
Im ˜ P ( σ ) = − (Im σ ) T ( σ ) + W ( σ ) with T ( σ ) = T = (cid:88) j T j + (cid:88) j T j T (cid:48) j + (cid:88) j T (cid:48) j T j + T (cid:48)(cid:48) with T j = T ∗ j ∈ Ψ , , − , b ( X ) (where T j is | σ | − times the T j of (3.8) ), T (cid:48) j = ( T (cid:48) j ) ∗ ∈ Ψ , − , − , b ( X ) , T (cid:48)(cid:48) j = ( T (cid:48)(cid:48) j ) ∗ ∈ Ψ , − , − , b ( X ) , W = W ∗ ∈ Ψ , − , − , b ( X ) , so T (cid:48) j , W areone order lower than T in terms of sc-decay, T (cid:48)(cid:48) two orders lower, and where, withthe notation of (3.4) and (3.5) , W ( σ ) has principal symbol Im (cid:16) x ( σ − a + b ) τ b + (cid:88) j x ( σ − a j + b j )( µ b ) j − xα + ( σ ) (cid:17) . Proof.
This is an immediate consequence of (3.3), (3.8) and (3.9). (cid:3)
We now prove
Proposition 4.20.
For l < − / α + ( σ )) and Im σ > , as well as for l > − / α + ( σ )) and Im σ < , with ˜ r, r arbitrary in either case, we have theestimates (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l − δ b ) and (4.26) (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N,l − δ b ) . These estimates hold in the sense that if u ∈ H ˜ r (cid:48) − / ,l b , resp. u ∈ H s (cid:48) ,r (cid:48) ,l sc , b for some s (cid:48) , ˜ r (cid:48) , r (cid:48) , and if ˆ P ( σ ) u is in the space indicated on the right hand side, then u is inthe space indicated on the left hand side and the estimate holds.Remark . The proof below directly strengthens the norm on the left hand sideof (4.26) to (cid:107) u (cid:107) H s,r,l sc , b + (cid:107) u (cid:107) H s,r +1 / ,l − / , b thanks to the (cid:107) T j A u (cid:107) terms in (4.34): ateach point on the lift of sc T ∗ ∂X X to the second microlocal space [ sc T ∗ X ; o ∂X ], oneof the T j A is an elliptic element of Ψ ∗ , ˜ r + l,l − / , b . This estimate could be furtherstrengthened by estimating ˜ P ( σ ) u on the left hand side of (4.34) in a correspondingdual space.In fact, perhaps the most systematic way of approaching this problem is to blowup the corner of [ sc T ∗ X ; o ∂X ] at the intersection of the front face (the b-face) andthe lift of sc T ∗ ∂X X , or equivalently, and in an analytically better manner for the samereasons as discussed regarding second microlocalization in Section 2, the corner of[ b T ∗ X ; ∂ T ∗ X ] at the intersection of the lift of b T ∗ ∂X X and the front face (thesc-face). The symbol, pseudodifferential and Sobolev spaces will have four orders,with a new order arising from the symbolic orders at the new front face. A simplecomputation shows that the vector fields x / xD x , x / D y j are altogether ellipticin the interior of the new front face, and thus this new front face corresponds to thescattering algebra in x / . Notice, however, that the above order convention makes x / order − / x / in the x / -scattering algebra. Then˜ P ( σ ) is in the set of pseudodifferential operators of order (without giving a name tothe space) 2 , , − , −
1. Then (cid:107) u (cid:107) H s,r,l sc , b + (cid:107) u (cid:107) H s,r +1 / ,l − / , b is equivalent to the normwith orders s, r + 1 / , r + l, l (the order at the new face is the sum of the sc-b ordersat the two adjacent faces, and for both terms these are the same, so microlocallythere the two terms are equivalent), and the result is an estimate (without givinga name to the space) (cid:107) u (cid:107) s,r +1 / ,r + l,l ≤ C ( (cid:107) ˜ P ( σ ) u (cid:107) s − ,r +1 / ,r + l,l +1 + (cid:107) u (cid:107) H − N,l b ) . Note that for an elliptic operator (in every sense) of the same order as ˜ P ( σ ) the normof the first term on the right hand side would be of type s − , r + 1 / , r + l + 1 , l + 1,so the only sense in which the estimate is not an elliptic estimate is at the new frontface, where there is a loss of an order corresponding to real principal type estimates;indeed, T ( σ ) is a subprincipal term there, so the two terms of (4.25) have the sameorder at the front face. A useful feature then is that the characteristic set at the new IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 35 face is purely described by Re ˜ P ( σ ), and is independent of σ , so in this formulationthe Fredholm theory is on fixed spaces for all σ with Im σ > σ , which is our goalin the subsequent proof of Theorem 1.1, so we do not develop this theory furtherhere. Proof.
We start by considering the term(4.27) Im ˜ P ( σ ) A + A Im ˜ P ( σ )of (4.25) and use Lemma 4.19, first dealing with the T ( σ ) term, namely(4.28) − (Im σ )( T ( σ ) A + AT ( σ )) . As before, we will apply this to u and take the inner product with u , resulting in − (Im σ ) (cid:104) ( T ( σ ) A + AT ( σ )) u, u (cid:105) . We again write A = A , A = A ∗ , A ∈ Ψ ˜ r − / , ˜ r + l,l +1 / , b which one can certainlydo by choosing A first, with the desired principal symbol, as in Proposition 4.10.Then T A + AT = T A + A T = 2 A T A + [ T, A ] A + A [ A , T ]= 2 A T A + [[ T, A ] , A ] , and now the second term on the right hand side is two orders lower than the firstdue to the double commutator. On the other hand, by Lemma 4.19,2 A T A = 2 (cid:88) j A ∗ T ∗ j T j A + 2 A ∗ T ∗ j T (cid:48) j A + 2 A ∗ ( T (cid:48) j ) ∗ T j A + 2 A ∗ T (cid:48)(cid:48) A , with the first term non-negative. The second and third terms are lower order byone order of sc-decay. This is not sufficient to regard them as error terms sincethey are of the same order as the main commutator term; the same is true for theother (namely, other than (4.28)) term W ( σ ) A + AW ( σ ) of (4.27). However, thisis not surprising: recall the factor x − iα + ( σ ) above in the asymptotics; for non-real α + the real part of the exponent is potentially large, so the constraints on l needto change just as in the real σ case. Now,(4.29) − (Im σ ) (cid:104) ( T ( σ ) A + AT ( σ )) u, u (cid:105) = − σ ) (cid:88) j (cid:107) T j A u (cid:107) − σ ) (cid:88) j (cid:104) A ∗ T ∗ j T (cid:48) j A u, u (cid:105)− σ ) (cid:88) j (cid:104) A ∗ ( T (cid:48) j ) ∗ T j A u, u (cid:105) − σ ) (cid:104) A ∗ T (cid:48)(cid:48) A u, u (cid:105) The estimate(4.30) |(cid:104) A ∗ T ∗ j T (cid:48) j A u, u (cid:105)| ≤ (cid:15) (cid:107) T j A u (cid:107) + (cid:15) − (cid:107) T (cid:48) j A u (cid:107) allows, for small (cid:15) >
0, to absorb the first term of its right hand side into (cid:107) T j A u (cid:107) ,while the second one now corresponds to (cid:104) A ( T (cid:48) j ) ∗ T (cid:48) j A u, u (cid:105) , and ( T (cid:48) j ) ∗ T (cid:48) j has thesame order as T (cid:48)(cid:48) , so it can be treated the same way, namely it is simply part ofthe error term.We now turn to the term W ( σ ) A + AW ( σ ) of (4.27) as well as to the other term(other than (4.27)), i [Re ˜ P ( σ ) , A ], of (4.25). The operator W ( σ ) has principal sym-bol − x Im( α + ( σ )) at the zero section by Lemma 4.19, which can be handled just as in the real σ case. Indeed, the second term of (4.25) plus the W ( σ ) contributionto the first term, i.e. W ( σ ) A + AW ( σ ) + i [Re ˜ P ( σ ) , A ] , is in Ψ −∞ , r + l ) − , l sc , b ( X ) and has principal symbol, modulo terms controlled byelliptic estimates (arising from dχ ),(4.31) x − l +˜ r )+1 ( τ + µ ) ˜ r − / (cid:16) (cid:0) ( l + ˜ r − Im α + ( σ )) τ + ( l + 1 / − Im α + ( σ )) µ (cid:1) − σ | σ | ( l + ˜ r ) τ ( τ + µ ) (cid:17) χ. Now, with χ chosen as discussed prior to the statement of Lemma 4.19, so inparticular with χ having sufficiently small support, the first and third terms of(4.31) can be absorbed into the second, and thus we can write (4.31) as b and take B ∈ Ψ ∗ ,l +˜ r − / ,l sc , b with principal symbol b so that(4.32) W ( σ ) A + AW ( σ ) + i [Re ˜ P ( σ ) , A ] = ∓ B ∗ B + E + F, where ∓ corresponds to ∓ ( l + 1 / − Im α + ( σ )) > B ∈ Ψ ∗ ,l +˜ r − / ,l sc , b has principalsymbol b , E ∈ Ψ ∗ , l +˜ r ) − ,l sc , b arising from the dχ terms, and F ∈ Ψ ∗ , l +˜ r ) − ,l sc , b islower order in the sc-decay sense. Applying to u and pairing with u gives(4.33) (cid:107) Bu (cid:107) ≤ |(cid:104) ˜ P ( σ ) u, Au (cid:105)| + |(cid:104) Eu, u (cid:105)| + |(cid:104) F u, u (cid:105)| , and the E term is controlled by elliptic estimates.Combining (4.29) and (4.32), we deduce that(4.34) (cid:104) i ( ˜ P ( σ ) ∗ A − A ˜ P ( σ )) u, u (cid:105) = − σ ) (cid:88) j (cid:107) T j A u (cid:107) ∓ (cid:107) Bu (cid:107) − σ ) (cid:88) j (cid:104) A ∗ T ∗ j T (cid:48) j A u, u (cid:105)− σ ) (cid:88) j (cid:104) A ∗ ( T (cid:48) j ) ∗ T j A u, u (cid:105) + (cid:104) Eu, u (cid:105) + (cid:104) F u, u (cid:105) − σ ) (cid:104) A ∗ T (cid:48)(cid:48) A u, u (cid:105) , and the first two terms on the right hand side have matching signs under thehypotheses of the proposition, while the T (cid:48)(cid:48) term can be absorbed into the F termby modifying F while keeping its order.Now, b is an elliptic multiple of x / a , so (cid:107) x / A u (cid:107) is controlled by (cid:107) Bu (cid:107) modulo terms that can be absorbed into |(cid:104) F u, u (cid:105)| (by modifying F without changingits order). Thus, modulo terms absorbed into the F term,(4.35) (cid:104) ˜ P ( σ ) u, Au (cid:105) = (cid:104) x − / A ˜ P ( σ ) u, x / A u (cid:105) is controlled by(4.36) (cid:107) Bu (cid:107)(cid:107) x − / A ˜ P ( σ ) u (cid:107) ≤ (cid:15) (cid:107) Bu (cid:107) + (cid:15) − (cid:107) x − / A ˜ P ( σ ) u (cid:107) , and now the first term on the right hand side can be absorbed into the second term ofthe right hand side of (4.34). This gives, using the controlled E terms, and simplydropping the term − σ ) (cid:80) j (cid:107) T j A u (cid:107) (after absorbing the third and fourthterms on the right hand side of (4.34) into the first, using (4.30)) which matches IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 37 the sign of ∓(cid:107) Bu (cid:107) , with elliptic estimates for the scattering differentiability order,and with r = ˜ r + l − / (cid:107) u (cid:107) H s,r,l sc , b ≤ C ( (cid:107) ˜ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b + (cid:107) u (cid:107) H − N,r − / ,l sc , b ) . Since (cid:107) u (cid:107) H − N,r − / ,l sc , b can be bounded by a small multiple of (cid:107) u (cid:107) H − N,r,l sc , b plus a largemultiple of (cid:107) u (cid:107) H − N, − N,l sc , b , with the former being absorbable into the left hand side,this proves the estimates of Proposition 4.20 with l − δ replaced by l in the last termon the right hand side. Again, a regularization argument shows that the estimateshold in the stronger sense that if the right hand side is finite, so is the left handside.Finally, we can use the normal operator estimate of Lemma 4.13 (with the fac-tored out σ being irrelevant) as in the proof of Proposition 4.16 to prove the propo-sition. (cid:3) Again, as soon as the nullspace is trivial, the usual argument allows the lastrelatively compact term to be dropped, so that (cid:107) u (cid:107) H ˜ r,l b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r,l +1b , as well as (cid:107) u (cid:107) H s,r,l sc , b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H s − ,r +1 ,l +1sc , b , and this is uniform for σ in compact sets in { Im σ > } when l is sufficiently nega-tive. Since taking adjoints changes the sign of Im σ , thus if l > − / α + ( σ )),but ˜ r is arbitrary, we still have analogous estimates for ˆ P ( σ ) ∗ , and thus Fredholmand invertibility (in the latter case under nullspace assumptions) statements forˆ P ( σ ).Allowing Im σ ≥ σ away from 0), only very minor changes are needed to theproof of Proposition 4.20, as we show below. Proof of the general case of Theorem 1.1.
The cutoff near the zero section now be-comes important, just as for real σ . We again proceed to compute with χ = χ ( µ ) χ ( τ ) χ (cid:16) τ τ + µ (cid:17) , with χ , χ , χ identically 1 near 0 of compact support sufficiently close to 0 andwith χ having relatively large support so that supp χ ( . ) ∩ supp dχ ( . ) is disjointfrom the zero set of Re ˆ p ( σ ) as above, so elliptic scattering estimates control the dχ term, and χ also chosen so that on the one hand elliptic sc-b estimates control thesupp dχ ( . ) ∩ supp χ ( . ) ∩ supp dχ ( . ) region and on the other hand in Lemma 4.18the ( l + 1 / µ term dominates the others as discussed after that lemma. On theother hand, doing the computation in the b-notation, (cid:110) Re σ | σ | x ( τ + µ ) − xτ b , χ ( x µ ) (cid:111) = 2 (cid:16) σ | σ | x τ b − x (cid:17) x µ χ (cid:48) ( x µ )= − x (1 − Re σ | σ | τ ) µ χ (cid:48) ( µ ) , so if χ is arranged to have sufficiently small support, say in [ − (Re σ ) / , (Re σ ) / − χ (cid:48) is a square, this simply adds anotherterm of the correct, positive, sign to our symbolic computation if Im σ ≤ l + 1 / >
0; it adds a term of the wrong sign if Im σ ≥ l + 1 / < r + l > W ( σ ) terms.The full computation proceeds exactly as above when Im σ was bounded awayfrom 0; now the terms − σ ) (cid:80) j (cid:107) T j A u (cid:107) in (4.34) are of no use (unlike before,when they could have been used to give a stronger result, see Remark 4.21). Thenet result is again an estimate(4.38) (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l b ) . Now, the last term can be estimated using the normal operator as above, yielding (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C ( (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b + (cid:107) u (cid:107) H − N,l − δ b ) . Again, as soon as the nullspace is trivial, the usual argument allows the last rela-tively compact term to be dropped, so that (cid:107) u (cid:107) H ˜ r − / ,l b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b , and this is uniform for σ in compact sets in R \ { } , times [0 , R ] along the imagi-nary direction with l as above. The second microlocal version is, under the sameassumptions, with r = ˜ r + l − / (cid:107) u (cid:107) H s,r,l sc , b ≤ C (cid:107) ˆ P ( σ ) u (cid:107) H s,r +1 ,l +1b . This proves Theorem 1.1. (cid:3) High energy/semiclassical results
In this final section we consider high energy scattering, which in turn can berescaled to a semiclassical problem. Since the arguments are very similar to thebounded non-zero σ ones, we only sketch the proofs.For the high energy estimates we need to be more specific on the σ -dependenceof P ( σ ). Recall from (3.3) that the conjugated operator takes the formˆ P ( σ ) = ˆ P (0) + σ ˆ Q − σ (cid:16) x D x + i n − x + 12 x (cid:0) − a σ + a + b σ − σ − a (cid:48) − b (cid:48) (cid:1)(cid:17) withˆ P (0) = P (0) − xa (cid:48) ∈ x Diff ( X ) + S − − δ Diff ( X ) , ˆ Q = Q − b (cid:48) x − xa ( x D x ) − (cid:88) j xa j ( xD y j ) ∈ x Diff ( X ) + S − − δ Diff ( X ) , where we allowed smooth dependence of b , b j , b (cid:48) on σ . From now we assume that b , b j , b (cid:48) are symbolic in σ , with b , b j order and b (cid:48) order , while their imaginarypart is order − , resp. . This is the natural order: when ˆ P ( σ ) is the temporalFourier transform of a wave operator, we expect these orders, with the imaginarypart statement coming from the formal self-adjointness of wave operators. IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 39
The semiclassical rescaling is arrived at by multiplying ˆ P ( σ ) by h , where h ∈ (0 ,
1] is understood as comparable to | σ | − , so h | σ | is in a compact subset of C \{ } .The rescaling gives, with z = hσ ,(5.1)ˆ P h ( z ) = h ˆ P ( h − z )= h ˆ P (0) + zh ˆ Q − z (cid:16) hx D x + ih n − x + 12 x (cid:0) − a z + a h + b z − h z − a (cid:48) − hb (cid:48) (cid:1)(cid:17) , and now h ˆ P (0), h ˆ Q are semiclassical differential operatorsˆ P h (0) = h ˆ P (0) = h P (0) − h xa (cid:48) ∈ x Diff , (cid:126) ( X ) + S − − δ Diff , (cid:126) ( X ) , ˆ Q h = h ˆ Q = hQ − hb (cid:48) x − xa ( hx D x ) − (cid:88) j xa j ( hxD y j ) ∈ x Diff , (cid:126) ( X ) + S − − δ Diff , (cid:126) ( X ) , and the parenthetical final term in (5.1) is in x Diff , (cid:126) ( X ) + S − − δ Diff , (cid:126) ( X ). Notethat terms with an extra h (beyond the h incorporated into the derivatives) aresemiclassically subprincipal, while those with an extra h are sub-subprincipal, sofor instance, modulo semiclassically sub-subprincipal terms, ˆ P h (0) = h P (0), i.e. inthe high energy limit, the potential term a (cid:48) becomes, in this sense, irrelevant (withthe analogous conclusion also holding for the parenthetical final term in (5.1)). Weremark also that the symbolic order of b , b j means that the corresponding terms inˆ Q h , as well as in the final term of (5.1), are semiclassically principal, as is b (cid:48) sinceit comes with an extra h − factor, cancelling out the overall h , thus not completelynegligible. However, all these terms are subprincipal in terms of the scatteringdecay, i.e. they vanish to leading order at x = 0.Now, ˆ P h ( z ), just like the similarly defined P h ( z ) = h P ( h − z ), is not semiclas-sically elliptic even in x >
0. Indeed, the semiclassical principal symbol of P h ( z )is p (cid:126) ( z ) = G + z ( b τ (cid:126) + (cid:88) j b j ( µ (cid:126) ) j ) + zxhb (cid:48) − z , while that of ˆ P h ( z ) is then, corresponding to the conjugation being a symplectomor-phism at the phase space level (namely: translation in the fibers by the differential − σx − dx of the phase, σ/x ), can be arrived at by replacing τ (cid:126) by τ (cid:126) − z ; theseare elliptic at fiber infinity, but vanish for appropriate finite τ (cid:126) , µ (cid:126) for z real, andindeed at the zero section even if z is complex. In particular,ˆ p (cid:126) ( z ) | x =0 = τ (cid:126) + µ (cid:126) − zτ (cid:126) . The semiclassical flow structure in the scattering setting was discussed in [20];the symplectomorphism corresponding to the conjugation simply translates this by − σx − dx . Thus, for ˆ P h ( z ), there are two radial sets at x = 0, one of which is atthe zero section; one of these is a source, the other is a sink, including in the senseof dynamics from the interior, x > ∂X . Thus, when the scattering decay order r is above the thresholdvalue, i.e. one treats the radial set other than the zero section as the incoming one, at the radial set other than the zero section, one gets automatic semiclassical esti-mates there and one can propagate them towards the outgoing radial set, stopping(since we need to discuss 2-microlocal estimates) before arriving at the resolved zerosection at ∂X . Since we no longer have ellipticity over X ◦ , it is important that forall points away from the b-front face, at h = 0, the flow in the backward (if theradial set outside the zero section is a source) or forward (if the radial set outsidethe zero section is a sink) direction tends to the radial set outside the zero section.This follows from the non-trapping hypothesis in the scattering setting. At thispoint it remains to do an estimate at the zero section, acting as the outgoing radialset. For this we can use essentially the same commutant as in the non-semiclassicalsetting for the potentially non-zero Im σ (recall that the real σ argument used amore delicate cancellation that is not in general robust), namely the weights, a cutoff in a neighborhood of the blown up zero section, which now must include a cutoffin the interior as well. Thus, we take χ = χ ( µ (cid:126) ) χ ( τ (cid:126) ) χ ( x ) χ ( τ (cid:126) / ( τ (cid:126) + µ (cid:126) ))= χ ( x µ , (cid:126) ) χ ( x τ , (cid:126) ) χ ( x ) χ ( τ , (cid:126) / ( τ , (cid:126) + µ , (cid:126) )) . The cutoff contributions before came from χ ( µ ) in view of the support of thedifferential of χ ; now χ ( µ (cid:126) ) plays an analogous role. Since we computed Poissonbrackets using the b-structure, the contribution came from the x dependence of χ ( x µ , (cid:126) ), and this x dependence is completely analogous to that of χ , with bothcutoffs being identically 1 near 0. Correspondingly, they contribute with the samesign, meaning they both need to be controlled (i.e. they have a sign opposite to thatgiven by the weight, where now l is to be below the threshold value), as they arefrom the estimate propagated from the incoming radial point; hence one obtainsthe zero section outgoing radial estimates.A completely analogous argument works when at the radial set outside the zerosection one has scattering decay order r below the threshold value, and correspond-ingly at the zero section, one has b-decay above the threshold regularity. In thiscase, one needs to start at the zero section (which is the incoming radial set), takingadvantage of the cutoffs mentioned above having the correct sign (matching that ofthe weights), and then propagate the estimates using the non-trapping assumptionsand the standard semiclassical scattering propagation results as in [20].In combination, these prove both semiclassical high and low b-decay statements,namely the following analogue of Proposition 4.20 (without using the normal op-erator argument, thus no gain of decay in the error term, but with a gain in thesemiclassical parameter h ), with the strengthened (in that real σ is allowed) state-ment that arises in the proof of Theorem 1.1 given after the proposition, see (4.38): Proposition 5.1.
For l < − / h Im( α + ( h − z )) and Im z ≥ , as well as for l > − / h Im( α + ( h − z )) and Im z ≤ , with ˜ r, r arbitrary in either case, wehave the estimates (cid:107) u (cid:107) H ˜ r − / ,l b , (cid:126) ≤ Ch − ( (cid:107) ˆ P h ( z ) u (cid:107) H ˜ r − / ,l +1b , (cid:126) + h N (cid:107) u (cid:107) H − N,l b , (cid:126) ) and (cid:107) u (cid:107) H s,r,l sc , b , (cid:126) ≤ Ch − ( (cid:107) ˆ P h ( z ) u (cid:107) H s − ,r +1 ,l +1sc , b , (cid:126) + h N (cid:107) u (cid:107) H − N,l b , (cid:126) ) . Due to the h N factors, the last term on the right hand side of both estimatesof Proposition 5.1 can be absorbed into the left hand side. This again gives direct IMITING ABSORPTION PRINCIPLE, A LAGRANGIAN APPROACH 41 and adjoint estimates, and one concludes that, for h ∈ (0 , (cid:107) u (cid:107) H ˜ r − / ,l b , (cid:126) ≤ Ch − (cid:107) ˆ P h ( z ) u (cid:107) H ˜ r − / ,l +1b , (cid:126) , which translates to (cid:107) u (cid:107) H ˜ r − / ,l b , (cid:126) ≤ Ch (cid:107) ˆ P ( h − z ) u (cid:107) H ˜ r − / ,l +1b , (cid:126) , thus (cid:107) u (cid:107) H ˜ r − / ,l b , (cid:126) ≤ C | σ | − (cid:107) ˆ P ( σ ) u (cid:107) H ˜ r − / ,l +1b , (cid:126) , with ˜ r, l as before, provided that the non-trapping assumption holds.The second microlocal version is completely analogous, as is the complex spectralparameter version, proving Theorem 1.5. References [1] Jean-Michel Bony. Second microlocalization and propagation of singularities for semilinearhyperbolic equations. In
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Department of Mathematics, Stanford University, CA 94305-2125, USA
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