Strichartz estimates for the one-dimensional wave equation
aa r X i v : . [ m a t h . A P ] D ec STRICHARTZ ESTIMATES FOR THE ONE-DIMENSIONAL WAVEEQUATION
ROLAND DONNINGER AND IRFAN GLOGI ´C
Abstract.
We study the hyperboloidal initial value problem for the one-dimensionalwave equation perturbed by a smooth potential. We show that the evolution decomposesinto a finite-dimensional spectral part and an infinite-dimensional radiation part. Forthe radiation part we prove a set of Strichartz estimates. As an application we studythe long-time asymptotics of Yang-Mills fields on a wormhole spacetime. Introduction
Strichartz estimates were originally discovered in the context of the Fourier restrictionproblem [13] but only later their true power was exploited in the study of nonlinear waveequations [9]. To illustrate this point, consider for instance the Cauchy problem for thecubic wave equation in three spatial dimensions, ( ( ∂ t − ∆ x ) u ( t, x ) = u ( t, x ) ( t, x ) ∈ R × R u ( t, x ) = f ( x ) , ∂ t u ( t, x ) = g ( x ) ( t, x ) ∈ { } × R , (1.1)for given initial data f, g ∈ S ( R ), say. A weak formulation of Eq. (1.1) is provided byDuhamel’s formula u ( t, · ) = cos( t |∇| ) f + sin( t |∇| ) |∇| g + Z t sin(( t − t ′ ) |∇| ) |∇| (cid:0) u ( t ′ , · ) (cid:1) dt ′ (1.2)with the wave propagators cos( t |∇| ) and sin( t |∇| ) |∇| . The latter are the Fourier multipliersthat yield the solution to the free wave equation ( ∂ t − ∆ x ) u ( t, x ) = 0. The point is thatEq. (1.2) is a reformulation of Eq. (1.1) as a fixed point problem. Proving the existence ofsolutions to Eq. (1.1) therefore amounts to showing that the operator on the right-handside of Eq. (1.2) has a fixed point. The main issue then is to find suitable spaces thatare compatible with the free evolution and that allow one to control the nonlinear term.For the cubic equation (1.2) the Sobolev embedding ˙ H ( R ) ֒ → L ( R ) suffices but if oneincreases the power of the nonlinearity or the spatial dimension, a more sophisticatedargument is required. The crucial tool is provided by the Strichartz estimates which aremixed spacetime bounds on the wave propagators of the form k cos( t |∇| ) f k L pt ( R ) L q ( R d ) := (cid:18)Z R k cos( t |∇| ) f k pL q ( R d ) dt (cid:19) /p . k f k ˙ H s ( R d ) for certain admissible values of p , q , s , and d . For instance, the sine propagator satisfiesthe Strichartz estimate (cid:13)(cid:13)(cid:13)(cid:13) sin( t |∇| ) |∇| g (cid:13)(cid:13)(cid:13)(cid:13) L t ( R ) L ( R ) . k g k L ( R ) Both authors acknowledge support by the Austrian Science Fund FWF, Project P 30076, “Self-similarblowup in dispersive wave equations”. hich allows one to control a quintic nonlinearity in three dimensions. At the same time,the Strichartz estimates provide information on the long-time asymptotics which makesthem crucial in proving scattering.The physical effect that is responsible for the existence of Strichartz estimates is dis-persion. The latter refers to the observation that waves of different frequencies travel atdifferent speeds. In other words, a wave packet tends to spread out which leads to anaveraged decay that is quantified by the Strichartz estimates. The strength of the disper-sive decay depends strongly on the underlying spatial dimension: The higher the spacedimension, the more room there is for the wave to spread out. On the other hand, in theone-dimensional case, there is no dispersion at all and the evolution is a pure transportphenomenon. This precludes the existence of Strichartz estimates as is easily seen bynoting that u ( t, x ) = χ ( t − x ) for a χ ∈ C ∞ c ( R ) solves ( ∂ t − ∂ x ) u ( t, x ) = 0. By translationinvariance we have k u ( t, · ) k L q ( R ) = k χ k L q ( R ) and k u k L p ( R ) L q ( R ) = ∞ , unless p = ∞ . The weak dispersion in low dimensions causessevere difficulties in understanding the asymptotics of many models in quantum fieldtheory, see e.g. [8, 12, 7] for recent work.In this paper we show that one can recover Strichartz estimates even in the one-dimensional case if one studies a hyperboloidal evolution problem instead of the standardCauchy problem. The key observation is that the standard Cartesian coordinates arenot very well suited for describing radiation processes. The foliation induced by thestandard coordinates is singular at null infinity and therefore unnatural in this context,see e.g. [4] for a discussion on this. Consequently, as suggested in many physics pa-pers, e.g. [6, 14, 15, 1], we choose a hyperboloidal foliation instead, where the leavesare asymptotic to translated forward lightcones. In this setup we study the evolutionproblem for the one-dimensional wave equation with an arbitrary potential added (toavoid technicalities we restrict ourselves to smooth potentials). We show that the solu-tion decomposes into a finite dimensional part which is controlled by spectral theory andan infinite-dimensional “radiation” part which satisfies Strichartz estimates, provided acertain spectral assumption holds. We remark in passing that there are some technicalsimilarities with Strichartz estimates in the context of self-similar blowup established in[2, 3].As a first application we consider Yang-Mills fields on a wormhole geometry. Undera certain symmetry reduction, we study small-energy perturbations of an explicit Yang-Mills connection and prove its asymptotic stability in a Strichartz sense.1.1. Main results.
We use the hyperboloidal coordinates from [1] defined by Φ : R × ( − , → R , Φ( s, y ) := ( s − log p − y , artanh y ) . The map Φ is a diffeomorphism onto its image with inverseΦ − ( t, x ) = ( t − log cosh x, tanh x ) . In these coordinates, the one-dimensional wave equation( ∂ t − ∂ x ) v ( t, x ) = 0 (1.3)reads (cid:2) ∂ s + 2 y∂ s ∂ y + ∂ s − (1 − y ) ∂ y + 2 y∂ y (cid:3) u ( s, y ) = 0 , (1.4) here v ( t, x ) = u ( t − log cosh x, tanh x ). By testing with ∂ s u ( s, y ), we formally find theenergy identity12 dds (cid:20)Z − (1 − y ) | ∂ y u ( s, y ) | dy + Z − | ∂ s u ( s, y ) | dy (cid:21) = −| ∂ s u ( s, − | − | ∂ s u ( s, | (1.5)and this motivates the introduction of the following energy norm . Definition 1.1.
For functions ( f, g ) ∈ C ( − , × C ( − , energy norm k ( f, g ) k H by k ( f, g ) k H := Z − (1 − y ) | f ′ ( y ) | dy + Z − | g ( y ) | dy. Our main result is concerned with a more general class of wave equations, that is tosay, we study the initial value problem ( [ ∂ s + 2 y∂ s ∂ y + ∂ s − (1 − y ) ∂ y + 2 y∂ y + V ( y )] u ( s, y ) = 0 ( s, y ) ∈ (0 , ∞ ) × ( − , u ( s, y ) = f ( y ) , ∂ s u ( s, y ) = g ( y ) ( s, y ) ∈ { } × ( − , u : [0 , ∞ ) × ( − , → C , prescribed initial data f, g : ( − , → C , anda given potential V : ( − , → C . Definition 1.2.
Let V ∈ C ∞ ([ − , V ⊂ C by saying that λ ∈ C belongs to Σ V if Re λ ≥ f λ ∈ C ∞ ([ − , − (1 − y ) f ′′ λ ( y ) + 2( λ + 1) yf ′ λ ( y ) + λ ( λ + 1) f λ ( y ) + V ( y ) f λ ( y ) = 0for all y ∈ ( − , Theorem 1.3.
Let V : [ − , → C be smooth and even, p ∈ [2 , ∞ ] , q ∈ [1 , ∞ ) , and ǫ > . Then there exist constants C p,q , C ǫ > such that the following holds. (1) The set Σ + V := Σ V ∩ { z ∈ C : Re z > } consists of finitely many points. (2) For any given odd initial data f, g ∈ C ∞ ([ − , , there exists a unique solution u = u f,g ∈ C ∞ ([0 , ∞ ) × ( − , to the initial value problem (1.6) that satisfies,for each s ≥ , k ( u ( s, · ) , ∂ s u ( s, · )) k H < ∞ . (3) For each λ ∈ Σ + V there exists a number n ( λ ) ∈ N and a set { φ λ,kf,g ∈ C ∞ ( − ,
1) : k ∈ { , , . . . , n ( λ ) }} of odd functions satisfying k ( φ λ,kf,g , k H < ∞ and such thatthe solution u f,g decomposes according to u f,g ( s, y ) = X λ ∈ Σ + V e λs n ( λ ) X k =0 s k φ λ,kf,g ( y ) + e u f,g ( s, y ) . The map ( f, g ) φ λ,kf,g has finite rank and k ( e u f,g ( s, · ) , ∂ s e u f,g ( s, · )) k H ≤ C ǫ e ǫs k ( f, g ) k H for all s ≥ . (4) If Σ V ∩ i R = ∅ , we have the Strichartz estimates k e u f,g k L p (0 , ∞ ) L q ( − , ≤ C p,q k ( f, g ) k H . emark . With slightly more effort it is also possible to improve the energy bound to k ( e u f,g ( s, · ) , ∂ s e u f,g ( s, · )) k H . k ( f, g ) k H for all s ≥
0, provided Σ V ∩ i R = ∅ . To keep the paper at a reasonable length, however,we refrain from working out the details. Remark . The smoothness assumptions are imposed for convenience and can of coursebe considerably weakened. This produces some inessential technicalities but no newinsight.2.
Application: Asymptotics of Yang-Mills fields on wormholes
We give an application of Theorem 1.3 to Yang-Mills fields on wormholes studied in [1].2.1.
Setup.
As in [1], we consider M := R × R × (0 , π ) × (0 , π ). Let ( t, r, θ, ϕ ) : M → R be a chart on M . We define a Lorentzian metric g on M by g := − dt ⊗ dt + dr ⊗ dr + cosh( r ) ( dθ ⊗ dθ + sin( θ ) dϕ ⊗ dϕ ) . Then ( M , g ) is a Lorentzian manifold with 2 asymptotic ends (as r → ±∞ ), which phys-ically represents a wormhole spacetime. We would like to study Yang-Mills connectionson the principal bundle M × SU(2). That is to say, we are looking for su(2)-valuedone-forms A = A dt + A dr + A dθ + A dϕ on M that formally extremize the Yang-Mills action Z ( M ,g ) tr( F µν F µν ) , where F µν := ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] is the curvature two-form. The Euler-Lagrangeequation associated to the Yang-Mills action reads1 p | det g | ∂ µ (cid:16)p | det g | F µν (cid:17) + [ A µ , F µν ] = 0 (2.1)and is called the Yang-Mills equation. Here, Greek indices run from 0 to 3 and we useEinstein’s summation convention. As usual, indices are raised and lowered by the metric,i.e., F µν = g µα g νβ F αβ , where g µν = g ( ∂ µ , ∂ ν ) with ∂ = ∂ t , ∂ = ∂ r , ∂ = ∂ θ , ∂ = ∂ ϕ ,and g µν is defined by the requirement that g µα g αν = δ µν , where δ µν is the Kroneckersymbol. Furthermore, det g = − cosh( r ) sin( θ ) is the determinant of the matrix ( g µν ).We choose the basis { τ , τ , τ } for su(2), where τ := − i (cid:18) (cid:19) , τ := − i (cid:18) − ii (cid:19) , τ := − i (cid:18) − (cid:19) . Then cos θτ dϕ solves the Yang-Mills equation, as is easily checked. We would like tostudy the stability of the explicit solution cos θτ dϕ . Following [1], we consider theperturbation ansatz A = cos θτ dϕ + W ( t, r )( τ dθ + sin θτ dϕ )for a real-valued function W . By noting the commutator relations [ τ , τ ] = τ , [ τ , τ ] = − τ , and [ τ , τ ] = τ , we readily compute the nonvanishing components of F µν , which “Formally” here means that we are in fact looking for solutions of the Euler-Lagrange equationassociated to the Yang-Mills action. re given by F = ∂ W τ , F = ∂ W sin θτ , F = ∂ W τ , F = ∂ W sin θτ , and F = − (1 − W ) sin θτ . Consequently, F = − ∂ W sin θ p | det g | τ , F = − ∂ W p | det g | τ , F = ∂ W sin θ p | det g | τ ,F = ∂ W p | det g | τ , F = − − W cosh( r ) p | det g | τ , and for ν ∈ { , } , Eq. (2.1) reduces to( ∂ t − ∂ r ) W ( t, r ) = W ( t, r )(1 − W ( t, r ) )cosh( r ) , (2.2)whereas for ν ∈ { , } , Eq. (2.1) is satisfied identically. In particular, we observe that W = 0 is a solution, showing that cos θτ dϕ indeed solves the Yang-Mills equation.Consequently, under this particular symmetry reduction enforced by the perturbationansatz, the study of the stability of cos θτ dϕ as a solution to the Yang-Mills equationboils down to analyzing the stability of the trivial solution W = 0 of Eq. (2.2). Notethat Eq. (2.2) is effectively a one-dimensional semilinear wave equation and studying itsasymptotics might seem hard due to the lack of dispersion.Evidently, there are two more trivial solutions to Eq. (2.2), namely W = ±
1. In [1]it is shown that these solutions are linearly stable under general perturbations, whereasthe solution W = 0 has one unstable mode. In this paper, we restrict ourselves to oddsolutions. Under this assumption, the solutions W = ± W = 0 becomesstable, as we will see. The restriction to odd solutions is important for our method ofproof, where the free wave equation and the corresponding explicit solution formula istaken as a starting point. We believe that our results remain valid in the general casebut this would require a different proof. We plan to address this issue in the future.2.2. Hyperboloidal formulation.
The hyperboloidal initial value problem for the Yang-Mills equation (2.2) takes the form ( [ ∂ s + 2 y∂ s ∂ y + ∂ s − (1 − y ) ∂ y + 2 y∂ y − u ( s, y ) = − u ( s, y ) ( s, y ) ∈ (0 , ∞ ) × ( − , u ( s, y ) = f ( y ) , ∂ s u ( s, y ) = g ( y ) ( s, y ) ∈ { } × ( − , , (2.3)where W ( t, r ) = u ( t − log cosh r, tanh r ). Note that the linear part in (2.3) is Eq. (1.6)with V ( y ) = −
1. We compute Σ V . Lemma 2.1.
Let V ( y ) = − for all y ∈ [ − , . Then Σ V = ∅ .Proof. According to Definition 1.2, we have to solve the spectral problem − (1 − y ) f ′′ ( y ) + 2( λ + 1) yf ′ ( y ) + λ ( λ + 1) f ( y ) − f ( y ) = 0for f ∈ C ∞ ([ − , λ ≥
0. This ODE can be solved explicitly in terms ofhypergeometric functions and it follows that no solution other than f = 0 exists. Werefrain from giving the details here because this spectral problem was discussed at lengthin [1]. (cid:3) Definition 2.2.
Set V ( y ) := − y ∈ [ − ,
1] and let u f,g be the solution of Eq. (1.6)provided by Theorem 1.3. We define the wave propagators by C ( s ) f := u f, ( s, · ) , S ( s ) g := u ,g ( s, · ) . y Theorem 1.3 and Lemma 2.1 we have the bound k S ( s ) g k L ( − , . k g k L ( − , forany s ≥ S ( s ) uniquely extends to a bounded operator S ( s ) : L ( − , → L ( − , L q odd ( − ,
1) denotes the completion of { f ∈ C ∞ ([ − , f is odd } with respect to k · k L q ( − , . Then a weak formulation of Eq. (2.3) is given by u ( s, · ) = C ( s ) f + S ( s ) g − Z s S ( s − s ′ ) (cid:0) u ( s ′ , · ) (cid:1) ds ′ . (2.4) Theorem 2.3.
There exists a δ > such that for all odd functions f, g ∈ C ∞ ([ − , with k ( f, g ) k H < δ , Eq. (2.4) has a unique solution u in C ([0 , ∞ ) , L ( − , . Furthermore, u ∈ L p ((0 , ∞ ) , L ( − , for any p ∈ [3 , ∞ ] .Remark . Theorem 2.3 implies that the Yang-Mills connection cos θτ dϕ is asymptot-ically stable under odd small-energy perturbations on the hyperboloid (cid:8) ( − log(1 − y ) , artanh y ) ∈ R , : y ∈ ( − , (cid:9) . Proof of Theorem 2.3.
For
R > X R := { u ∈ C ([0 , ∞ ) , L ( − , k u k L (0 , ∞ ) L ( − , + k u k L ∞ (0 , ∞ ) L ( − , ≤ R } . Note that u ∈ X R implies u ( s, · ) ∈ L ( − ,
1) and thus, u ( s, · ) ∈ L ( − ,
1) for any s ≥
0. Consequently, K f,g ( u )( s ) := C ( s ) f + S ( s ) g − Z s S ( s − s ′ ) (cid:0) u ( s ′ , · ) (cid:1) ds ′ is well-defined as a map K f,g : X R → C ([0 , ∞ ) , L ( − , R >
Lemma 2.5.
There exist
M, δ > such that for all δ ∈ (0 , δ ] and any pair of oddfunctions f, g ∈ C ∞ ([ − , satisfying k ( f, g ) k H < δ , K f,g maps X Mδ to itself.Proof. Let u ∈ X R for some R > k ( f, g ) k H < δ . By Theorem 1.3 andLemma 2.1 we have kK f,g ( u ) k L (0 , ∞ ) L ( − , . k ( f, g ) k H + Z ∞ (cid:13)(cid:13) [0 ,s ] ( s ′ ) S ( s − s ′ ) (cid:0) u ( s ′ , · ) (cid:1)(cid:13)(cid:13) L s (0 , ∞ ) L ( − , ds ′ = k ( f, g ) k H + Z ∞ (cid:13)(cid:13) S ( s − s ′ ) (cid:0) u ( s ′ , · ) (cid:1)(cid:13)(cid:13) L s ( s ′ , ∞ ) L ( − , ds ′ = k ( f, g ) k H + Z ∞ (cid:13)(cid:13) S ( s ) (cid:0) u ( s ′ , · ) (cid:1)(cid:13)(cid:13) L s (0 , ∞ ) L ( − , ds ′ . k ( f, g ) k H + Z ∞ k u ( s ′ , · ) k L ( − , ds ′ = k ( f, g ) k H + Z ∞ k u ( s ′ , · ) k L ( − , ds ′ = k ( f, g ) k H + k u k L (0 , ∞ ) L ( − , . Analogously, kK f,g ( u ) k L ∞ (0 , ∞ ) L ( − , . k ( f, g ) k H + k u k L (0 , ∞ ) L ( − , and we obtain kK f,g ( u ) k L (0 , ∞ ) L ( − , + kK f,g ( u ) k L ∞ (0 , ∞ ) L ( − , ≤ Cδ + CR or some constant C >
0. Now we choose δ = (8 C ) − and R = 2 Cδ . Then we have Cδ + CR = Cδ + C (2 Cδ ) ≤ Cδ + 8 C δ Cδ = 2 Cδ and the claim follows with M = 2 C since the wave propagators preserve oddness. (cid:3) Now we set up an iteration by u := 0 and u n := K f,g ( u n − ) for n ∈ N . For brevity wedefine k u k X := k u k L (0 , ∞ ) L ( − , + k u k L ∞ (0 , ∞ ) L ( − , . Lemma 2.6.
There exist
M, δ > such that u n ∈ X Mδ for all n ∈ N and the sequence ( u n ) n ∈ N is Cauchy with respect to k · k X , provided that k ( f, g ) k H < δ .Proof. The first statement follows from Lemma 2.5. The algebraic identity a − b =( a − b )( a + ab + b ) and H¨older’s inequality yield k u n +1 − u n k L (0 , ∞ ) L ( − , = kK f,g ( u n ) − K f,g ( u n − ) k L (0 , ∞ ) L ( − , . Z ∞ (cid:13)(cid:13) S ( s ) (cid:2) u n ( s ′ , · ) − u n − ( s ′ , · ) (cid:3)(cid:13)(cid:13) L s (0 , ∞ ) L ( − , ds ′ . Z ∞ k u n ( s ′ , · ) − u n − ( s ′ , · ) k L ( − , ds ′ . Z ∞ k u n ( s ′ , · ) − u n − ( s ′ , · ) k L ( − , (cid:16) k u n ( s ′ , · ) k L ( − , + k u n − ( s ′ , · ) k L ( − , (cid:17) ds ′ . k u n − u n − k L (0 , ∞ ) L ( − , (cid:16) k u n k L (0 , ∞ ) L ( − , + k u n − k L (0 , ∞ ) L ( − , (cid:17) . Analogously, we obtain the bound k u n +1 − u n k L ∞ (0 , ∞ ) L ( − , . k u n − u n − k L (0 , ∞ ) L ( − , (cid:16) k u n k L (0 , ∞ ) L ( − , + k u n − k L (0 , ∞ ) L ( − , (cid:17) and in summary, k u n +1 − u n k X ≤ CM δ k u n − u n − k X for some constant C >
0. Thus,by choosing δ sufficiently small, we find k u n +1 − u n k X ≤ k u n − u n − k X for all n ∈ N andthis implies the claim. (cid:3) As a consequence of Lemma 2.6, the sequence ( u n ) n ∈ N converges to an element u ∈ C ([0 , ∞ ) , L ( − , ∩ L ((0 , ∞ ) , L ( − , ∩ L ∞ ((0 , ∞ ) , L ( − , , which satisfies Eq. (2.4). It remains to prove the uniqueness. Lemma 2.7.
Let f, g ∈ C ∞ ([ − , be odd. Then there exists at most one function u ∈ C ([0 , ∞ ) , L ( − , that satisfies Eq. (2.4) .Proof. Suppose u, e u ∈ C ([0 , ∞ ) , L ( − , s > s ∈ [0 , s ], we have k u ( s, · ) − e u ( s, · ) k L ( − , ≤ Z s (cid:13)(cid:13) S ( s − s ′ )[ u ( s ′ , · ) − e u ( s ′ , · ) ] (cid:13)(cid:13) L ( − , ds ′ . Z s k u ( s ′ , · ) − e u ( s ′ , · ) k L ( − , ds ′ . (cid:16) k u k L ∞ (0 ,s ) L ( − , + k e u k L ∞ (0 ,s ) L ( − , (cid:17) × Z s k u ( s ′ , · ) − e u ( s ′ , · ) k L ( − , ds ′ nd Gronwall’s inequality implies that k u ( s, · ) − e u ( s, · ) k L ( − , = 0 for all s ∈ [0 , s ]. (cid:3) The hyperboloidal initial value problem for the free wave equation
Now we turn to the proof of Theorem 1.3 and start with the hyperboloidal initial valueproblem for the free wave equation, i.e., we study ( [ ∂ s + 2 y∂ s ∂ y + ∂ s − (1 − y ) ∂ y + 2 y∂ y ] u ( s, y ) = 0 ( s, y ) ∈ (0 , ∞ ) × ( − , u ( s, y ) = f ( y ) , ∂ s u ( s, y ) = g ( y ) ( s, y ) ∈ { } × ( − ,
1) (3.1)for an unknown u : [0 , ∞ ) × ( − , → R and given data f, g : ( − , → R .3.1. Classical solution of the initial value problem.
The solution to (3.1) can begiven explicitly. This is a straightforward consequence of the fact that the general solutionof Eq. (1.3) is of the form v ( t, x ) = F ( t − x ) + G ( t + x ). Definition 3.1.
For f, g ∈ C ∞ ( − ,
1) and ( s, y ) ∈ [0 , ∞ ) × ( − ,
1) we set u f,g ( s, y ) := f (0) − Z − e − s (1+ y ) (1 − x ) f ′ ( x ) dx + 12 Z − e − s (1 − y )0 (1 + x ) f ′ ( x ) dx + 12 Z − e − s (1 − y ) − e − s (1+ y ) g ( x ) dx. Lemma 3.2 (Existence and uniqueness of smooth solutions) . Let f, g ∈ C ∞ ( − , .Then u f,g ∈ C ∞ ([0 , ∞ ) × ( − , and u = u f,g is a solution to (3.1) . Furthermore, thissolution is unique in C ∞ ([0 , ∞ ) × ( − , .Proof. Since − e − s (1 + y ) ∈ ( − ,
1) and 1 − e − s (1 − y ) ∈ ( − ,
1) for all s ≥ y ∈ ( − , u f,g ∈ C ∞ ([0 , ∞ ) × ( − , u = u f,g solves (3.1). In fact, the formula for u f,g is derived fromthe general solution v ( t, r ) = F ( t − r ) + G ( t + r ) of Eq. (1.3) and thus, u f,g is necessarilyunique in C ∞ ([0 , ∞ ) × ( − , (cid:3) Lemma 3.3 (Boundedness of the energy) . Let f, g ∈ C ∞ ( − , with k ( f, g ) k H < ∞ .Then we have k ( u f,g ( s, · ) , ∂ s u f,g ( s, · )) k H . k ( f, g ) k H for all s ≥ .Proof. This is a simple exercise. (cid:3)
Solution for odd data and Strichartz estimates.
The existence of the constantfinite-energy solution u ( s, y ) = 1 precludes the possibility of Strichartz estimates. Con-sequently, we restrict ourselves to odd data f, g ∈ C ∞ ( − , u f,g isgiven by u f,g ( s, y ) = 12 Z − e − s (1 − y )1 − e − s (1+ y ) [(1 + x ) f ′ ( x ) + g ( x )] dx. The following simple Sobolev embedding shows that the energy is strong enough to control L q , provided q < ∞ . Lemma 3.4.
Let q ∈ [1 , ∞ ) . Then we have the bound k f k L q ( − , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L ( − , for all odd f ∈ C ( − , such that the right-hand side is finite. roof. By the fundamental theorem of calculus and the oddness of f , we infer f ( y ) = Z y f ′ ( x ) dx for all y ∈ ( − ,
1) and Cauchy-Schwarz yields | f ( y ) | ≤ Z | y | | f ′ ( x ) | dx = Z | y | (1 − x ) − (1 − x ) | f ′ ( x ) | dx ≤ Z | y | (1 − x ) − dx ! (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L ( − , . Since Z | y | (1 − x ) − dx . | log(1 − y ) | + 1and the square root of the latter function belongs to L q ( − ,
1) for any q ∈ [1 , ∞ ), thestated bound follows. (cid:3) Proposition 3.5 (Strichartz estimates for the free equation) . Let p ∈ [2 , ∞ ] and q ∈ [1 , ∞ ) . Then we have the Strichartz estimates k u f,g k L p (0 , ∞ ) L q ( − , . k ( f, g ) k H for all odd f, g ∈ C ∞ ( − , with k ( f, g ) k H < ∞ .Proof. The case p = ∞ is a consequence of Lemmas 3.3 and 3.4. Thus, it suffices to provethe bound k u f,g k L (1 , ∞ ) L q ( − , . k ( f, g ) k H . We first consider the case g = 0. Then we have | u f, ( s, y ) | . Z − e − s (1 − y )1 − e − s (1+ y ) | f ′ ( x ) | dx = e − s Z y − y | f ′ (1 − e − s x ) | dx = Z [1 − y, y ] ( x ) | e − s f ′ (1 − e − s x ) | dx for all y ∈ [0 ,
1) and thus, by Minkowski’s inequality and the oddness of u f, ( s, · ), k u f, ( s, · ) k L q ( − , . (cid:13)(cid:13)(cid:13)(cid:13)Z [1 − y, y ] ( x ) | e − s f ′ (1 − e − s x ) | dx (cid:13)(cid:13)(cid:13)(cid:13) L qy (0 , . Z k [1 − y, y ] ( x ) k L qy (0 , | e − s f ′ (1 − e − s x ) | dx. Now note that 1 [1 − y, y ] ( x ) ≤ [1 − x, ( y ) for all x ∈ [0 ,
2] and y ∈ [0 , k [1 − y, y ] ( x ) k L qy (0 , . x q and thus, k u f, ( s, · ) k L q ( − , . Z x q | e − s f ′ (1 − e − s x ) | dx. Consequently, k u f, k L (1 , ∞ ) L q ( − , . (cid:13)(cid:13)(cid:13)(cid:13)Z x q | e − s f ′ (1 − e − s x ) | dx (cid:13)(cid:13)(cid:13)(cid:13) L s (1 , ∞ ) . Z x q k e − s f ′ (1 − e − s x ) k L s (1 , ∞ ) dx gain by Minkowski’s inequality. Now we have k e − s f ′ (1 − e − s x ) k L s (1 , ∞ ) = Z ∞ | f ′ (1 − e − s x ) | e − s ds = x − Z − e − x (1 − η ) | f ′ ( η ) | dη . x − k ( f, k H for all x ∈ (0 ,
2] and in summary, we obtain k u f, k L (1 , ∞ ) L q ( − , . k ( f, k H Z x q − dx . k ( f, k H . The case f = 0 is much simpler and it suffices to note that | u ,g ( s, y ) | ≤ Z − e − s (1 − y )1 − e − s (1+ y ) | g ( x ) | dx . e − s k g k L ( − , . e − s k (0 , g ) k H for all s ≥ y ∈ [0 , (cid:3) In particular, Proposition 3.5 shows that the zero solution is asymptotically stableunder odd perturbations in the energy space.3.3.
Semigroup formulation.
For later purposes it is desirable to translate the resultsobtained so far into semigroup language. First, we need to define proper function spacesand operators.
Definition 3.6.
We set e H := { f = ( f , f ) ∈ C ∞ ([ − , × C ∞ ([ − , f is odd } . The vector space e H equipped with the inner product( f | g ) H := Z − (1 − y ) f ′ ( y ) g ′ ( y ) dy + Z − f ( y ) g ( y ) dy is a pre-Hilbert space and we denote by H its completion. Furthermore, we consider the formal differential expression L f ( y ) := (cid:18) f ( y )(1 − y ) f ′′ ( y ) − yf ′ ( y ) − yf ′ ( y ) − f ( y ) (cid:19) and define the operator e L : D ( e L ) ⊂ H → H by D ( e L ) := e H and e L f := L f .By construction, e L is a densely-defined operator on H . With these definitions at hand,the initial value problem (3.1) can be written as ( ∂ s Φ( s ) = e L Φ( s ) for all s > f for Φ( s ) = ( u ( s, · ) , ∂ s u ( s, · )) and f = ( f, g ). The well-posedness of this initial valueproblem now means that (the closure of) e L generates a semigroup. Lemma 3.7.
The operator e L : D ( e L ) ⊂ H → H is closable and its closure L generatesa strongly continuous one-parameter semigroup { S ( s ) ∈ B ( H ) : s ≥ } . Furthermore,we have the estimate k S ( s ) f k H ≤ k f k H for all s ≥ and all f ∈ H . roof. A straightforward computation shows thatRe (cid:0)e L f (cid:12)(cid:12) f (cid:1) H = −| f ( − | − | f (1) | ≤ f = ( f , f ) ∈ D ( e L ) (this is just an instance of the energy identity Eq. (1.5)).Furthermore, for g = ( g , g ) ∈ e H we set g ( y ) := yg ′ ( y ) + g ( y ) + g ( y ) and f ( y ) := 11 + y Z y − (1 + x ) g ( x ) dx + 11 − y Z y (1 − x ) g ( x ) dx. Note that f is odd and belongs to C ∞ ([ − , f := ( f , f − g ). Then we have f ∈ D ( e L ) and a straightforward computation shows that (1 − e L ) f = g . Since g ∈ e H was arbitrary, we see that the range of 1 − e L is dense in H and an application of theLumer-Phillips theorem (see e.g. [5], p. 83, Theorem 3.15) completes the proof. (cid:3) Corollary 3.8.
We have σ ( L ) ⊂ { z ∈ C : Re z ≤ } .Proof. The statement is a consequence of the growth bound in Lemma 3.7 and [5], p. 55,Theorem 1.10. (cid:3)
Remark . In fact, we have σ p ( L ) = { z ∈ C : Re z < } (and hence σ ( L ) = { z ∈ C :Re z ≤ } ). This follows easily by noting that for any λ ∈ C , the function f = ( f , λf )with f ( y ) := (1 + y ) − λ − (1 − y ) − λ satisfies ( λ − L ) f = 0. However, we omit a formal proof of this result since it is notneeded in the following.4. The wave equation with a potential
Now we move on to the main problem and add a potential V ∈ C ∞ ([ − , V to be even. That is to say, we study the initialvalue problem ( [ ∂ s + 2 y∂ s ∂ y + ∂ s − (1 − y ) ∂ y + 2 y∂ y + V ( y )] u ( s, y ) = 0 ( s, y ) ∈ (0 , ∞ ) × ( − , u ( s, y ) = f ( y ) , ∂ s u ( s, y ) = g ( y ) ( s, y ) ∈ { } × ( − , . (4.1)4.1. Semigroup formulation.
We immediately switch to the semigroup picture. Notethat by Lemma 3.4, the operator ( f , f ) (0 , − V f ) is bounded on H . Definition 4.1.
Let V ∈ C ∞ ([ − , L ′ V : H → H by L ′ V f := (cid:18) − V f (cid:19) . Furthermore, we set L V := L + L ′ V , where L is the closure of e L , see Lemma 3.7.Eq. (4.1) can be written as ( ∂ s Φ( s ) = L V Φ( s ) for all s > f, g )and the abstract theory immediately tells us that this initial value problem is well-posed. emma 4.2. The operator L V : D ( L ) ⊂ H → H generates a strongly continuous one-parameter semigroup { S V ( s ) ∈ B ( H ) : s ≥ } and we have the bound k S V ( s ) f k H ≤ e k L ′ V k s k f k H for all s ≥ and all f ∈ H .Proof. The statement is a consequence of the bounded perturbation theorem, see e.g. [5],p. 158, Theorem 1.3. (cid:3)
Analysis of the generator.
In order to relate the semigroup formulation to theclassical picture, we need some technical results on the generator L V . The point is thatthe latter is only abstractly defined as the closure of e L V := e L + L ′ V . Lemma 4.3.
Let n ∈ N , δ ∈ (0 , , and I δ := ( − δ, − δ ) . Then we have the bound k f k W n, ∞ ( I δ ) + k f k W n − , ∞ ( I δ ) . k f k H + k e L nV f k H for all f = ( f , f ) ∈ e H .Proof. Since f and f are odd, we have f j ( y ) = R y f ′ j ( x ) dx , j ∈ { , } , and Cauchy-Schwarz yields k f k L ∞ ( I δ ) . k f ′ k L ( I δ ) . k f k H k f k L ∞ ( I δ ) . k f ′ k L ( I δ ) = k [ e L V f ] ′ k L ( I δ ) . k e L V f k H . Furthermore, f ′′ ( y ) = 11 − y h [ e L V f ] ( y ) + 2 yf ′ ( y ) + 2 yf ′ ( y ) + f ( y ) + V ( y ) f ( y ) i and thus, by the one-dimensional Sobolev embedding, k f ′ k L ∞ ( I δ ) . k f ′ k L ( I δ ) + k f ′′ k L ( I δ ) . k f k H + k e L V f k H . This settles the case n = 1 and from here we proceed inductively. (cid:3) Corollary 4.4.
Let n ∈ N . If f ∈ D ( L nV ) then f can be identified with an odd functionin C n ( − , × C n − ( − , .Proof. Fix δ ∈ (0 , I δ := ( − δ, − δ ), and let f ∈ D ( L nV ). Then there existsa sequence ( f k ) k ∈ N ⊂ e H such that f k → f and e L nV f k → L nV f . In particular, ( f k ) k ∈ N and( e L nV f k ) k ∈ N are Cauchy sequences with respect to k · k H . By Lemma 4.3 we see that f k converges to an odd function in C n ( I δ ) × C n − ( I δ ). Since δ ∈ (0 ,
1) was arbitrary, ( f k ) k ∈ N converges pointwise on ( − ,
1) to an odd function in C ( − , × C ( − , f . (cid:3) Remark . From now on we will implicitly make the identification suggested in Corollary4.4. Consequently, any f ∈ D ( L nV ) is an odd function in C n ( − , × C n − ( − ,
1) and wehave the inclusion D ( L nV ) ⊂ C n ( − , × C n − ( − , Corollary 4.6. On D ( L ) , L acts as a classical differential operator, i.e., if f ∈ D ( L ) ⊂ C ( − , × C ( − , , we have L f = L f on ( − , .Proof. Let f ∈ D ( L ). Then there exists a sequence ( f k ) k ∈ N with e L n f k → L n f for n ∈ { , , } . By the definition of e L and Lemma 4.3, we see that ( L f k ) k ∈ N convergespointwise on ( − ,
1) to L f ∈ C ( − , × C ( − , L f . (cid:3) Corollary 4.7.
Let f = ( f, g ) ∈ e H and set u ( s, · ) := [ S ( s ) f ] . Then we have u = u f,g . roof. We have f ∈ D ( L n ) for any n ∈ N . Thus, by [5], p. 124, Proposition 5.2, we obtain S ( s ) f ∈ D ( L n ) for all s ≥ n ∈ N . Furthermore, since ∂ ns S ( s ) f = S ( s ) L n f , itfollows that u ∈ C ∞ ([0 , ∞ ) × ( − , u is a smooth finite-energysolution of Eq. (3.1) and by Lemma 3.2, we must have u = u f,g . (cid:3) Spectral properties.
The special structure of the operator L ′ V allows us to obtainimportant spectral information, even at this level of generality. First, we need a simplecompactness result. Lemma 4.8.
Let ( f n ) n ∈ N ⊂ C ( − , be a sequence of odd functions that satisfy (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ n (cid:13)(cid:13)(cid:13) L ( − , . for all n ∈ N . Then there exists a subsequence of ( f n ) n ∈ N that is Cauchy in L ( − , .Proof. We mimic the classical proof of the Arzel`a-Ascoli theorem. The set ( − , ∩ Q is countable and dense in ( − ,
1) and we write ( − , ∩ Q = { y j : j ∈ N } . By thefundamental theorem of calculus and the oddness of f n , we have f n ( y ) = Z y f ′ n ( x ) dx and thus, by Cauchy-Schwarz, | f n ( y ) | ≤ Z y | f ′ n ( x ) | dx = Z y (1 − x ) − (1 − x ) | f ′ n ( x ) | dx ≤ (cid:18)Z y (1 − x ) − dx (cid:19) / (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ n (cid:13)(cid:13)(cid:13) L ( − , . (cid:12)(cid:12) log(1 − y ) (cid:12)(cid:12) + 1for all y ∈ ( − ,
1) and all n ∈ N . Since y j ∈ ( − , j ∈ N , the sequence ( f n ( y j )) n ∈ N ⊂ C is bounded. By Cantor’s classical diagonal argumentwe extract a subsequence ( f n k ) k ∈ N of ( f n ) n ∈ N such that for each j ∈ N , ( f n k ( y j )) k ∈ N isCauchy in C .Now note that for any δ ∈ (0 , | f n ( x ) − f n ( y ) | . δ − | x − y | for all x, y ∈ [ − δ, − δ ] and n ∈ N . Indeed, f n ( x ) − f n ( y ) = Z xy f ′ n ( t ) dt and thus, by Cauchy-Schwarz, | f n ( x ) − f n ( y ) | . | x − y | (cid:18)Z − δ − δ | f ′ n ( t ) | dt (cid:19) / . δ − | x − y | (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ n (cid:13)(cid:13)(cid:13) L ( − , . δ − | x − y | , as claimed. As a consequence of this estimate, ( f n ) n ∈ N is equicontinuous on [ − δ, − δ ]and the density of { y j : j ∈ N } implies that ( f n k ) k ∈ N is Cauchy in L ∞ ( − δ, − δ ).Now let ǫ ∈ (0 , N ǫ ∈ N such that k f n k − f n ℓ k L ∞ ( − ǫ, − ǫ ) ≤ ǫ or all k, ℓ ≥ N ǫ . Consequently, k f n k − f n ℓ k L ( − , = k f n k − f n ℓ k L ( − ǫ, − ǫ ) + k f n k − f n ℓ k L ( − , − ǫ ) + k f n k − f n ℓ k L (1 − ǫ, . ǫ for all k, ℓ ≥ N ǫ since k f n k L ( − , − ǫ ) . Z − ǫ − | log(1 − y ) | dy . ǫ for all n ∈ N and analogously for k f n k L (1 − ǫ, . (cid:3) We continue with a simple resolvent bound. Note that this bound is just a consequenceof the fact that the operator L ′ V maps the first component to the second component. Lemma 4.9.
Let ǫ > . Then we have the bound k L ′ V R L ( λ ) f k H . | λ | k f k H for all λ ∈ C with Re λ ≥ ǫ and all f ∈ H .Proof. To begin with, let f = ( f , f ) ∈ D ( L ) and set u := R L ( λ ) f . Then u = ( u , u ) ∈D ( L ) and ( λ − L ) u = f . By Corollary 4.4, u ∈ C ( − , × C ( − ,
1) and Corollary4.6 yields ( λ − L ) u = f . The first component of this equation reads λu − u = f or,equivalently, [ R L ( λ ) f ] = 1 λ ([ R L ( λ ) f ] + f ) . Consequently, k L ′ V R L ( λ ) f k H = k V [ R L ( λ ) f ] k L ( − , . | λ | (cid:0) k [ R L ( λ ) f ] k L ( − , + k f k L ( − , (cid:1) . | λ | ( k R L ( λ ) f k H + k f k H ) . | λ | (cid:18) λ k f k H + k f k H (cid:19) . | λ | k f k H by Lemma 3.4 and [5], p. 55, Theorem 1.10. Thus, the claim follows by density. (cid:3) Lemma 4.10.
The operator L ′ V : H → H is compact. As a consequence, the set σ ( L V ) ∩ { z ∈ C : Re z > } consists of finitely many eigenvalues of finite algebraic multiplicity.Proof. Let ( f n ) n ∈ N ⊂ H be a bounded sequence and write f n = ( f n, , f n, ). Then we have (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ n, (cid:13)(cid:13)(cid:13) L ( − , . n ∈ N and Lemma 4.8 implies that ( f n, ) n ∈ N has a subsequence, again denoted by( f n, ) n ∈ N , that is Cauchy in L ( − , k L ′ V f m − L ′ V f n k H = k V ( f m, − f n, ) k L ( − , . k f m, − f n, k L ( − , and thus, ( L ′ V f n ) n ∈ N has a convergent subsequence. This shows that L ′ V is compact.By Corollary 3.8, R L is holomorphic on the open right half-plane H + := { z ∈ C :Re z > } and the obvious identity λ − L V = [ I − L ′ V R L ( λ )]( λ − L ) shows that λ ∈ H + belongs to ρ ( L V ) if and only if I − L ′ V R L ( λ ) is bounded invertible. By Lemmas 4.2 and 4.9 e immediately see that σ ( L V ) ∩ H + is bounded. Furthermore, the map λ L ′ V R L ( λ )is holomorphic on H + and has values in the set of compact operators on the Hilbert space H . Consequently, the analytic Fredholm theorem (see e.g. [11], p. 194, Theorem 3.14.3)implies that the inverse λ [ I − L ′ V R L ( λ )] − has finitely many poles of finite order withfinite rank residues. For every λ ∈ σ ( L V ) ∩ H + we therefore have 1 ∈ σ ( L ′ V R L ( λ )) andthus, 1 ∈ σ p ( L ′ V R L ( λ )). Let f λ ∈ H be an associated eigenfunction. Then R L ( λ ) f λ ∈D ( L V ) is an eigenfunction of L V and we see that every λ ∈ σ ( L V ) ∩ H + is an eigenvalueof L V and the corresponding spectral projection has finite rank. (cid:3) Lemma 4.10 allows us to remove the unstable part of the spectrum by a finite-rankprojection.
Definition 4.11.
Let γ : [0 , π ] → ρ ( L V ) be a positively oriented, regular, smooth,simple closed curve that encircles the set σ ( L V ) ∩ { z ∈ C : Re z > } (the existence ofsuch a curve is guaranteed by Lemma 4.10). Then we define P V := 12 πi Z γ R L V ( λ ) dλ. Our goal now is to prove a set of Strichartz estimates for the reduced semigroup S V ( s )( I − P V ) under a suitable spectral assumption on L V . To this end, we first needto clarify the relation between the abstract Hilbert space H and the standard Lebesguespaces. Definition 4.12.
For q ∈ [1 , ∞ ), we define the Banach space L q odd ( − ,
1) as the comple-tion of { f ∈ C ∞ ([ − , f odd } with respect to k · k L q ( − , . Lemma 4.13.
Let q ∈ [1 , ∞ ) . Then there exists a linear, bounded, and injective map i : H → L q odd ( − , × L ( − , .Proof. For f ∈ e H we set i ( f ) := f . By Lemma 3.4 we obtain the bound k i ( f ) k L q ( − , × L ( − , . k f k H for all f ∈ e H and by density, i extends to a linear and bounded map i : H → L q odd ( − , × L ( − , i ( f ) = 0 for f ∈ H . Then there existsa sequence ( f n ) n ∈ N ∈ e H such that f n → f in H and i ( f n ) → i ( f ) = 0 in L q odd ( − , × L ( − , n → ∞ . In particular, f n ⇀ f in H . Furthermore, since Z − (1 − y ) f ′ ( y ) g ′ ( y ) dy = − Z − (1 − y ) f ( y ) g ′′ ( y ) dy + 2 Z − yf ( y ) g ′ ( y ) dy for all f, g ∈ C ∞ ([ − , f n = i ( f n ) → L q odd ( − , × L ( − ,
1) implies f n ⇀ H and the uniqueness of weak limits yields f = 0. (cid:3) Remark . By Lemma 4.13, we may identify f ∈ H with i ( f ) ∈ L q odd ( − , × L ( − ,
1) and this yields the continuous embedding H ֒ → L q odd ( − , × L ( − , Theorem 4.15.
Let V ∈ C ∞ ([ − , be even, p ∈ [2 , ∞ ] , and q ∈ [1 , ∞ ) . Furthermore,suppose that the operator L V has no eigenvalues on the imaginary axis. Then we havethe Strichartz estimates k [ S V ( s )( I − P V ) f ] k L ps (0 , ∞ ) L q ( − , . k ( I − P V ) f k H for all f ∈ H . .4. Explicit representation of the semigroup.
First, we show that the reducedsemigroup S V ( s )( I − P V ) inherits the decay from the free semigroup S , up to an ǫ -loss.This follows from the celebrated Gearhart-Pr¨uss theorem and the simple resolvent boundfrom Lemma 4.9. Lemma 4.16.
Let ǫ > . Then there exists a C ǫ > such that k S V ( s )( I − P V ) f k H ≤ C ǫ e ǫs k ( I − P V ) f k H for all s ≥ and all f ∈ H .Proof. We denote by L st V the part of L V in ker P V . Then R L st V ( λ ) is the part of R L V ( λ )in ker P V . By construction, σ ( L st V ) ∩ { z ∈ C : Re z > } = ∅ and Lemma 4.9 togetherwith the identity λ − L V = [ I − L ′ V R L ( λ )]( λ − L ) shows thatsup {k R L st V ( λ ) k H : Re λ ≥ ǫ } < ∞ . Consequently, the Gearhart-Pruess Theorem, see e.g. [5], p. 302, Theorem 1.11, impliesthe claim. (cid:3)
In the following, we denote by S st V : [0 , ∞ ) → B (ker P V ) the reduced semigroup, i.e., S st V ( s ) f := S V ( s ) f for all f ∈ ker P V . The generator of the semigroup S st V is L st V , thepart of L V in ker P V . From Lemma 4.16 and [5], p. 234, Corollary 5.15, we obtain therepresentation S st V ( s ) f = 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs R L st V ( λ ) f dλ for any ǫ > f ∈ D ( L st V ). If we set u := ( λ − L st V ) − f , we obtain ( λ − L V ) u = f .Formally at least, this equation is equivalent to λu ( y ) − u ( y ) = f ( y ) λu ( y ) − (1 − y ) u ′′ ( y ) + 2 yu ′ ( y ) + 2 yu ′ ( y ) + u ( y ) + V ( y ) u ( y ) = f ( y )and inserting the first equation into the second one yields − (1 − y ) u ′′ ( y ) + 2( λ + 1) yu ′ ( y ) + λ ( λ + 1) u ( y ) + V ( y ) u ( y ) = F λ ( y ) (4.2)with F λ ( y ) := 2 yf ′ ( y ) + ( λ + 1) f ( y ) + f ( y ). Consequently, our next goal is to solveEq. (4.2). 5. The Green function
In order to solve Eq. (4.2), we need to first construct a suitable fundamental system forthe homogeneous equation − (1 − y ) u ′′ ( y ) + 2( λ + 1) yu ′ ( y ) + λ ( λ + 1) u ( y ) + V ( y ) u ( y ) = 0 . (5.1)5.1. Construction of a fundamental system.
In terms of v ( y ) := (1 − y ) ( λ +1) u ( y ),Eq. (5.1) reads v ′′ ( y ) + 1 − λ (1 − y ) v ( y ) = V ( y )1 − y v ( y ) . (5.2) Definition 5.1.
For y ∈ ( − ,
1) and λ ∈ C we set ψ ( y, λ ) := (1 − y ) (1+ λ ) (1 + y ) (1 − λ ) . ote that ∂ y ψ ( y, λ ) + 1 − λ (1 − y ) ψ ( y, λ ) = 0 (5.3)for all y ∈ ( − ,
1) and λ ∈ C . We construct a perturbative solution to Eq. (5.2) withgood control of the error near the singularity at y = 1. Proposition 5.2.
There exists a solution v ( y ) = v ( y, λ ) to Eq. (5.2) of the form v ( y, λ ) = ψ ( y, λ )[1 + a ( y, λ )] , where the function a satisfies | a ( y, λ ) | . (1 − y ) h λ i − for all y ∈ [0 , and λ ∈ C with Re λ ≥ − . Furthermore, for all k, ℓ, m ∈ N , there exists a constant C k,ℓ,m > suchthat | ∂ mκ ∂ ℓω ∂ ky a ( y, κ + iω ) | ≤ C k,ℓ,m (1 − y ) − k h ω i − − ℓ for all y ∈ [0 , , ω ∈ R , and κ ∈ [ − , ] .Proof. To begin with, we assume λ = 0 and define ψ ( y, λ ) := ψ ( y, λ ) − ψ ( y, − λ ) . Note that W ( ψ ( · , λ ) , ψ ( · , λ )) = 2 λ . Consequently, by the variation of parameters for-mula and Eq. (5.3), v has to satisfy the integral equation v ( y, λ ) = ψ ( y, λ ) + Z y ψ ( y, λ ) ψ ( x, λ ) − ψ ( x, λ ) ψ ( y, λ )2 λ V ( x )1 − x v ( x, λ ) dx (5.4)for all y ∈ [0 , C ([0 , v = ψ h . Then, Eq. (5.4) is equivalent to the Volterraequation h ( y, λ ) = 1 + Z y K ( y, x, λ ) h ( x, λ ) dx (5.5)with the kernel K ( y, x, λ ) = 12 λ (cid:20) ψ ( y, λ ) ψ ( y, λ ) ψ ( x, λ ) − ψ ( x, λ ) ψ ( x, λ ) (cid:21) V ( x )1 − x = 12 λ " − (cid:18) − y y (cid:19) − λ (cid:18) − x x (cid:19) λ V ( x ) . (5.6)We have the bound | K ( y, x, λ ) | . (1 − x ) − | λ | − for all 0 ≤ y ≤ x < λ ∈ C \ { } with Re λ ≥ − . If, in addition, | λ | ≥
1, this yields Z sup y ∈ (0 ,x ) | K ( y, x, λ ) | dx . | λ | − . h ( · , λ ) ∈ L ∞ (0 ,
1) satisfying k h ( · , λ ) k L ∞ (0 , . λ ∈ C with Re λ ≥ − and | λ | ≥
1. It follows that h ( · , λ ) ∈ C ([0 , | h ( y, λ ) − | . Z y | K ( x, y, λ ) || h ( x, λ ) | dx . | λ | − k h ( · , λ ) k L ∞ (0 , Z y (1 − x ) − dx . (1 − y ) | λ | − . (1 − y ) h λ i − , which implies the claimed estimate on a .The difficulty in proving the derivative bounds in the regime | λ | ≥ λ = κ + iω appears in the exponent in Eq. (5.6). Thus, it seems that differentiating ith respect to ω does not improve the decay in ω . This problem can be dealt withby a suitable change of variables. More precisely, we consider the diffeomorphism ϕ :(0 , ∞ ) → (0 , ϕ ( ξ ) := − e − ξ e − ξ , with inverse ϕ − ( x ) = − log − x x . We write λ = κ + iω and it suffices to consider the case ω ≥
1. Then we may rewrite Eq. (5.5) as h ( ϕ ( η ) , λ ) = Z ∞ η K ( ϕ ( η ) , ϕ ( ξ ) , λ ) h ( ϕ ( ξ ) , λ ) ϕ ′ ( ξ ) dξ = 1 ω Z ∞ K ( ϕ ( η ) , ϕ ( ω − ξ + η ) , λ ) ϕ ′ ( ω − ξ + η ) h (cid:0) ϕ ( ω − ξ + η ) , λ (cid:1) dξ = 12 λω Z ∞ (cid:16) − e − ( κω − + i ) ξ (cid:17) V (cid:0) ϕ ( ω − ξ + η ) (cid:1) ϕ ′ ( ω − ξ + η ) × h (cid:0) ϕ ( ω − ξ + η ) , λ (cid:1) dξ and from this representation the derivative bounds follow inductively.In the case | λ | ≤ K ( y, x, λ ) at λ = 0. In fact, this singularity is removable because ψ ( y,
0) = 0. In orderto exploit this, we first note that ∂ t ψ ( y, tλ ) = λ (cid:18) − y y (cid:19) ψ ( y, tλ )for t ∈ R and then we use the fundamental theorem of calculus to write ψ ( y, λ ) = Z ∂ t ψ ( y, tλ ) dt = λ e ψ ( y, λ )with e ψ ( y, λ ) := 12 log (cid:18) − y y (cid:19) Z [ ψ ( y, tλ ) + ψ ( y, − tλ )] dt. We have the bound | e ψ ( y, λ ) | . | log(1 − y ) | Z h (1 − y ) (1+ t Re λ ) + (1 − y ) (1 − t Re λ ) i dt . | log(1 − y ) | h (1 − y ) (1+Re λ ) + (1 − y ) (1 − Re λ ) i and thus, | K ( y, x, λ ) | . | log(1 − x ) | (1 − x ) − . (1 − x ) − for all 0 ≤ y ≤ x < λ ∈ C with | λ | ≤
1, Re λ ≥ − . Consequently, a Volterraiteration yields the stated estimate for a . For the derivative bounds it suffices to note thateach derivative with respect to ω or κ produces a singular term log(1 − x ) which, however,is harmless since | log(1 − x ) | n (1 − x ) − ≤ C n (1 − x ) − for any n ∈ N . Consequently,the derivative bounds follow inductively. (cid:3) Definition 5.3.
In the following, v always refers to the solution constructed in Propo-sition 5.2.Proposition 5.2 shows that the error a improves upon differentiation with respect to ω . On the other hand, when differentiating with respect to y , the bounds get worse.Both operations have in common that taking a derivative results in the loss of one powerof the respective variable in the estimate. This is a crucial property and we introduce amore economical notation to keep track of this behavior. efinition 5.4. For α, β ∈ R , we write f ( y, ω ) = O ((1 − y ) α h ω i β ) if for all k, ℓ ∈ N there exist constants C k,ℓ > | ∂ ℓω ∂ ky f ( y, ω ) | ≤ C k,ℓ (1 − y ) α − k h ω i β − ℓ in a range of the variables y and ω that is specified explicitly or follows from the context.In other words, the O -terms may be formally differentiated. Such functions are said tobe of symbol type . We also use self-explanatory variants of this notation. Remark . By Proposition 5.2 we have, with ω = Im λ , W ( v ( · , λ ) , v ( · , − λ )) = W ( ψ ( · , λ ) , ψ ( · , − λ ))[1 + O ((1 − y ) h ω i − )]+ ψ ( y, λ ) ψ ( y, − λ ) O ((1 − y ) − h ω i − )= 2 λ [1 + O ((1 − y ) h ω i − )] + O ((1 − y ) h ω i − )for all y ∈ [0 ,
1) and λ ∈ C with | Re λ | ≤ . This expression is in fact independent of y and thus, we may evaluate it at y = 1 which yields W ( v ( · , λ ) , v ( · , − λ )) = 2 λ. The bounds on the derivatives of v are sufficient for our purposes and easy to workwith but certainly not optimal, as the following result shows. Lemma 5.6.
Let Re λ ≥ − . Then we have v ( · , λ ) ψ ( · , λ ) ∈ C ∞ ([0 , . Proof.
As in the proof of Proposition 5.2 we write v = ψ h and from Eqs. (5.5) and(5.6), we obtain ∂ y h ( y, λ ) = − − y (cid:18) − y y (cid:19) − λ Z y (cid:18) − x x (cid:19) λ V ( x ) h ( x, λ ) dx. The change of variables x = y + t (1 − y ) yields ∂ y h ( y, λ ) = − (1 + y ) λ − Z (cid:18) − t y + t (1 − y ) (cid:19) λ V (cid:0) y + t (1 − y ) (cid:1) h (cid:0) y + t (1 − y ) , λ (cid:1) dt and from this expression the statement follows inductively. (cid:3) The solution v is sufficient to construct the Green function for Eq. (4.2). Definition 5.7.
For y ∈ [0 ,
1) and λ ∈ C with Re λ ≥ − we set u ( y, λ ) := (1 − y ) − (1+ λ ) v ( y, λ ) . Furthermore, for | Re λ | ≤ , we define u ( y, λ ) := (1 − y ) − (1+ λ ) [ v (0 , − λ ) v ( y, λ ) − v (0 , λ ) v ( y, − λ )] . Regularity theory.
We take up the opportunity to establish the link between Σ V ,see Definition 1.2, and the spectrum of L V . The key observation in this respect is aregularity result for the operator L V . Lemma 5.8.
For any λ ∈ C with Re λ ≥ , we have ker( λ − L V ) ⊂ e H . roof. Let f ∈ ker( λ − L V ), i.e., f ∈ D ( L ) and L V f = λ f . Inductively, this implies that f ∈ D ( L nV ) for any n ∈ N . By Corollary 4.4 and Remark 4.5, f ∈ C ∞ ( − , × C ∞ ( − , λ − L − L ′ V ) f = 0. As a consequence, f = λf and f isan odd solution of Eq. (5.1) on ( − , f ∈ C ∞ ([ − , u ( y, λ ) = (1 − y ) − (1+ λ ) ψ ( y, λ ) v ( y, λ ) ψ ( y, λ ) = (1 + y ) − λ v ( y, λ ) ψ ( y, λ )and Lemma 5.6 shows that u ( · , λ ) ∈ C ∞ ([0 , u ( y, λ ) = (1 + y ) − λ [1 + O ((1 − y ) h λ i − )]. In particular, there exists a c ∈ (0 ,
1) suchthat | u ( y, λ ) | > y ∈ [ c,
1] and we set e u ( y, λ ) := u ( y, λ ) Z yc (1 − x ) − − λ u ( x, λ ) dx. Then { u ( · , λ ) , e u ( · , λ ) } is a fundamental system for Eq. (5.1) on [ c, a, b ∈ C such that f ( y ) = au ( y, λ ) + b e u ( y, λ )for y ∈ [ c, | ∂ y e u ( y, λ ) | & (1 − y ) − − Re λ for y ∈ [ c,
1) and thus, Z c (1 − y ) | ∂ y e u ( y, λ ) | dy & Z c (1 − y ) − − λ dy = ∞ . Consequently, since k f k H < ∞ , we must have b = 0 and therefore, f ∈ C ∞ ([ − , (cid:3) Lemma 5.9.
We have Σ V = σ p ( L V ) ∩ { z ∈ C : Re z ≥ } . Proof.
Let λ ∈ Σ V . Then Re λ ≥ f λ ∈ C ∞ ([ − , y ∈ ( − , f := ( f λ , λf λ ). Then f ∈ e H and( λ − e L − L ′ V ) f = 0, which implies that λ ∈ σ p ( L V ).Conversely, if Re λ ≥ λ ∈ σ p ( L V ), Lemma 5.8 implies that there exists a non-trivial f = ( f , f ) ∈ e H such that ( λ − e L − L ′ V ) f = 0. In other words, f = λf and f isa nontrivial solution of Eq. (5.1). Consequently, λ ∈ Σ V . (cid:3) Next, we relate the point spectrum of L V to the value of u ( y, λ ) at y = 0. Lemma 5.10.
Let λ ∈ C with Re λ ≥ . If u (0 , λ ) = 0 then λ ∈ σ p ( L V ) .Proof. The function u ( · , λ ) satisfies Eq. (5.1) for all y ∈ [0 ,
1) and by evaluation at y = 0, we find inductively that ∂ ky u ( y, λ ) | y =0 = 0 for all k ∈ N (here the assumption u (0 , λ ) = 0 enters). We extend u ( · , λ ) to [ − ,
1] by setting u ( − y, λ ) := − u ( y, λ )for y ∈ [0 , u ( · , λ ) ∈ C ∞ ([ − , u ( · , λ ) is odd and satisfies Eq. (5.1) for all y ∈ ( − , λ ∈ Σ V and Lemma 5.9 finishes the proof. (cid:3) Construction of the Green function.
In order to construct the Green function,we need a more explicit expression for the Wronskian of u and u . Note carefully thatthis is the place where the spectral assumption enters. Lemma 5.11.
We have W ( u ( · , λ ) , u ( · , λ ))( y ) = 2 λu (0 , λ )(1 − y ) − − λ for all λ ∈ C with | Re λ | ≤ . Furthermore, if L V has no eigenvalues on the imaginaryaxis, there exists an ǫ > such that | u (0 , λ ) | & for all λ ∈ C with Re λ ∈ [0 , ǫ ] . roof. By definition and Remark 5.5, we have W ( u ( · , λ ) , u ( · , λ ))( y ) = (1 − y ) − − λ W ( v (0 , − λ ) v ( · , λ ) − v (0 , λ ) v ( · , − λ ) , v ( · , λ ))= − v (0 , λ ) W ( v ( · , − λ ) , v ( · , λ ))(1 − y ) − − λ = 2 λv (0 , λ )(1 − y ) − − λ = 2 λu (0 , λ )(1 − y ) − − λ for all λ ∈ C with | Re λ | ≤ .By assumption and Lemma 4.10, there exists an ǫ > L V in the strip { z ∈ C : Re z ∈ [0 , ǫ ] } . Consequently, by Lemma 5.10, u (0 , λ ) = 0for all λ ∈ C with Re λ ∈ [0 , ǫ ] and, since u (0 , λ ) = 1 + O ( h λ i − ) by Proposition 5.2,the claim follows. (cid:3) Definition 5.12.
For any λ ∈ C with Re λ ∈ (0 , ] and λ σ p ( L V ), we set G V ( y, x, λ ) := − − x ) W ( u ( · , λ ) , u ( · , λ ))( x ) ( u ( y, λ ) u ( x, λ ) 0 ≤ y ≤ x < u ( y, λ ) u ( x, λ ) 0 ≤ x ≤ y < . Lemma 5.13.
There exists an ǫ > such that any λ ∈ C with Re λ ∈ (0 , ǫ ] belongs to ρ ( L V ) and for any f = ( f , f ) ∈ e H we have R L V ( λ ) f ( y ) = (cid:18) R G V ( y, x, λ )[2 xf ′ ( x ) + ( λ + 1) f ( x ) + f ( x )] dxλ R G V ( y, x, λ )[2 xf ′ ( x ) + ( λ + 1) f ( x ) + f ( x )] dx − f ( y ) (cid:19) for y ∈ [0 , .Proof. By Lemma 4.10, it follows that there exists an ǫ > λ ∈ (0 , ǫ ]implies λ ∈ ρ ( L V ). Recall from the proof of Lemma 5.8 that u ( · , λ ) ∈ C ∞ ([0 , − y ) − (1+ λ ) v ( y, − λ ) = (1 − y ) − (1+ λ ) ψ ( y, − λ ) v ( y, − λ ) ψ ( y, − λ ) = (1 − y ) − λ v ( y, − λ ) ψ ( y, − λ )and by Lemma 5.6 we see that u ( y, λ ) = v (0 , − λ ) u ( y, λ ) + (1 − y ) − λ h ( y, λ ) , where h ( · , λ ) ∈ C ∞ ([0 , F λ ( x ) := 2 xf ′ ( x ) + ( λ + 1) f ( x ) + f ( x ).Then, by Lemma 5.11, we have Z G V ( y, x, λ ) F λ ( x ) dx = − λu (0 , λ ) X k =1 I k ( y )with I ( y ) := v (0 , − λ ) u ( y, λ ) Z y (1 − x ) λ u ( x, λ ) F λ ( x ) dxI ( y ) := h ( y, λ )(1 − y ) − λ Z y (1 − x ) λ u ( x, λ ) F λ ( x ) dxI ( y ) := v (0 , − λ ) u ( y, λ ) Z y (1 − x ) λ u ( x, λ ) F λ ( x ) dxI ( y ) := u ( y, λ ) Z y (1 + x ) λ h ( x, λ ) F λ ( x ) dx. y assumption, F λ ∈ C ∞ ([0 , I ∈ C ∞ ([0 , I ( y ) + I ( y ) = v (0 , − λ ) u ( y, λ ) Z (1 − x ) λ u ( x, λ ) F λ ( x ) dx and thus, I + I ∈ C ∞ ([0 , x = y + t (1 − y ) yields I ( y ) = h ( y, λ )(1 − y ) Z (1 − t ) λ [1 + y + t (1 − y )] λ u ( y + t (1 − y ) , λ ) F λ ( y + t (1 − y )) dt and thus, I ∈ C ∞ ([0 , w λ ( y ) := R G V ( y, x, λ ) F λ ( x ) dx be-longs to C ∞ ([0 , w λ satisfies Eq. (4.2) for y ∈ [0 , w λ (0) = 0 and by the oddness of F λ , we find inductively from Eq. (4.2) that w (2 k ) λ (0) = 0for all k ∈ N . This means that w λ extends to an odd function in C ∞ ([ − , w λ satisfies Eq. (4.2) for all y ∈ [ − , u := ( w λ , λw λ − f ) belongs to e H and satisfies ( λ − L V ) u = f , which means that u = R L V ( λ ) f . (cid:3) Time evolution on the unstable subspace.
By now we have collected enoughinformation so that we can prove the first part of Theorem 1.3, which is a consequenceof the following result combined with Lemmas 4.16 and 5.9.
Lemma 5.14.
We have rg P V ⊂ C ∞ ( − , × C ∞ ( − , and for every λ ∈ σ ( L V ) ∩ { z ∈ C : Re z > } =: σ u ( L V ) there exists a number n ( λ ) ∈ N such that S V ( s ) f = X λ ∈ σ u ( L V ) e λs n ( λ ) X k =0 s k k ! ( L V − λ ) k f for all f ∈ rg P V and all s ≥ .Proof. By Lemma 4.10, σ u ( L V ) is finite and consists of eigenvalues with finite algebraicmultiplicities. For each λ ∈ σ u ( L V ), let P V,λ be the corresponding spectral projection.Then rg P V = M λ ∈ σ u ( L V ) rg P V,λ . Denote by L V,λ the part of L V in the finite-dimensional subspace rg P V,λ . Clearly,rg P V,λ ⊂ D ( L V ) and for any f ∈ rg P V,λ , we have L V,λ f = L V f ∈ rg P V,λ ⊂ D ( L V ).Inductively, this implies rg P V,λ ⊂ D ( L nV ) for any n ∈ N and Corollary 4.4 shows thatrg P V,λ ⊂ C ∞ ( − , × C ∞ ( − , S V,λ ( s ) be the part of S V ( s ) in rg P V,λ and set e S V,λ ( s ) := e − λs S V,λ ( s ). Then e S V,λ ( s ) is a semigroup on rg P V,λ with generator L V,λ − λ . Since σ ( L V,λ − λ ) = { } anddim rg P V,λ < ∞ , it follows that L V,λ − λ is nilpotent and there exists an n ( λ ) ∈ N suchthat ( L V,λ − λ ) n ( λ ) = . Note that ∂ ns e S V,λ ( s ) f = e S V,λ ( s )( L V,λ − λ ) n f for all n ∈ N and f ∈ rg P V,λ . Consequently, ∂ n ( λ ) s e S V,λ ( s ) f = e S V,λ ( s )( L V,λ − λ ) n ( λ ) f = 0and integrating this equation yields e S V,λ ( s ) f = n ( λ ) X k =0 s k k ! ( L V,λ − λ ) k f = n ( λ ) X k =0 s k k ! ( L V − λ ) k f . Summation over all λ ∈ σ u ( L V ) finishes the proof. (cid:3) . Strichartz estimates
In order to separate the free evolution from the effect of the potential, we introducesuitable operators that account for the difference.
Definition 6.1.
For any ǫ > f ∈ C ([0 , T ǫ ( s ) f ]( y ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs Z [ G V ( y, x, λ ) − G ( y, x, λ )] f ( x ) dxdλ [ ˙ T ǫ ( s ) f ]( y ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN λe λs Z [ G V ( y, x, λ ) − G ( y, x, λ )] f ( x ) dxdλ The key result for the Strichartz estimates are the following bounds on T ǫ and ˙ T ǫ . Theorem 6.2.
Let p ∈ [2 , ∞ ] and q ∈ [1 , ∞ ) . Then there exists an ǫ > such that k e − ǫs T ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L (0 , k e − ǫs ˙ T ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L (0 , + k f k L (0 , for all ǫ ∈ (0 , ǫ ] and f ∈ C ([0 , . We now reduce the proof of Theorem 4.15 to Theorem 6.2. The rest of this section isthen devoted to the proof of Theorem 6.2.
Lemma 6.3.
Assume that Theorem 6.2 holds. Then Theorem 4.15 follows.Proof.
Let f ∈ D ( L V ). Then ( I − P V ) f ∈ D ( L V ) and by [5], p. 234, Corollary 5.15, weobtain S V ( s )( I − P V ) f = S st V ( s )( I − P V ) f = 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs R L st V ( λ )( I − P V ) f dλ for all ǫ >
0. By Lemma 5.13 there exists an ǫ > ǫ ∈ (0 , ǫ ], S V ( s )( I − P V ) f = 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs R L V ( λ )( I − P V ) f dλ. Now we set H := (ker P V ∩ D ( L V )) + e H , Y := L p loc ((0 , ∞ ) , L q (0 , ǫ ∈ (0 , ǫ ]and f ∈ H we defineΦ ǫ ( f )( s ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs [ R L V ( λ ) f ] dλ − [ S ( s ) f ] . We claim that Φ ǫ maps H into Y . Indeed, for f ∈ ker P V ∩ D ( L V ), we have Φ ǫ ( f )( s ) =[ S V ( s ) f ] − [ S ( s ) f ] and thus, Φ ǫ ( f ) ∈ Y by Remark 4.14. Furthermore, for f = ( f , f ) ∈ e H , Lemma 5.13 shows thatΦ ǫ ( f )( s ) = T ǫ ( s ) (cid:0) | · | f ′ + f + f (cid:1) + ˙ T ǫ ( s ) f and Theorem 6.2 yields the bound k e − ǫs Φ ǫ ( f )( s ) k L ps (0 , ∞ ) L q (0 , . k f k H . (6.1)Consequently, Φ ǫ ( f ) ∈ Y for all f ∈ H , as claimed. By density, Φ ǫ uniquely extends toa map Φ ǫ : H → Y and the bound (6.1) holds for all f ∈ H . For f ∈ ker P V ∩ D ( L V ) weobtain k e − ǫs [ S V ( s ) f ] k L ps (0 , ∞ ) L q (0 , ≤ k e − ǫs Φ ǫ ( f )( s ) k L ps (0 , ∞ ) L q (0 , + k e − ǫs [ S ( s ) f ] k L ps (0 , ∞ ) L q (0 , . k f k H y Proposition 3.5 and monotone convergence yields k [ S V ( s ) f ] k L ps (0 , ∞ ) L q (0 , . k f k H which, by density, extends to all f ∈ ker P V . (cid:3) Analysis of the operator T ǫ . First, we identify the integral kernel of T ǫ . Lemma 6.4.
Let λ ∈ C \ { } with | Re λ | ≤ and ω = Im λ . Then we have G V ( y, x, λ ) − G ( y, x, λ ) = X j =1 G V,j ( y, x, λ ) = X j =1 e G V,j ( y, x, λ ) , where G V, ( y, x, λ ) = 1 [0 , ( x − y ) λ − (1 + y ) − λ (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) G V, ( y, x, λ ) = 1 [0 , ( x − y ) λ − (1 − y ) − λ (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) G V, ( y, x, λ ) = 1 [0 , ( y − x ) λ − (1 + y ) − λ (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) G V, ( y, x, λ ) = 1 [0 , ( y − x ) λ − (1 + y ) − λ (1 + x ) λ O ((1 − y ) (1 − x ) h ω i − ) as well as e G V, ( y, x, λ ) = 1 [0 , ( x − y )(1 − x ) λ Z (1 + y ) − tλ O ((1 − y ) (1 − x ) h ω i t ) dt e G V, ( y, x, λ ) = 1 [0 , ( x − y ) h log(1 − y ) i (1 − x ) λ × Z (1 − y ) − tλ O ((1 − y ) (1 − x ) h ω i t ) dt e G V, ( y, x, λ ) = 1 [0 , ( y − x )(1 + y ) − λ (1 − x ) λ × Z (1 + x ) (1 − t ) λ O ((1 − y ) (1 − x ) h ω i t ) dt e G V, ( y, x, λ ) = 1 [0 , ( y − x ) h log(1 − x ) i (1 + y ) − λ (1 + x ) λ × Z (1 − x ) (1 − t ) λ O ((1 − y ) (1 − x ) h ω i t ) dt for all y, x ∈ [0 , .Proof. By definition and Proposition 5.2, u ( y, λ ) = (1 − y ) − (1+ λ ) v ( y, λ ) = (1 − y ) − (1+ λ ) ψ ( y, λ )[1 + O ((1 − y ) h ω i − )]= (1 + y ) − λ [1 + O ((1 − y ) h ω i − )]as well as u ( y, λ ) = (1 − y ) − (1+ λ ) [ v (0 , − λ ) v ( y, λ ) − v (0 , λ ) v ( y, − λ )]= (1 − y ) − (1+ λ ) ψ ( y, λ )[1 + O ((1 − y ) h ω i − )] − (1 − y ) − (1+ λ ) ψ ( y, − λ )[1 + O ((1 − y ) h ω i − )]= (1 + y ) − λ [1 + O ((1 − y ) h ω i − )] − (1 − y ) − λ [1 + O ((1 − y ) h ω i − )] . Finally, by Lemma 5.11,1(1 − x ) W ( u ( · , λ ) , u ( · , λ ))( x ) = (1 − x ) λ λu (0 , λ ) = (1 − x ) λ λ [1 + O ( h ω i − )] nd the first representation follows.For the second representation, we need to exploit the fact that u ( y,
0) = 0 to git ridof the apparent singularity of the Green function at λ = 0. By the fundamental theoremof calculus we obtain u ( y, λ ) = Z ∂ t u ( y, tλ ) dt = λ Z (1 + y ) − tλ O ((1 − y ) h tω i ) dt + λ h log(1 − y ) i Z (1 − y ) − tλ O ((1 − y ) h tω i ) dt. Inserting this expression for u in the definition of the Green function yields the secondrepresentation. (cid:3) In order to estimate the kernel of the operators T ǫ and ˙ T ǫ , we make frequent use of thefollowing elementary bound. Lemma 6.5.
We have h a − b i & h a i − h b i for all a, b ∈ R .Proof. If | b | ≤ | a | we have h a i − h b i . . h a − b i and if | b | ≥ | a | , h a − b i ≃ | a − b | ≥ | b | − | a | ≥ | b | ≃ h b i & h a i − h b i . (cid:3) Proposition 6.6.
There exists an ǫ > such that K ǫ ( s, y, x ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN e λs [ G V ( y, x, λ ) − G ( y, x, λ )] dλ exists for any ( s, y, x ) ∈ [0 , ∞ ) × [0 , × [0 , and ǫ ∈ (0 , ǫ ] and we have T ǫ ( s ) f ( y ) = Z K ǫ ( s, y, x ) f ( x ) dx. Furthermore, | K ǫ ( s, y, x ) | . e ǫs h log(1 − y ) i h s + log(1 − x ) i − for all ( s, y, x ) ∈ [0 , ∞ ) × [0 , × [0 , and all ǫ ∈ (0 , ǫ ] .Proof. Lemma 6.4 yields the rough bound | G V ( y, x, λ ) − G ( y, x, λ ) | . (1 − y ) − (1 − x ) − h λ i − for all x, y ∈ [0 ,
1) and λ ∈ C with Re λ ∈ (0 , ǫ ], provided ǫ > K ǫ ( s, y, x ) follows and Fubini’s theorem yields the statedexpression for T ǫ ( s ) f .To prove the bound on K ǫ , we need to distinguish between λ small and λ large. Tothis end, we use a standard cut-off χ : R → [0 ,
1] that satisfies χ ( t ) = 1 if | t | ≤ χ ( t ) = 0 if | t | ≥
2. Then we split K ǫ ( s, y, x ) = e ǫs π Z R e iωs [ G V ( y, x, ǫ + iω ) − G ( y, x, ǫ + iω )] dω =: e ǫs π [ I ǫ ( s, y, x ) + J ǫ ( s, y, x )] nd use Lemma 6.4 to decompose I ǫ ( s, y, x ) = Z R χ ( ω ) e iωs [ G V ( y, x, ǫ + iω ) − G ( y, x, ǫ + iω )] dω = X j =1 Z R χ ( ω ) e iωs e G V,j ( y, x, ǫ + iω ) dω | {z } =: I ǫ,j ( s,y,x ) and J ǫ ( s, y, x ) = Z R [1 − χ ( ω )] e iωs [ G V ( y, x, ǫ + iω ) − G ( y, x, ǫ + iω )] dω = X j =0 Z R [1 − χ ( ω )] e iωs G V,j ( y, x, ǫ + iω ) dω | {z } =: J ǫ,j ( s,y,x ) . By Lemma 6.4 we have[1 − χ ( ω )] G V, ( y, x, ǫ + iω ) = [1 − χ ( ω )](1 + y ) − ǫ (1 − x ) ǫ (1 + y ) − iω (1 − x ) iω × ( ǫ + iω ) − O ((1 − y ) (1 − x ) h ω i − )= e iω ( − log(1+ y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − )and thus, | J ǫ, ( s, y, x ) | . (cid:12)(cid:12)(cid:12)(cid:12)Z R e iω ( s − log(1+ y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − ) dω (cid:12)(cid:12)(cid:12)(cid:12) . h s − log(1 + y ) + log(1 − x ) i − . h s + log(1 − x ) i − by means of two integrations by parts. For J ǫ, we note that[1 − χ ( ω )] G V, ( y, x, ǫ + iω ) = 1 [0 , ( x − y )[1 − χ ( ω )](1 − y ) − ǫ (1 − x ) ǫ (1 − y ) − iω (1 − x ) iω × ( ǫ + iω ) − O ((1 − y ) (1 − x ) h ω i − )= e iω ( − log(1 − y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − )and we obtain | J ǫ, ( s, y, x ) | . h s − log(1 − y ) + log(1 − x ) i − . h log(1 − y ) i h s + log(1 − x ) i − . The terms J ǫ, and J ǫ, are handled analogously and in summary, we obtain | J ǫ ( s, y, x ) | . h log(1 − y ) i h s + log(1 − x ) i − . Now we turn to the low-frequency part I ǫ . By Lemma 6.4 we have χ ( ω ) e G V, ( y, x, ǫ + iω ) = Z (1 + y ) − itω (1 − x ) iω O ((1 − y ) (1 − x ) h ω i − t ) dt = Z e iω ( − t log(1+ y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − t ) dt nd thus, by Fubini and two integrations by parts, | I ǫ, ( s, y, x ) | ≤ Z (cid:12)(cid:12)(cid:12)(cid:12)Z R e iω ( s − t log(1+ y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − t ) dω (cid:12)(cid:12)(cid:12)(cid:12) dt . Z h s − t log(1 + y ) + log(1 − x ) i − dt . Z h t log(1 + y ) i h s + log(1 − x ) i − dt . h s + log(1 − x ) i − . For e G V, we have χ ( ω ) e G V, ( y, x, ǫ + iω ) = Z h log(1 − y ) i (1 − y ) − itω (1 − x ) iω O ((1 − y ) (1 − x ) h ω i − t ) dt and thus, | I ǫ, ( s, y, x ) |h log(1 − y ) i ≤ Z (cid:12)(cid:12)(cid:12)(cid:12)Z R e iω ( s − t log(1 − y )+log(1 − x )) O ((1 − y ) (1 − x ) h ω i − t ) dω (cid:12)(cid:12)(cid:12)(cid:12) dt . Z h s − t log(1 − y ) + log(1 − x ) i − dt . Z h t log(1 − y ) i h s + log(1 − x ) i − dt . h log(1 − y ) i h s + log(1 − x ) i − . The corresponding bound for I ǫ, follows analogously and for I ǫ, we note that χ ( ω ) e G V, ( y, x, ǫ + iω ) = Z [0 , ( y − x ) h log(1 − x ) i (1 + y ) − iω (1 + x ) iω (1 − x ) i (1 − t ) ω × O ((1 − y ) (1 − x ) h ω i − t ) dt = Z [0 , ( y − x ) h log(1 − y ) i e iω ( − log(1+ y )+log(1+ x )+(1 − t ) log(1 − x )) × O ((1 − y ) (1 − x ) h ω i − t ) dt and thus, | I ǫ, ( s, y, x ) |h log(1 − y ) i ≤ Z (cid:12)(cid:12)(cid:12) Z R [0 , ( y − x ) e iω ( s − log(1+ y )+log(1+ x )+(1 − t ) log(1 − x )) × O ((1 − y ) (1 − x ) h ω i − t ) dω (cid:12)(cid:12)(cid:12) dt . Z [0 , ( y − x ) h s − log(1 + y ) + log(1 + x ) + (1 − t ) log(1 − x ) i − dt . Z [0 , ( y − x ) h− log(1 + y ) + log(1 + x ) − t log(1 − x ) i × h s + log(1 − x ) i − dt . h log(1 − y ) i h s + log(1 − x ) i − . (cid:3) Now we can conclude the desired bound for the operator T ǫ . emma 6.7. Let p ∈ [2 , ∞ ] and q ∈ [1 , ∞ ) . Then there exists an ǫ > such that k e − ǫs T ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L (0 , for all f ∈ C ([0 , and ǫ ∈ (0 , ǫ ] .Proof. By Proposition 6.6 we have | e − ǫs T ǫ ( s ) f ( y ) | ≤ Z e − ǫs | K ǫ ( s, y, x ) || f ( x ) | dx = Z ∞ e − ǫs | K ǫ ( s, y, − e − η ) || f (1 − e − η ) | e − η dη . h log(1 − y ) i Z R h s − η i − [0 , ∞ ) ( η ) | f (1 − e − η ) | e − η dη and thus, k e − ǫs T ǫ ( s ) f k L q (0 , . Z R h s − η i − | [0 , ∞ ) ( η ) f (1 − e − η ) | e − η dη. Consequently, Young’s inequality yields k e − ǫs T ǫ ( s ) f k L s (0 , ∞ ) L q (0 , . kh·i − k L ( R ) Z ∞ | f (1 − e − η ) | e − η dη . Z (1 − x ) | f ( x ) | dx. On the other hand, by Cauchy-Schwarz, we also have k e − ǫs T ǫ ( s ) f k L q (0 , . Z (1 − x ) | f ( x ) | dx for all s ≥ k e − ǫs T ǫ ( s ) f k L ∞ s (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L (0 , . (cid:3) Analysis of the operator ˙ T ǫ . The treatment of the operator ˙ T ǫ is very similar. Proposition 6.8.
Let p ∈ [2 , ∞ ] and q ∈ [1 , ∞ ) . Then there exists an ǫ > such that k e − ǫs ˙ T ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L (0 , + k f k L (0 , for all f ∈ C ([0 , and ǫ ∈ (0 , ǫ ] .Proof. Let e χ : R → [0 ,
1] be a smooth cut-off that satisfies e χ ( x ) = 1 if | x | ≤ e χ ( x ) = 0 if | x | ≥
2. We define χ : C → [0 ,
1] by χ ( z ) := e χ (Re z, Im z ). We split˙ T ǫ ( s ) = ˙ T ♭ǫ ( s ) + ˙ T ♯ǫ ( s ), where˙ T ♭ǫ ( s ) f ( y ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN χ ( λ ) λe λs [ G V ( y, x, λ ) − G ( y, x, λ )] f ( x ) dxdλ ˙ T ♯ǫ ( s ) f ( y ) := 12 πi lim N →∞ Z ǫ + iNǫ − iN [1 − χ ( λ )] λe λs [ G V ( y, x, λ ) − G ( y, x, λ )] f ( x ) dxdλ. or the low-frequency part ˙ T ♭ǫ ( s ) f , the additional factor of λ (compared to T ǫ ( s ) f ) ishelpful as it cancels the singularity of the Green function at λ = 0. Consequently, weimmediately infer the bound k e − ǫs ˙ T ♭ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L (0 , . k f k L (0 , by proceeding as in the proofs of Proposition 6.6 and Lemma 6.7.The high-frequency part ˙ T ♯ǫ ( s ) f is more delicate as the additional λ destroys the inversesquare decay of the Green function as | Im λ | → ∞ . Consequently, we need to performan integration by parts with respect to x in order to recover the decay. More precisely,by Lemma 6.4, we have˙ T ♯ǫ ( s ) f ( y ) = 12 πi lim N →∞ X j =1 Z ǫ + iNǫ − iN [1 − χ ( λ )] e λs Z λG V,j ( y, x, λ ) f ( x ) dxdλ and an integration by parts yields Z λG V, ( y, x, λ ) f ( x ) dx = (1 + y ) − λ Z y (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) f ( x ) dx = (1 + y ) − λ (1 − y ) λ O ((1 − y ) h ω i − ) f ( y )+ (1 + y ) − λ Z y (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) f ( x ) dx + (1 + y ) − λ Z y (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) f ′ ( x ) dx =: (1 + y ) − λ (1 − y ) λ O ((1 − y ) h ω i − ) f ( y ) + Z H V, ( y, x, λ ) f ( x ) dx + Z H ′ V, ( y, x, λ ) f ′ ( x ) dx. Note that the kernels H V, and H ′ V, are of the same type as G V, . By the same procedurewe obtain an analogous representation of R λG V, ( y, x, λ ) f ( x ) dx . The remaining twocontributions produce an additional boundary term, i.e., Z λG V, ( y, x, λ ) f ( x ) dx = (1 + y ) − λ Z y (1 − x ) λ O ((1 − y ) (1 − x ) h ω i − ) f ( x ) dx = (1 + y ) − λ O ( h ω i − ) f (0) + (1 + y ) − λ (1 − y ) λ O ((1 − y ) h ω i − ) f ( y )+ Z H V, ( y, x, λ ) f ( x ) dx + Z H ′ V, ( y, x, λ ) f ′ ( x ) dx nd analogously for G V, . This means that e − ǫs ˙ T ♯ǫ ( s ) f ( y ) = f (0) Z R e iω ( s − log(1+ y )) O ( h ω i − ) dω + f ( y ) Z R e iω ( s − log(1+ y )+log(1 − y )) O ((1 − y ) h ω i − ) dω + f ( y ) Z R e iωs O ((1 − y ) h ω i − ) dω + [ A ǫ ( s ) f ]( y ) + [ B ǫ ( s ) f ′ ]( y )= O ( h s i − ) f (0) + O ( h s i − (1 − y ) ) f ( y ) + [ A ǫ ( s ) f ]( y ) + [ B ǫ ( s ) f ′ ]( y ) , where the operators A ǫ ( s ) and B ǫ ( s ) satisfy the bound for T ǫ ( s ) from Lemma 6.7. Con-sequently, we find k e − ǫs ˙ T ♯ǫ ( s ) f k L ps (0 , ∞ ) L q (0 , . | f (0) | + (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L q (0 , + (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L (0 , + (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L (0 , . (cid:13)(cid:13)(cid:13) (1 − | · | ) f (cid:13)(cid:13)(cid:13) L ∞ (0 , + (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L (0 , + k f k L (0 , and the simple estimate(1 − y ) | f ( y ) | . Z y (1 − x ) | f ′ ( x ) | dx + Z y (1 − x ) − | f ( x ) | dx . (cid:13)(cid:13)(cid:13) (1 − | · | ) f ′ (cid:13)(cid:13)(cid:13) L (0 , + k f k L (0 , for all y ∈ [0 ,
1] finishes the proof. (cid:3)
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Universit¨at Wien, Fakult¨at f¨ur Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vi-enna, Austria
E-mail address : [email protected] Universit¨at Wien, Fakult¨at f¨ur Mathematik, Oskar-Morgenstern-Platz 1, 1090 Vi-enna, Austria
E-mail address : [email protected]@univie.ac.at