Limiting absorption principle on L p -spaces and scattering theory
aa r X i v : . [ m a t h . A P ] M a r LIMITING ABSORPTION PRINCIPLE ON L p -SPACES ANDSCATTERING THEORY KOUICHI TAIRA
Abstract.
In this paper, we study the mapping property form L p to L q ofthe resolvent of the Fourier multipliers and scattering theory of generalizedSchr¨odinger operators. Though the first half of the subject is studied in [4],we extend their result to away from the duality line and we also study theH¨older continuity of the resolvent. Introduction
In this note, we study L p -estimates for resolvents of the Fourier multipliers andthe scattering theory of the discrete Schr¨odinger operator, the fractional Schr¨odingeroperators and the Dirac operators.One of the interest in the scattering theory of the Schr¨odinger operator is toprove the asymptotic completeness of the wave operators: W ± = s − lim t →±∞ e it ( − ∆+ V ) e − it ( − ∆) , i.e. that W ± are surjections onto the absolutely continuous subspace of L ( R d ).Through the Kato’s smooth perturbation theory, the asymptotic completeness ofthe wave operators is closely related to the limit absorption principle:sup z ∈ I ± \ I k| V | ( − ∆ − z ) − | V | k B ( L ( R d )) < ∞ , (1) sup z ∈ I ± \ I k| V | ( − ∆ + V − z ) − | V | k B ( L ( R d )) < ∞ , (2)where I ⊂ (0 , ∞ ) is an interval and I ± = { z ∈ C | ± Im z ≥ } and V is a real-valued function. A strong tool for proving (1) and (2) is the Mourre theory [23],which gives sufficient conditions that (1) and (2) hold.On the other hands, Kenig, Ruiz and Sogge [19] establish the L p -type limitingabsorption principle for the free Schr¨odinger operator: k ( − ∆ − z ) − k B ( L p ( R d ) ,L q ( R d )) ≤ C p,q | z | d ( p − q ) − , z ∈ C \ [0 , ∞ ) , d ≥ C p,q > z ∈ C \ [0 , ∞ ) and (1 /p, /q ) ∈ (0 , × (0 , / ( d + 1) ≤ /p − /q ≤ /d , ( d + 1) / d < /p and 1 /q < ( d − / (2 d ).(3) is also proved by Kato and Yajima [18] independently when 1 /p + 1 /q = 1,and applied to the scattering theory of the Schr¨odinger operator − ∆ + V , where V ∈ L p ( R d ), d/ ≤ p < ( d + 1) / V ∈ L p ( R d ) for d/ ≤ p ≤ ( d +1) / Mathematics Subject Classification.
Primary 47A10, Secndary 47A40.
Key words and phrases. discrete Schr¨odinger operators, resolvents, limiting absorptionprinciple. proved the L p -type limiting absorption principle for Schr¨odinger operator − ∆ + V with a real-valued potential V ∈ L r ( R d ) ∩ L / ( R d ), r > / Re z ≥ λ , < ± Im z ≤ k ( − ∆ + V − z ) − k B ( L p ( R d ) ,L q ( R d )) ≤ C (Re z ) d ( p − q ) − , where λ > d = 3, p = 4 / q = 4. The strategy of the proof in [7] isto replace the L -trace theorem in the proof of the classical Agmon-Kato-Kurodatheorem [24, Theorem XIII. 33] by Stein-Tomas L p -restriction theorem for thesphere [31]. Ionescu and Schlag [13] extends the result of [7] to a large class ofpotentials V , which contains L p ( R d ), d/ ≤ p ≤ ( d + 1) /
2, the global Kato classpotentials and some perturbations of first order operators. See also the recent worksby Huang, Yao, Zheng [11] and Mizutani [23]. Moreover, in [13], it is also provedthat existence and asymptotic completeness of the wave operators. We note thatthere are no positive eigenvalues of − ∆ + V when V ∈ L p ( R d ), d/ ≤ p ≤ ( d + 1) / p > ( d + 1) / T ( D ) on X d , we study uniform resol-vent estimates, H¨older continuity of the resolvent and Carleman type inequalitiesfor Fourier multipliers on X d , where X = R or X = Z . The uniform resolventestimates for a Fourier multipliers are investigated in [4] and [5] in the dualityline when X = R in order to study the Lieb-Thirring type bounds for fractionalSchr¨odinger operators and Dirac operators. One of the purpose is to prove theuniform resolvent estimates away form the duality line and to extend to the case of X = Z . To prove this, we follow the argument in [9, Appendix] for the Laplacianon the Euclidean space, however, the argument in [9] does not cover the generalcase since in the proof of [9, Theorem 6], the spherical symmetry and the Stein-Tomas theorem for the sphere are crucial. Moreover, we study the scattering theoryof the discrete Schr¨odinger operator, the fractional Schr¨odinger operators and theDirac operators. We note that the limiting absorption principle for free discreteSchr¨odinger operators is studied in [14], [21] and [30]. In [21], the scattering theoryof the discrete Schr¨odinger operators perturbed by L p -potentials are studied for arange of p . In [30], it is proved that the range of ( p, q ) which the uniform resolventestimate holds for the discrete Schr¨odinger operators differs from the one for thecontinuous Schr¨odinger operators when d ≥ L p,r by real interpolation. For simplicity we do not mention this below.Throughout this paper, we denote X d = Z d or R d for an integer d ≥
2. Wedenote µ by the Lebesgue measure if X d = R d by the counting measure if X d = Z d .Moreover, we write c X d = R d if X d = R d and c X d = T d = ( R / Z ) d if X d = Z d . Weoften use [ − / , / d ⊂ R d as a fundamental domain of T d .Let T ∈ C ∞ ( c X d , R ). Moreover, we assume T ∈ S ′ ( R d ) if X = R . We denotethe set of all critical values of T by Λ c ( T ) and set M λ = { ξ ∈ c X d | T ( ξ ) = λ } for λ ∈ R . We denote the induced surface measure by µ λ away from the critical pointsof T . Moreover, for I ⊂ R , we write I ± = { z ∈ C | Re z ∈ I, ± Im z ≥ } .Set S k = { ( 1 p , q ) ∈ [0 , × [0 , | q ≤ p − k + 1 , k k < p , q < k k } . (4) Assumption A.
Let U ⊂ c X d be a relativity compact open set and I ⊂ R bean compact interval. Suppose ∂ ξ T ( ξ ) = 0 for ξ ∈ ¯ U . The Fourier transform of IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 3 the induced surface measure satisfies the following estimate: For any χ ∈ C ∞ c ( c X d )supported in U , there exists C > | Z M λ e πix · ξ χ ( ξ ) dµ λ ( ξ ) | ≤ C (1 + | x | ) − k , x ∈ X d , λ ∈ I. (5) Remark . If ∂ ξ d T = 0 on supp χ and supp χ is small enough, (5) is rewritten as | Z \ X d − e πi ( x ′ · ξ ′ + x d h λ ( ξ ′ )) χ ( ξ ′ , h λ ( ξ ′ )) dξ ′ | ≤ C ′ (1 + | x | ) − k , x ∈ X d , λ ∈ I where ξ = ( ξ ′ , ξ d ) and M λ = { ( ξ ′ , ξ d ) ∈ c X d | ξ d = h λ ( ξ ′ ) } . Moreover, if (5) holds,then there exits N ≥ | Z \ X d − e πi ( x ′ · ξ ′ + x d h λ ( ξ ′ )) b ( ξ ′ ) dξ ′ | ≤ C X | α |≤ N sup ξ ′ ∈ \ X d − | ∂ αξ ′ b ( ξ ′ ) | where b ∈ C ∞ c ( \ X d − ) which is supported in { ξ ′ | ( ξ ′ , h λ ( ξ ′ )) ∈ supp χ } and C isindependent of b . Example 1.
Suppose that M λ ∩ supp χ has at least m nonvanishing principalcurvature curvature at every point, then (5) holds for k = m/ R ± ( z ) = ( T ( D ) − z ) − for z ∈ { z ∈ C | ± Im z > } . Moreover, fora signature ± , we define χ ( D ) R ± ( λ ± i
0) if ∂ ξ T = 0 on supp χ by the Fouriermultiplier with its symbol χ ( ξ )( T ( ξ ) − λ ± i − . For 1 ≤ p ≤ ∞ , L p ( X d ) denotesthe Lebesgue space with the Lebesgue measure if X = R and with the countingmeasure if X = Z .Our first result is the following: Theorem 1.2.
Let T ∈ C ∞ ( c X d , R ) and let I be a compact interval of R . Supposethat T − ( I ) is compact. Fix a signature ± . Let χ ∈ C ∞ c ( c X d ) . Suppose that (5) holds for λ ∈ I and supp χ ⊂ U . ( i ) There exists such that sup z ∈ I ± k χ ( D ) R ± ( z ) k B ( L p ( X d ) ,L q ( X d )) < ∞ , for (1 /p, /q ) ∈ S k . ( ii ) Set k δ = k − δ for < δ ≤ and β δ = (2 /p − δ . Then sup z,w ∈ I ± , | z − w |≤ | z − w | − β δ k χ ( D )( R ± ( z ) − R ± ( w )) k B ( L p ( X d ) ,L p ∗ ( X d )) < ∞ , for (1 /p, /p ∗ ) ∈ S k δ , where p ∗ = p/ ( p − . ( iii ) Suppose X = R . Under Assumption A, for (1 /p, /q ) ∈ S k , there exists C N,p,q > such that k µ N,γ ( x ) χ ( D ) u k L q ( R d ) ∩B ∗ ≤ C N,p,q k µ N,γ ( x )( T ( D ) − λ ) χ ( D ) u k L p ( R d )+ B for u ∈ S ( R d ) . KOUICHI TAIRA
Applications to the fractional Schr¨odinger operators and the Diracoperators.
Let n = 2 d/ if d is even and n = 2 ( d +1) / if d is odd. We define theDirac operators on R d : D = d X j =1 α j D j , D = d X j =1 α j D j + α d +1 , where α j are n × n Hermitian matrix and satisfy the Clifford relations: α j α k + α k α j = − δ jk I n × n and D j = ∂ x j / (2 πi ). Note that if we define D d +1 = mI n × n , then D = − ( d X j =1 I n × n D j ) = − ∆ · I n × n , D = ( − ∆ + 1) · I n × n , where we denote ∆ = ( P dj =1 ∂ x j ) / (4 π ). In this subsection, we suppose that T ( D )is the one of the following operators: T ( D ) = ( − ∆) s/ , T ( D ) = ( − ∆ + 1) s/ − , T ( D ) = D , T ( D ) = D , where 0 < s < d . We use the convention that s = 1 when T ( D ) = D or T ( D ) = D .Moreover, we denote the product space Z n for a function space Z by simply Z when T ( D ) = D or T ( D ) = D . As is noted in [4, § c (( − ∆) s/ ) = ( { } if s > , ∅ if s ≤ , Λ c (( − ∆ + 1) s/ −
1) = { } , and Λ c ( D ) = { } , Λ c ( D ) = {− , } . Moreover, T ( D ) is self-adjoint on its domain H s ( R d ) by the elliptic regularity.Let Y , Y be Banach spaces such that( Y , Y ) ∈ [ ( p , q ) ∈ S d − { L p ( R d ) } × { L q ( R d ) } , (6)if 2 d/ ( d + 1) ≤ s < d and( Y , Y ) ∈ [ ( p , q ) ∈ S d − , p − q ≤ sd { L p ( R d ) + L p ( R d ) } × { L q ( R d ) ∩ L q ( R d ) } , (7)if 0 < s < dd +1 .A part of the following estimate is a generalization of [4, Theorem3.1]. Theorem 1.3.
Let I ⊂ R \ Λ c ( T ( D )) be a compact interval. We define R ± ( λ ) for λ ∈ I by the Fourier multiplier of the distribution ( T ( ξ ) − ( λ ± i − , where thisdistribution is well-defined since T ( ξ ) has no critical points in T − ( I ) . ( i ) We have sup z ∈ I ± k R ± ( z ) k B ( Y ,Y ) < ∞ . IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 5 ( ii ) Let ( Y , Y ) be satisfying p = q in (6) if d/ ( d + 1) ≤ s < d and p = q in (7) if < s < d/ ( d + 1) . Let < δ ≤ and β δ = (2 /p − δ . Then sup z,w ∈ I ± , | z − w |≤ | z − w | − β δ k ( R ± ( z ) − R ± ( w )) k B ( Y ,Y ) < ∞ . ( iii ) Let V ∈ L ( d +1) / ( R d ) ∩ L ∞ ( R d ) . Assume V is a self-adjoint matrix if T ( D ) = D or D . Set H = T ( D ) and H = H + V denotes the unique self-adjoint exten-sions of T ( D ) | C ∞ c ( R d ) and T ( D ) + V | C ∞ c ( R d ) respectively. Then the wave operators W ± = s − lim t →±∞ e itH e − itH exist and are complete, i.e. the ranges of W ± are the absolutely continuous subspace H ac ( H ) of H . ( iv ) Let V ∈ L ( d +1) / ( R d ) ∩ L ∞ ( R d , R ) . Assume s > / only when T ( D ) = ( − ∆) s/ with s / ∈ N . Then the set of nonzero eigenvalues σ pp ( H ) \{ } is discrete in R \{ } .Moreover, each eigenvalue in σ pp ( H ) \ { } has finite multiplicity.Remark . ( i ) is proved in [4] if 1 /p + 1 /q = 1. In [11], ( i ) is proved when T ( D ) = ( − ∆) s/ for 2 d/ ( d + 1) ≤ s < d . Remark . In ( iii ) and ( iv ), the condition V ∈ L ∞ ( R d ) is expected to be relaxedif we consider the appropriate selj-adjoint extension of T ( D )+ V . However, in orderto avoid the technical difficulty, we assume V ∈ L ∞ ( R d ). Remark . When T ( D ) = D or T ( D ) = ( − ∆) s/ , by a scaling argument as in[4, Remark 4.2], we have the uniform bound of R ± ( z ) with z ∈ C ± . Even when T ( D ) = D or T ( D ) = ( − ∆ + 1) s/ −
1, the author expects to obtain the uniformbound of R ± ( z ) with z ∈ C ± by further analysis. Remark . When T ( D ) = ( − ∆) s/ or T ( D ) = ( − ∆ + 1) s/ −
1, under the as-sumption of part ( iv ), we can provesup z ∈ I ± k ( H − z ) − k B ( X,X ∗ ) < ∞ (8)for any compact set I ⊂ R \ ( σ pp ∪ { } ). In particular, the singular continuousspectrum of T ( D ) is empty. For its proof, we may mimic the argument in [13,Section 4]. However, when T ( D ) = D or T ( D ) = D , the author do not knowwhether (8) holds or not since the difference of the outgoing resolvent and incomingresolvent is not always positive definite: R +0 ( λ ) − R − ( λ ) =( D + λ )( R +0 ( λ ) − R − ( λ )) , if T ( D ) = D ,R +0 ( λ ) − R − ( λ ) =( D + λ )( R +1 ( λ ) − R − ( λ )) , if T ( D ) = D , where R ± ( λ ) = ( − ∆ − ( λ ± i ) − and R ± ( λ ) = ( − ∆ + 1 − ( λ ± i ) − . See thearguments in [13, Proof of Theorem 1.3 (d) and (e)] or [24, Lemma 8 in the proofof Theorem XIII.33]. Remark . Under the assumption of ( iv ), we can prove that each eigenfunction u of H associated with eigenvalue λ ∈ R \ { } satisfies(1 + | x | ) N u ∈ H ( R d ) , N ≥ N < s − / T ( D ) = ( − ∆) s/ with s / ∈ N . The restriction N < s − / T ( D ) = ( − ∆) s/ with s / ∈ N is needed due to the singularityof the symbol T ( ξ ) = | ξ | s at ξ = 0. KOUICHI TAIRA
Scattering theory for the discrete Schr¨odinger oeprators.
The scatter-ing theory of the discrete Schr¨odinger operators is studied in [21] for the potential V ∈ L p ( Z d ), with 1 ≤ p < / d = 3 and 1 ≤ p < d/ (2 d + 1) if d ≥
4. In thissubsection, we extend their results to when V ∈ L p ( Z d ) for 1 ≤ p ≤ d/ d ≥ H u ( x ) = − X | x − y | =1 ,y ∈ Z d ( u ( x ) − u ( y )) , x ∈ Z d . Note that H is a bounded self-adjoint operator on L ( Z d ). We write h ( ξ ) = 4 d X j =1 sin πξ j for ξ ∈ T d , H = h ( D )and hence the spectrum σ ( H ) of H is equal to [0 , d ]. Moreover, σ ac ( H ) = [0 , d ],where σ ac ( H ) is the absolutely continuous spectrum of H . Set R ± ( z ) = ( H − z ) − for ± Im z >
0. Note that Λ c ( h ( D )) = { k } dk =0 , where we recall that Λ c ( h ( D ))is the set of all critical values of h ( ξ ). Moreover, if V ∈ L p ( Z d , R ) for some1 ≤ p < ∞ , H = H + V is a bounded self-adjoint operator and σ ess ( H ) = [0 , d ]since V ∈ L p ( Z d ) ⊂ L ∞ ( Z d ) and V ( x ) → ∞ as | x | → ∞ . Here σ ess ( H ) denotesthe essential spectrum of H .We define R ± ( λ ) for λ ∈ I by the Fourier multiplier of the distribution ( h ( ξ ) − ( λ ± i − , where this distribution is well-defined by virtue of [30, Theoerem 1.8].Note that we may take λ as a critical value. We recall thatsup z ∈ C \ R k R ± ( z ) k B ( L p ( Z d ) ,L p ∗ ( Z d )) < ∞ , holds for 1 ≤ p ≤ dd +3 ([30, Proposition 3.3]) and d ≥ Theorem 1.9.
Fix a signature ± and let d ≥ . ( i ) Let ≤ p ≤ dd +3 . Then sup z ∈ C ± k R ± ( z ) k B ( L p ( Z d ) ,L p ∗ ( Z d )) < ∞ . ( ii ) Let ≤ p < dd +3 . Take < δ ≤ such that p < / (3 δ/d + ( d + 3) /d ) . Then sup z,w ∈ C ± , | z − w |≤ | z − w | − β δ k ( R ± ( z ) − R ± ( w )) k B ( L p ( Z d ) ,L q ( Z d )) < ∞ . ( iii ) Let V ∈ L p ( Z d ) for ≤ p < d/ and set V / = sgn V | V | / . Then, amap z ∈ I ±
7→ | V | / R ± ( z ) | V | / is H¨older continuous. Moreover, for V ∈ L d/ ( Z d ) , it follows that a map z ∈ I ±
7→ | V | / R ± ( z ) | V | / iscontinuous. ( iv ) Let V ∈ L d/ ( Z d , R ) and set H = H + V . Then the wave operators W ± = s − lim t →±∞ e itH e − itH exist and are complete, i.e. the ranges of W ± are the absolutely continuoussubspace H ac ( H ) of H .Remark . In Proposition 4.10, we prove that the range of p can be extended inthe low energy or the high energy. IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 7 We fix some notations. For an integer k ≥ C ∞ c ( X k ) denotes C ∞ c ( R k ) if X = R and the set of all finitely supported functions if X = Z . For 1 ≤ p ≤ ∞ , we write p ∗ = p/ ( p − t + = max ( t,
0) for t ∈ R . We define the Bezov space B and B ∗ by k u k B = k u k L ( | x |≤ + ∞ X j =1 j/ k u k L (2 j − ≤| x | < j ) , k u k B ∗ = k u k L ( | x |≤ + sup j ≥ − j/ k u k L (2 j − ≤| x | < j ) , B = { u ∈ L loc ( X d ) | k u k B < ∞} , B ∗ = { u ∈ L loc ( X d ) | k u k B ∗ < ∞} , B ∗ = { u ∈ B ∗ | lim sup R →∞ R Z | x |≤ R | u ( x ) | dx = 0 } . Acknowledgment.
The author was supported by JSPS Research Fellowship forYoung Scientists, KAKENHI Grant Number 17J04478 and the program FMSP atthe Graduate School of Mathematics Sciences, the University of Tokyo. The authorwould like to thank his supervisors Kenichi Ito and Shu Nakamura for encouragingto write this paper. The author also would like to gratefully thank Haruya Mizu-tani and Yukihide Tadano for helpful discussions. Moreover, the author would beappreciate Evgeny Korotyaev informing the paper [21] and Jean-Claude Cuenin forpointing out a mistake of the first draft.2.
Abstract theorem
In this section, we state abstract theorems which give estimates for some integraloperators. Let K ∈ L ∞ ( X d × X d ). For x, y ∈ X d , we denote K ( x, y ) = K ( x ′ , y ′ , x d , y d ) = K x d ,y d ( x ′ , y ′ ) , x = ( x ′ , x d ) , y = ( y ′ , y d ) , where x ′ , y ′ ∈ X d − and x d , y d ∈ X . Moreover, we denote Kf ( x ) = Z X d K ( x, y ) f ( y ) dy, T x d ,y d g ( x ′ ) = Z X d − K x d ,y d ( x ′ , y ′ ) f ( y ′ ) dy ′ for f ∈ C ∞ c ( X d ) and g ∈ C ∞ c ( X d − ).2.1. Estimates for integral operators on duality line.
We consider the fol-lowing assumptions:
Assumption B.
There exists C , C > x d , y d ∈ X and g ∈ C ∞ c ( X d − ) k T x d ,y d g k L ( X d − ) ≤ C k g k L ( X d − ) , (9) k T x d ,y d g k L ∞ ( X d − ) ≤ C (1 + | x d − y d | ) − k k g k L ( X d − ) . (10) Remark . Suppose that we can write K ( x, y ) = K ( x ′ − y ′ , x d , y d ) for some K ∈ L ∞ ( X d +1 ). Then Assumption B directly follows from the following estimates: k Z X d − K ( x ′ , x d , y d ) e − πix ′ · ξ ′ dx ′ k L ∞ ( \ X d − ξ ′ ) ≤ C , sup x ′ ∈ X d − | K ( x ′ , x d , y d ) | ≤ C (1 + | x d − y d | ) − k . KOUICHI TAIRA
Remark . By the Riesz-Thorin interpolation theorem, (9) and (10) imply k T x d ,y d g k L p ∗ ( X d − ) ≤ C − p C p − (1 + | x d − y d | ) − k ( p − k g k L p ( X d − ) , (11)for 1 ≤ p ≤ Proposition 2.3.
Suppose Assumption B . Then there exists a universal constant M d > and M p,k > such that ( sup R> ,x ∈ R d R Z | x − x |≤ R | Kf ( x ) | dx ) ≤ M d C k f k B , f ∈ B , (12) k Kf k L p ∗ ( X d ) ≤ M p,k C − p C p − k f k L p ( X d ) , f ∈ L p ( X d )(13) for ≤ p ≤ k + 1) / ( k + 2) .Remark . (13) follows from Proposition 2.8 below under the assumption ofProposition 2.8. However, the proof below is simpler than the proof of Proposition2.8. Proof.
By a density argument, we may assume f ∈ C ∞ c ( X d ). We observesup R> ,x ∈ X d R Z | x − x |
0. Using the Minkowski inequality and (9), weobtain (12).Next, we prove (13). We set L p = C − p C p − . By the Minkowski inequalityand (11), we have k Kf k L p ∗ ( X d ) = kk Z X T x d ,y d ( f ( · , y d )) dy d k L p ∗ ( X d − x ′ ) k L p ∗ ( X xd ) ≤ L p k Z X (1 + | x d − y d | ) − k ( p − k f ( · , y d ) k L p ∗ ( X d − y ′ ) dy d k L p ∗ ( X xd ) ≤ M p,k L p k f k L p ( X d ) , where we use the fractional integration theorem in the last line. This gives (13). (cid:3) Estimates for integral operators away from duality line.
For x d ∈ X ,we define T x d and T ∗ x d by T x d f ( x ′ ) = Kf ( x ′ , x d ) = Z X d K ( x, y ) f ( y ) dy, T ∗ x d g ( y ) = Z X d − ¯ K ( x, y ) g ( x ′ ) dx ′ . We define S x d ( y d , z d ) g ( y ′ ) = Z X d − Z X d − ¯ K ( x, y ) K ( x, z ) g ( z ′ ) dz ′ dx ′ . Note that T ∗ x d T x d f ( y ) = Z X ( S x d ( y d , z d ) f ( · , z d ))( y ′ ) dz d . Next, we consider the following assumption.
IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 9 Assumption C.
There exists C , C > x d , y d , z d ∈ X k S x d ( y d , z d ) g k L ( X d − ) ≤ C k g k L ( X d − ) , (16) k S x d ( y d , z d ) g k L ∞ ( X d ) ≤ C (1 + | y d − z d | ) − k k g k L ( X d − ) . (17) Remark . Suppose that we can write K ( x, y ) = K ( x ′ − y ′ , x d , y d ) for some K ∈ L ∞ ( X d +1 ). Then Assumption C directly follows from the following estimates: k Z X d − Z X d − e πiy ′ · ξ ′ ¯ K ( x ′ , x d , y d ) K ( x ′ − y ′ , x d , z d ) dx ′ dy ′ k L ∞ ( \ X d − ) ≤ C , sup y ′ ,z ′ ∈ X d − | Z X d − ¯ K ( x ′ − y ′ , x d , y d ) K ( x ′ − z ′ , x d , z d ) dx ′ | ≤ C (1 + | y d − z d | ) − k . Remark . By the Riesz-Thorin interpolation theorem, (16) and (17) imply k S x d ( y d , z d ) g k L p ∗ ( X d − ) ≤ ( C − p C p − ) (1 + | y d − z d | ) − k ( p − k g k L p ( X d − ) , (18)for 1 ≤ p ≤ Proposition 2.7.
Suppose that K satisfies Assumption C . Then there exists auniversal constant M ′ p,k > such that ( sup R> ,x ∈ R d R Z | x − x |≤ R | Kf ( x ) | dx ) ≤ M ′ p,k C − p C p − k f k L p ( X d ) , f ∈ L p ( X d ) , (19) for ≤ p ≤ k + 1) / ( k + 2) . Moreover, if K ∗ ( x, y ) = ¯ K ( y, x ) satisfies Assumption C , then it follows that k K ∗ f k L q ( X d ) ≤ M ′ q/ ( q − ,k C q C − q k f k B , f ∈ B , (20) for k + 1) /k ≤ q ≤ ∞ .Proof. By a density argument, we may assume f ∈ C ∞ c ( X d ). First, we prove (19).Due to (14), it suffices to prove k T x d f k L ( X d − ) ≤ M ′ p,k C − p C p − k f k L p ( X d ) , f ∈ C ∞ c ( X d ) . (21)By the standard T ∗ T argument, this estimate is equivalent to k T ∗ x d T x d f k L p ∗ ( X d ) ≤ ( M ′ p,k C − p C p − ) k f k L p ( X d ) . We set L p = ( C − p C p − ) . Using the Minkowski inequality and (18), we have k T ∗ x d T x d f k L p ∗ ( X d ) = kk Z X ( S x d ( y d , z d ) f ( · , z d ))( y ′ ) dz d k L p ∗ ( X d − y ′ ) k L p ∗ ( X yd ) ≤ L p k Z X (1 + | y d − z d | ) − k ( p − k f ( · , y d ) k L p ∗ ( X d − y ′ ) dy d k L p ∗ ( X yd ) ≤ ( M ′ p,k ) L p k f k L p ( X d ) , where we use the fractional integration theorem (the Hardy-Littlewood-Sobolevtheorem) in the last line. This proves (19).Next, we prove (20). Replacing K in (21) by K ∗ , we have k Z X d ¯ K ( y, x ) f ( y ) dy k L ( X d − x ′ ) ≤ M ′ p,k C − p C p − k f k L p ( X d ) , f ∈ C ∞ c ( X d ) . By duality, we have k Z X d − K ( y, x ) g ( x ′ ) dx ′ k L q ( X dy ) ≤ M ′ q/ ( q − ,k C q C − q k g k L ( X d − ) , x d ∈ X, where q = p ∗ . By (15) and the Minkowski inequality, we obtain k Kf k L q ( X d ) ≤ Z X k Z X d − K ( x, y ) f ( y ) dy ′ k L q ( X dx ) dy d ≤ M ′ q/ ( q − ,k C q C − q Z X k f ( · , y d ) k L ( X d − y ′ ) dy d ≤ M ′ q/ ( q − ,k C q C − q k f k B . (cid:3) We impose the additional assumption.
Assumption D.
There exists C > | K ( x, y ) | ≤ C (1 + | x − y | ) − k , x ∈ X d . Under Assumption C and D , we obtain the estimates similar to (13) away fromthe H¨older exponent. Proposition 2.8.
Suppose that K and K ∗ ( x, y ) = ¯ K ( y, x ) satisfy Assumption C and D . Then there exists a universal constant L ′ p,,q,k > such that k Kf k L q ( X d ) ≤ L ′ p,q,k C p,q,k,l k f k L p ( X d ) , f ∈ L p ( X d ) , where /p − /q = 1 /l and C p,q,k,l = C p ∗ C p − C − q , if ≤ p ≤ ( k +1)(2 k +1) k +3 k +1 , q > kk , k +1 p ∗ k ≤ q ,C k +1)2 k +1 (1 − l )2 C k +1) − l (2 k +1) l C l +2 k (2 k +1) l , if ≤ l ≤ k + 1 , k ( k +1) q < p ∗ < k +1 kq ,C q C q ∗ − C − p ∗ , if ≤ p < k k , q ≥ (2 k +1)( k +1) k , k +1 kq ≤ p ∗ . We prove this proposition by a series of lemmas.
Lemma 2.9.
Suppose that K satisfies Assumption C . Let ψ ∈ C ∞ c ( R ) . Define K j ( x, y ) = ψ ((2 x d − z d ) / j +1 , (2 y d − z d ) / j +1 ) K ( x − y ) for j and z d ∈ X . Thenfor ≤ p ≤ k + 1) / ( k + 2) k K j f k L ( X d ) ≤ L ′ M ′ p,k C − p C p − j/ k ψ k L ∞ ( X ) k f k L p ( X d ) with L ′ > independent of z d ∈ X and j .Proof. We take
L > ψ ⊂ B L , where B L ⊂ X is an open ball withradius L and with center 0. We observe k K j f k L ( X d ) = Z | x d − z d / |≤ L j k K j f ( · , x d ) k L ( X d − ) dx d Replacing K in (21) with K j , we have k K j f ( · , x d ) k L ( X d − ) ≤ M ′ p,k C − p C p − k ψ k L ∞ ( X ) k f k L p ( X d ) . We note that there exists L ′ > z d and j such that( Z | x d − z d / |≤ L j dx d ) / ≤ L ′ j/ . IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 11 Combining the above three inequality, we obtain the desired result. (cid:3)
We need the following technical lemma in order to prove Lemma 2.11 below.
Lemma 2.10.
Let F ∈ C ∞ c ( R ) . Then there exists ψ ∈ C ∞ c ( R ) such that F ( x d − y d j ) = L j Z X ψ ( 2 x d − z d j , y d − z d j ) dz d , x d , y d ∈ R , where L j = 2 − j if X = R and − j − ≤ L j ≤ − j if X = Z .Proof. We define ψ ∈ C ∞ c ( R ) as follows: Take χ ∈ C ∞ c ( R , [0 , R R χ ( x ) dx = 2 and supp χ ⊂ ( − / , /
2) if X = R and such that χ ( t ) = 1 on | t | ≤ χ ( t ) = 0 on | t | ≥ X = Z . We define ψ ( z, z ′ ) = F ( z − z ′ ) χ ( z + z ′ ),Then F ( x d ) = R X ψ ( x d + z, z ) dz if X = R and Z X ψ ( 2 x d − z d j +1 , y d − z d j +1 ) dz d = X z d ∈ Z ψ ( 2 x d − z d j +1 , y d − z d j +1 )= F ( x d − y d j ) X z d ∈ Z χ ( z d j ) . if X = Z . We note 2 j ≤ X z d ∈ Z χ ( z d j ) ≤ j +2 . We set L j = 1 if X = R and L j = P z d ∈ Z χ ( z d j ) if X = Z and we are done. (cid:3) The following lemma is a consequence of Lemma 2.9, however its proof is a bittechnical due to the convolution type cut-off. The conclusion of the following lemmais same as [9, Lemma 1], where the uniform resolvent estimate of the Laplacian isstudied. However, since their proof strongly depends on the spherical symmetryof the Laplacian and the Stein-Tomas theorem for the sphere, we cannot directlyapply their argument to our cases. In order to overcome this difficulty, we borrowan idea from the proof of the Carleson-Sj¨olin theorem [10, Theorem 2.1].
Lemma 2.11.
Suppose that K satisfies Assumption C . Let F ∈ C ∞ c ( R ) . Define K j, conv ( x, y ) = F (( x d − y d ) / j ) K ( x, y ) for non-negative integer j . Then for ≤ p ≤ k + 1) / ( k + 2) , there exists a universal constant M ′′ p,k such that k K j, conv f ( x ) k L ( X d ) ≤ M ′′ p,k C − p C p − j k f k L p ( X d ) (22) Proof.
By Lemma 2.10, we have | K j, conv f ( x ) | ≤ − j | Z X K j,z d ( x, y ) dz d | , where we set K j,z d ( x, y ) = K ( x, y ) ψ ((2 x d − z d ) / j +1 , (2 y d − z d ) / j +1 ). Take ϕ ∈ C ∞ c ( R ) such that ψ ( x d , y d ) = ψ ( x d , y d ) ϕ ( y d ). We take L > ψ ⊂ B L , where B L ⊂ X is an open ball with radius L and with center 0. We note |{ z d ∈ X | ψ ( 2 x d − z d j +1 , y d − z d j +1 ) = 0 }| ≤ L j +1 . Set M = ( M ′ p,k C − p C p − ) . Using the Cauchy-Schwarz inequality and Lemma2.9, we have Z X d | Z X K j,z d f ( x ) dz d | dx ≤ L j +1 Z X Z X d | K j,z d f ( x ) | dxdz d ≤ LL ′ M j Z X k ϕ ( 2 · − z d j +1 ) f k L p ( X d ) dz d . Since p ≤
2, by using the Minkowski inequality, we have Z X k ϕ ( 2 · − z d j +1 ) f k L p ( X d ) dz d ≤k ϕ ( 2 · − z d j +1 ) k L ( X ) k f k L p ( X d ) ≤ L ′′ j k ϕ k L ( X ) k f k L p ( X d ) with L ′′ depends only on ϕ . Thus we obtain Z X d | K j,conv f ( x ) | dx ≤ ( M ′′ p,k C − p C p − ) j k f k L p ( X d ) , where ( M ′′ p,k ) = 2 LL ′ L ′′ ( M ′ p,k ) k ϕ k L ( X ) . (cid:3) Corollary 2.12.
Suppose that K satisfies Assumption D. Then there exists aconstant L > which depends only on F , d and k such that k K j,conv f k L ∞ ( X d ) ≤ L C − jk k f k L ( X d ) . (23) In addition, we suppose that K and K ∗ ( x, y ) = ¯ K ( y, x ) satisfy Assumption C. Set /p = 1 − q/ p ∗ and L ,p,q = ( M ′′ p ,k ) /q L − /q . Then k K j,conv f k L q ( X d ) ≤ L ,p,q C p ∗ C p − C − q j kq − jk k f k L p ( X d ) (24) if q ≥ and ( k + 1)(1 − /p ) /k ≤ /q and k K j,conv f k L q ( X d ) ≤ L ,q ∗ ,p ∗ C q C q ∗ − C − p ∗ j (1+2 k ) p ∗ − jk k f k L p ( X d ) (25) if p ≤ and ( k + 1) / ( kq ) ≤ − /p .Proof. (23) follows from k K j,conv f k L ∞ ( X d ) ≤k F ( · / j ) K k L ∞ ( X d ) k f k L ( X d ) ≤ L C − jk k f k L ( X d ) with some constant L > K ∗ also satisfies Assumption C and D, by duality, (25)holds. (cid:3) Proof of Proposition 2.8.
Take η ∈ C ∞ c ( R , [0 , η ( t ) = 1 on 0 ≤ t ≤ η = 0 on t ≥
2. Set F ( x ) = η ( | x | ) − η ( | x | / IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 13 ( k + 1)(1 − /p ) /k ≤ /q , q > (1 + 2 k ) /k , we have k Kf k L q ( X d ) = k ∞ X j =0 K j,conv f k L q ( X d ) ≤ ∞ X j =0 k K j,conv f k L q ( X d ) ≤ L ,p,q C p ∗ C p − C − q ∞ X j =0 j/ jd (1 /q − / k f k L p ( X d ) ≤ L ′ ,p,q C p ∗ C p − C − q k f k L p ( X d ) , where L ′ ,p,q = L ,p,q P ∞ j =0 j/ jd (1 /q − / . Similarly, for ( k + 1) / ( kq ) ≤ − /p , p < (1 + 2 k ) / (1 + k ), we have k Kf k L q ( X d ) ≤ L ′ ,q ∗ ,p ∗ C p ∗ C p − C − p ∗ k f k L p ( X d ) . In order to prove the end point estimates, we use Bourgain’s interpolation trick([2], [3, § ≤ p ≤ ∞ and 1 ≤ r ≤ ∞ by L p,r ( X d ): k f k L p,r ( X d ) = ( p r ( R ∞ µ ( { x ∈ X d | | f ( x ) | > α } ) rp α r − dα ) r , r < ∞ , sup α> αµ ( { x ∈ X d | | f ( x ) | > α } ) p , r = ∞ ,L p,r ( X d ) = { f : X d → C | f : measurable , k f k L p,r ( X d ) < ∞} . Bourgain’s interpolation trick with (24) and (25) implies that for 1 ≤ p ≤ ( k +1)(2 k + 1) / ( k + 3 k + 1) , q = (1 + 2 k ) /k , it follows that k Kf k L q, ∞ ( X d ) ≤ L ′ ,p,q C p ∗ C p − C − q k f k L p, ( X d ) with a universal constant L ′ ,p,q . Similarly, for p = (1 + 2 k ) / (1 + k ), q ≥ (2 k +1)( k + 1) /k , we have k Kf k L q, ∞ ( X d ) ≤ L ′ ,q ∗ ,p ∗ C q C q ∗ − C − p ∗ k f k L p, ( X d ) . By real interpolating above estimates, we complete the proof. (cid:3) Uniform resolvent estimates
Proof of Theorem 1.2 ( i ) and ( ii ) . Proof of Theorem 1.2 ( i ) and ( ii ) . We follows the argument as in [4, Lemma 3.3].By using a partition of unity and a linear coordinate change, we may assume that ∂ ξ d T = 0 on supp χ . Moreover, by the implicit function theorem, we may assumethat for λ ∈ I , M λ has the following graph representation: M λ ∩ supp χ ⊂ { ( ξ ′ , h λ ( ξ ′ )) ∈ c X d | ξ ′ ∈ U } for some relativity compact open set U ⊂ \ X d − and h λ which is smooth withrespect to ξ ′ ∈ U and λ ∈ I and T ( ξ ) − λ = e ( ξ, λ )( ξ d − h λ ( ξ ′ )) , (26) where e ( ξ, λ ) = R ( ∂ ξ d T )( ξ ′ , tξ d + (1 − t ) h λ ( ξ ′ )) dt . Furthermore, we may assumemin ξ ∈ supp χ,λ ∈ A e ( ξ, λ ) > χ small. Set K z, ± ( x ) = Z c X d e πix · ξ χ ( ξ ) T ( ξ ) − z dξ,K z,w, ± ( x ) = K z, ± ( x ) − K w, ± ( x ) , where λ = Re z, µ = Re w ∈ I and ± Im z, ± Im w ≥
0. In order to prove Theorem1.2, it suffices to show that K z, ± satisfies Assumptions C and D, and that K z,w, ± satisfies Assumption B. Lemma 3.1.
Fix a signature ± . For any ≤ δ ≤ , there exists C , C , C ,δ > such that for x = ( x ′ , x d ) ∈ X d , z, w ∈ I ± with | z − w | ≤ , we have sup ξ ′ ∈ \ X d − | Z X d − K z, ± ( y ′ , x d ) e − πiy ′ · ξ ′ dy ′ | ≤ C , | K z, ± ( x ) | ≤ C (1 + | x | ) − k sup ξ ′ ∈ \ X d − | Z X d − K z,w, ± ( y ′ , x d ) e − πiy ′ · ξ ′ dy ′ | ≤ C , | K z,w, ± ( x ) | ≤ C ′ | z − w | δ (1 + | x | ) − k + δ Proof.
Note that R X d − K z, ± ( y ′ , x d ) e − πiy ′ · ξ ′ dy ′ = R b X e πixdξd χ ( ξ ) T ( ξ ) − z dξ d . If necessarywe take supp χ is small, it suffices to replace the integration region by R . Thus by(26), we have Z b X e πix d ξ d χ ( ξ ) T ( ξ ) − z dξ d = Z R e πix d ξ d χ ( ξ ) T ( ξ ) − z dξ d = Z R e πix d ( ξ d + h λ ( ξ ′ )) χ ( ξ ′ , ξ d + h λ ( ξ ′ )) e ( ξ ′ , ξ d + h λ ( ξ ′ ) , ξ d ) ξ d − i Im z dξ d = : e πix d h λ ( ξ ′ ) γ z, ± ( ξ ′ , x d ) . By using [4, (3.10)] for ± Im z > ± Im z = 0, we have | ∂ αξ ′ γ z, ± ( ξ ′ , x d ) | ≤ C α (27)for α ∈ N d − . We will prove (27) in Lemma A.3. Thus the first inequality holds.Moreover, we note that K z, ± ( x ) = Z \ X d − γ z, ± ( ξ ′ ) e πi ( x ′ · ξ ′ + x d h λ ( ξ ′ )) dξ ′ . Since γ z, ± is compactly supported in ξ ′ -variable, then (5) and (27) imply the secondinequality. The estimates for K z,w, ± ( x ) follow from the estimates | ∂ αξ ′ γ z, ± ( ξ ′ , x d ) | ≤ C ′ α | z − w | δ (1 + | x d | ) δ , which is also proved after Lemma A.3: (51). (cid:3) Lemma 3.2.
There exists C > such that | Z X d − Z X d − e πiy ′ · ξ ′ ¯ K z, ± ( x ′ , x d − y d ) K z, ± ( x ′ − y ′ , x d − z d ) dx ′ dy ′ | ≤ C | Z X d − ¯ K z, ± ( x ′ − y ′ , x d − y d ) K z, ± ( x ′ − z ′ , x d − z d ) dx ′ | ≤ C (1 + | y d − z d | ) − k where C > is as in the proof of Lemma 3.1. IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 15 Proof.
Note that Z X d − Z X d − e πiy ′ · ξ ′ ¯ K z, ± ( x ′ , x d − y d ) K z, ± ( x ′ − y ′ , x d − z d ) dx ′ dy ′ = e πi ( y d − z d ) h λ ( ξ ′ ) γ z, ± ( ξ ′ , x d − z d ) γ z, ± ( ξ ′ , x d − y d ) , where γ z, ± is as in the proof of Lemma 3.1. Moreover, we have Z X d − ¯ K z, ± ( x ′ − y ′ , x d − y d ) K z, ± ( x ′ − z ′ , x d − z d ) dx ′ = Z b X d − e πi ( y ′ − z ′ ) · ξ ′ +2 πi ( y d − z d ) h λ ( ξ ′ ) γ z, ± ( ξ ′ , x d − z d ) γ z, ± ( ξ ′ , x d − y d ) dξ ′ . Thus (5) and (27) imply the conclusion. (cid:3)
Lemma 3.1 and 3.2 imply that K z, ± satisfies Assumptions C and D and K z,w, ± satisfies Assumption B. This completes the proof of Theorem 1.2. (cid:3) Remark . In order to prove ( i ), it is sufficient to prove ( i ) for ± Im z = 0 byusing the Phragm´en-Lindel¨of principle as in [27, Section 5.3]. See also [4, AppendixA] for the estimates of the Shatten norm of the resolvent. Here we avoid using thePhragm´en-Lindel¨of principle. Corollary 3.4.
Let r , r ∈ (1 , k + 2] satisfying /r + 1 /r ≥ / ( k + 1) . Then sup z ∈ I ± k W χ ( D ) R ± ( z ) W k B ( L ( X d )) ≤ C k W k L r ( X d ) k W k L r ( X d ) (28) for W ∈ L r ( X d ) and W ∈ L r ( X d ) . Moreover, let W ∈ L r ( X d ) and W ∈ L r ( X d ) . Then it follows that W χ ( D ) R ± ( z ) W belongs to B ∞ ( L ( X d )) and amap z ∈ I ± W χ ( D ) R ± ( z ) W ∈ B ∞ ( L ( X d )) is continuous in z ∈ I ± . Inaddition, for r = r = r ∈ (1 , k δ + 2) , we have k W χ ( D )( R ± ( z ) − R ± ( w )) W k B ( L ( X d )) ≤ C | z − w | β δ k W k L r ( X d ) k W k L r ( X d ) (29) for z, w ∈ I ± , | z − w | ≤ .Proof. (28) and (29) follow from Theorem 1.2 and the H¨older inequality. For prov-ing the other statements, we may assume W , W ∈ C ∞ c ( X d ) by ε/ W and W are compactly supported and since the integral kernel of χ ( D ) R ± is in L ∞ by Lemma 3.1, then the integral kernel of W χ ( D ) R ± ( z ) W issquare integrable and hence Hilbert-Schmidt. Thus it follows that W χ ( D ) R ± ( z ) W is compact. Moreover, by (29), we see that W χ ( D ) R ± ( z ) W is continuous in z ∈ I ± . The case of W ∈ L r ( X d ) and W ∈ L r ( X d ) follows from the ε/ (cid:3) Supersmoothing, Proof of Theorem 1.2 ( iii ) . In this subsection, we as-sume X = R . The author expect that the following proposition with X = Z holds.However, we prove this with with X = R for possibly technical reason. We recall µ N,γ ( x ) = (1 + | x | ) N (1 + γ | x | ) − N . We restate Theorem 1.2 ( iii ): Proposition 3.5.
Let I ⊂ R be a compact interval. Suppose T − ( I ) is compact.Let χ ∈ C ∞ c ( R d ) be supported in T − ( I ) . Under Assumption A, for (1 /p, /q ) ∈ S k ,there exists C N,p,q > such that k µ N,γ ( x ) χ ( D ) u k L q ( R d ) ∩B ∗ ≤ C N,p,q k µ N,γ ( x )( T ( D ) − λ ) χ ( D ) u k L p ( R d )+ B (30) for u ∈ S ( R d ) . Lemma 3.6.
Suppose that m ∈ C ∞ ( R d ) satisfies | ∂ αξ m ( ξ ) | ≤ C α (1 + | ξ | ) −| α | / for α ∈ N d . Let < p < ∞ . We set ˜ µ N,γ ( x d ) = (1 + | x d | ) N (1 + γ | x | ) − N . Thenwe have k µ ( x ) m ( D ) µ ( x ) − k B ( L p ( R d )) ≤ C N,m,p , k µ ( x ) m ( D ) µ ( x ) − k B ( B ( R d )) ≤ C N,m , k µ ( x ) m ( D ) µ ( x ) − k B ( B ∗ ( R d )) ≤ C N,m if µ ( x ) ∈ { µ N,γ ( x ) , µ − N,γ ( x ) , ˜ µ N,γ ( x d ) , ˜ µ − N,γ ( x d ) } , where C N,m,p and C N,m are in-dependent of < γ ≤ and depends only on d , N and finite number of C M .Proof. The proof is same as in the proof of [13, (3.7)]. In fact, though the range of p is restricted in [13], the proof succeeds even when 1 < p < ∞ . (cid:3) Lemma 3.7. ( i ) For α ∈ N d , we have ∂ αx µ N,γ ( x ) = b α ( x ) µ N,γ ( x )(31) ∂ αx µ N,γ ( x ) − = b ′ α ( x ) µ N,γ ( x ) − (32) for some functions b α , b ′ α ∈ C ∞ ( R d ) such that for β ∈ N d , | (1 + | x | ) ( | α | + | β | ) / ∂ βx b α ( x ) | ≤ C α,β,N , | (1 + | x | ) ( | α | + | β | ) / ∂ βx b ′ α ( x ) | ≤ C α,β,N with some constant C α,β,N which is independent of < γ ≤ . ( ii ) There exists C N > independent of < γ ≤ such that µ N,γ ( x ) µ N,γ ( y ) − + µ N,γ ( y ) µ N,γ ( x ) − ≤ C N (1 + | x − y | ) N , x, y ∈ R d . Proof. ( i ) We prove (31) only. The proof of (32) is similar. We prove (31) byinduction in | α | . If α = 0, then (31) is trivial. Let M > | α | ≤ M . If | α | = M , by the induction hypothesis, we have ∂ x j ∂ αx µ N,α ( x ) =( ∂ x j b α ( x )) µ N,γ ( x ) + b α ( x ) ∂ x j µ N,c ( x )=(( ∂ x j b α )( x ) + b α ( x ) b e j ( x )) µ N,γ ( x ) , where ( e , ..., e d ) is a standard basis in R d . Thus, if we set b α + e j ( x ) = ( ∂ x j b α )( x ) + b α ( x ) b e j ( x ), then | (1 + | x | ) ( | α | + | β | ) / ∂ βx b α ( x ) | ≤ C α,β,N follows. This proves (31)for | α | = M + 1. ( ii ) is easily proved. (cid:3) Corollary 3.8.
For k ∈ R we define Λ k = ( I − ∆) k/ . Then k µ Λ k µ − Λ − k k B ( L p ( R d )) + k µ Λ k µ − L − k k B ( B ) + k µ Λ k µ − L − k k B ( B ∗ ) ≤ C N,k,p , k Λ k µ Λ − k µ − k B ( L p ( R d )) + k Λ k µ Λ − k µ − k B ( B ) + k Λ k µ Λ − k µ − k B ( B ∗ ) ≤ C N,k,p , with some C N,k,p > independent of < γ ≤ for µ ∈ { µ N,γ , µ − N,γ } and < p < ∞ .Proof. The proof is same as in [13, Lemma 3.2] by virtue of Lemma 3.6 and 3.7. (cid:3)
IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 17 Proof of Proposition 3.5.
Let Y ∈ { L p ( R d ) , B} and Y ∈ { L q ( R d ) , B ∗ } . If neces-sary, we may assume supp χ is small enough. In fact, by using a partition of unity { χ j } Mj =1 such that P Mj =1 χ j = 1 on supp χ , we have k µ N,γ ( x ) χ ( D ) u k Y ≤ M X j =1 k µ N,γ ( x )( χ j χ )( D ) u k Y , M X j =1 k µ N,γ ( x )( T ( D ) − λ )( χ j χ )( D ) u k Y ≤ C N,m,p k µ N,γ ( x )( T ( D ) − λ ) χ ( D ) u k Y , where we use the triangle inequality in the first line and Lemma 3.6 in the secondline. Thus we may replace χ ( D ) by ( χ j χ )( D ) in (30).We may suppose ˆ u and ˆ f are supported in supp χ and we may suppose ∂ ξ d T = 0on supp χ by rotating the coordinate and by taking supp χ small enough. We set ξ + j = ε e j + p − ε e d for j = 1 , ..., d − ξ + d = ξ d , where ε > e , ..., e d ) is the standard basis of R d . Since ( ξ +1 , ..., ξ + d ) is the basisof R d , then C − d X j =1 ˜ µ N,γ ( x · ξ + j ) ≤ µ N,γ ( x ) ≤ C d X j =1 ˜ µ N,γ ( x · ξ + j )with some constant C > γ , where˜ µ N,γ ( t ) = (1 + t ) N (1 + γt ) − N . Thus it suffices to prove that k ˜ µ N,γ ( x · ξ + j ) u k Y ≤ C N k ˜ µ N,γ ( x · ξ + j )( T ( D ) − λ ) u k Y for each j = 1 , ..., d . If ε > ∂ ξ d T = 0 implies ξ + j · ∇ T ( ξ ) = ε ∂ ξ T + p − ε ∂ ξ d T = 0 on supp χ . Thus by rotating the coordinate, we mayreduce to prove k ˜ µ N,γ ( x d ) u k Y ≤ C N k ˜ µ N,γ ( x d )( T ( D ) − λ ) u k Y . We remark that this reduction is the only part to miss proving this Propositionwhen X = Z . In fact, there are no basis containing the normal vector of x · ξ + j -direction when X = Z .Set f = ( T ( D ) − λ ) u . By the implicit function theorem, we have T ( ξ ) − λ = e ( ξ, λ )( ξ d − h λ ( ξ ′ )) as in (26). Then we have e ( ξ, λ ) − ˆ f ( ξ ) = ( ξ d − h λ ( ξ ′ ))ˆ u ( ξ ) onsupp χ . We denote ˜ f ( ξ ′ , x d ) is the Fourier transform of f with respect to ξ , ..., ξ d − -variables and set ˆ g ( ξ ) = e ( ξ, λ ) − ˆ f ( ξ ). Here e ( ξ, λ ) − is well-defined on supp ˆ f sincesupp f ⊂ supp χ . Then ( D x d − h λ ( ξ ′ ))˜ u ( ξ ′ , x d ) = ˜ g ( ξ ′ , x d ) , Since ˜ u and ˜ g are smooth, by using variation of parameters, we can write˜ u ( ξ ′ , x d ) = Z x d −∞ e πi ( x d − y d ) h λ ( ξ ′ ) ˜ g ( ξ ′ , y d ) dy d = − Z ∞ x d e πi ( x d − y d ) h λ ( ξ ′ ) ˜ g ( ξ ′ , y d ) dy d . Note that we use the first line of the above representation if x d ≤ x d ≥
0. Taking the inverse Fourier transform and multiplying ˜ µ N,γ ( x d ), wehave ˜ µ N,γ ( x d ) u ( x ) = Z R Z R d − K N,γ ( x ′ − y ′ , x d , y d )˜ µ N,γ ( y d ) g ( y ) dy ′ dy d where K N,γ ( x ′ − y ′ , x d , y d ) = ˜ µ N,γ ( x d )˜ µ N,γ ( y d ) ( χ x d < χ x d ≤ y d − χ x d > χ x d ≤ y d ) × Z [R d − e πi ( x ′ − y ′ ) · ξ ′ +2 πi ( x d − y d ) h λ ( ξ ′ ) ψ ( ξ ′ ) dξ ′ . Note that ˜ µ N,γ ( x d )˜ µ N,γ ( y d ) ( χ x d < χ x d ≤ y d − χ x d > χ x d ≤ y d ) ≤
1. Let R be the linear op-erator on R d with the integral kernel K N,γ . We recall supp ˆ f ⊂ supp χ andˆ g = e ( ξ, λ ) − ˆ f ( ξ ). Hence we can write˜ µ N,γ ( x d ) u ( x ) = K N,γ ( x ′ − y ′ ) ∗ (˜ µ N,γ ( y d ) ϕ ( D ) e ( D, λ ) − ˜ µ − N,γ ( y d )˜ µ N,γ ( y d ) f )( x )where ϕ ∈ C ∞ c ( R d ) such that ϕ = 1 on supp χ . By virtue of Lemma 3.6, itfollows that the operator norms of ˜ µ N,γ ( y d ) χ ( D ) e ( D, λ ) − ˜ µ N,γ ( y d ) − on L p ( R d )(1 < p < ∞ ), B and B ∗ are uniformly bounded in λ ∈ I .By virtue of Propositions 2.7 and 2.8, it suffices to K N,γ and K ∗ N,γ ( x, y ) =¯ K N,γ ( y, x ) satisfies Assumptions C and D. To see this, we may mimic the proof ofLemma 3.2. We omit the detail. (cid:3) Applications
Fractional Schr¨odinger operators and Dirac operators.
In this subsec-tion, we suppose that T ( D ) is the one of the following operators: T ( D ) = ( − ∆) s/ , T ( D ) = ( − ∆ + 1) s/ − , T ( D ) = D , T ( D ) = D , where 0 < s ≤ d . Proof of Theorem 1.3.
We consider the case when T ( D ) = ( − ∆) s/ or T ( D ) =(1 − ∆) s/ only. The case when T ( D ) = D or T ( D ) = D is similarly proved if wenotice D = − ∆ I n × n , D = ( − ∆ + 1) I n × n as in the proof of [4, Theorem 3.1]. We take a real-valued function χ ∈ C ∞ c ( R d , [0 , χ = 1 on T − ( I ) and supp χ ⊂ R \ Λ c ( T ( D )). Note that M λ = { T ( ξ ) = λ } is sphere and hence has non vanishing Gaussian curvature. if λ ∈ σ ( T ( D )) \ Λ c ( T ( D )). Then we apply Theorem 1.2 with k = ( d − / z ∈ I ± k χ ( D ) R ± ( z ) k B ( L p ( R d ) ,L q ( R d )) < ∞ (33)for ( p, q ) ∈ S d − . On the other hand, by the support property of χ and the Hardy-Littlewood-Sobolev inequality, we havesup z ∈ I ± k (1 − χ ( D )) R ( z ) k B ( L p ( R d ) ,L q ( R d )) < ∞ (34) IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 19 if 1 /p − /q ≤ s/d . In fact, if 2 α = − d/ d/p and 2 β = − d/q + d/
2, then k (1 − χ ( D )) R ( z ) k B ( L p ( R d ) ,L q ( R d )) ≤k ( I − ∆) − α k B ( L p ( R d ) ,L ( R d )) k (1 − χ ( D ))( I − ∆) α + β R ( z ) k B ( L ( R d )) × k ( I − ∆) − β k B ( L ( R d ) ,L q ( R d )) . Thus (34) follows from the the Hardy-Littlewood-Sobolev inequality. Combining(33) with (34), we obtain ( i ). ( ii ) is similarly proved. Lemma 4.1. ( i ) Suppose d/ ( d + 1) ≤ s < d . Let < δ ≤ , r ∈ (2 d/s, d + 1) − δ ] and r , r ∈ (1 , d + 1)] satisfying d + 1 ≤ r + 1 r ≤ sd . Then sup z ∈ I ± k W R ± ( z ) W k B ( L ( R d )) ≤ C k W k L r ( R d ) k W k L r ( R d ) k W ( R ± ( z ) − R ± ( w )) W k B ( L ( R d )) ≤ C | z − w | β δ k W k L r ( R d ) k W k L r ( R d ) for z, w ∈ I ± with | z − w | ≤ and W ∈ L r ( R d ) , W ∈ L r ( R d ) , W , W ∈ L r ( R d ) .Moreover, if W ∈ L r ( R d ) and W ∈ L r ( R d ) , then W R ± ( z ) W ∈ B ∞ ( L ( R d )) follows for z ∈ I ± and a map z ∈ I ± W R ± ( z ) W is continuous. ( ii ) Suppose < s < d/ ( d + 1) . Let < δ ≤ , r ∈ (1 , d + 1) − δ ] , r , r , ∈ (1 , d + 1)] and r ′ , r ′ , r ′ ∈ [2 d/s, ∞ ) satisfying d + 1 ≤ r + 1 r , r ′ + 1 r ′ ≤ sd . The all results in Lemma 4.1 part ( i ) hold if we replace L r ( R d ) , L r ( R d ) and L r ( R d ) by L r ( R d ) ∩ L r ′ ( R d ) , L r ( R d ) ∩ L r ′ ( R d ) and L r ( R d ) ∩ L r ′ ( R d ) respectively.Proof. Note that for W , W ∈ C ∞ c ( R d ), it follows that W (1 − χ ( D )) R ± ( z ) W is compact and smooth in z ∈ I ± by using dR ( z ) /dz = R ( z ) and the Rellich-Kondrachov theorem. The other parts of the proof are same as in the proof ofCorollary 3.4. (cid:3) Part ( iii ): Existence and completeness of the wave operators are similarly provedas in the proof of Theorem 1.9 ( iv ) in subsection 4.3 by using Lemma 4.1.Proof of Part ( iv ) is proved in subsection 4.2. (cid:3) Carleman estimate, Proof of Theorem 1.3 ( iv ) . First, we give the Car-leman estimate for T ( D ). We recall µ N,γ ( x ) = (1 + | x | ) N (1 + γ | x | ) − N andΛ l = ( I − ∆) l/ . For 1 < p < ∞ and l ∈ R , we introduce the standard Sobolevspaces W l,p = { u ∈ S ′ ( R d ) | Λ l u ∈ L p ( R d ) } , k u k W l,p = k Λ l u k L p ( R d ) . We set p d = 2( d + 1) / ( d + 3), p ∗ d = 2( d + 1) / ( d − l d = s/ − d/ ( d + 1), X s = ( W − l d ,p d + Λ s/ B , if 2 d/ ( d + 1) ≤ s < d, ( L p d ( R d ) ∩ L d/ ( d + s ) ( R d )) + Λ s/ B , if 0 < s < d/ ( d + 1) , and X ∗ s = ( W l d ,p ∗ d ∩ Λ − s/ B ∗ , if 2 d/ ( d + 1) ≤ s < d, ( L p ∗ d ( R d ) + L d/ ( d − s ) ( R d )) ∩ Λ − s/ B ∗ , if 0 < s < d/ ( d + 1) . By the Sobolev embedding theorem, we have X s ֒ → W − s/ , , W s/ , ֒ → X ∗ s . (35) Proposition 4.2.
Let N ≥ be a real number satisfying N < s/ , if T ( D ) = ( − ∆) s/ with s / ∈ N . (36) Then there exists C N,d > independent of < γ ≤ such that k µ N,γ ( x ) u k X ∗ s ≤ C N,d k µ N,γ ( x )( T ( D ) − λ ) u k X s for u ∈ B ∗ and | λ | ∈ I .Remark . The condition (36) is needed due to the singularity of the symbol T ( ξ ) = | ξ | s at ξ = 0. Proof.
First, we assume u ∈ S ( R d ). Let χ , χ , χ ∈ C ∞ ( R d ) be smooth functionssuch that χ , χ ∈ C ∞ c ( R d ) and χ + χ + χ = 1 , χ ( ξ ) = 1 near ξ = 0 , χ ( ξ ) = 1 on supp T − ( I ) . By Lemma 3.6, it suffices to prove k µ N,γ ( x ) ψ ( D ) u k X ∗ s ≤ C N,d k µ N,γ ( x ) ψ ( D )( T ( D ) − λ ) u k X s (37)for ψ ∈ { χ , χ , χ } . The case when ψ = χ directly follows from Proposition 3.5and Corollary 3.8. The case when ψ = χ follows from Corollary 3.8 and (35): k µ N,γ ( x ) χ ( D ) u k X ∗ s ≤ C k µ N,γ ( x ) χ ( D ) u k W s/ , = C k Λ s/ µ N,γ ( x ) u k L ( R d ) , Λ s/ µ N,γ =(Λ s/ µ N,γ Λ − s/ µ − N,γ ) × ( µ N,γ Λ s/ χ ( D )( T ( D ) − λ ) − µ − N,γ Λ s/ ) × Λ − s/ µ N,γ χ ( D )( T ( D ) − λ ) , where χ ∈ C ∞ ( R d ) satisfies χ = 1 on supp χ and supp χ ∩ T − ( I ) = ∅ . More-over, the L -boundedness of Λ s/ µ N,γ Λ − s/ µ − N,γ follows from Corollary 3.8 and L -boundedness of µ N,γ Λ s/ χ ( D )( T ( D ) − λ ) − µ − N,γ Λ s/ is proved by mimickingthe proof of Corollary 3.8.Finally, we deal with the case of ψ = χ . (37) with T ( D ) = ( − ∆) s/ or T ( D ) =( − ∆) s/ for s ∈ N is similarly proved as in the proof of (37) with ψ = χ . Thuswe may assume T ( D ) = ( − ∆) s/ with s / ∈ N . For its proof, we need some lemmas. Lemma 4.4.
Let s > and m ∈ C ∞ ( R d \ { } ) ∩ C c ( R d ) satisfying | ∂ αξ m ( ξ ) | ≤ C α | ξ | M α , M α = ( , if α = 0 ,s − N, if | α | ≥ . Then m ( D )( x ) = R R d e πix · ξ m ( ξ ) dξ satisfies | m ( D )( x ) | ≤ C (1 + | x | ) − s − d . IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 21 Proof.
Since m is compactly supported, we may assume | x | ≥
1. Let χ ∈ C ∞ c ( R )satisfying χ ( t ) = 1 on | t | ≤ χ ( t ) = 0 on | t | ≥
2. Set ¯ χ = 1 − χ . For δ >
0, byintegrating by parts, we have m ( D )( x ) = x | x | · Z R d e πix · ξ ( − D ξ m ( ξ )) dξ = x | x | · Z R d e πix · ξ ( χ ( | ξ | /δ ) + ¯ χ ( | ξ | /δ ))( − D ξ m ( ξ )) dξ =: m ( x ) + m ( x ) . We simply obtain | m ( x ) | ≤ C | x | − Z | ξ |≤ δ | ξ | s − dξ ≤ C | x | − δ d + s − . For M ≥ s + d + 2, by integrating by parts, we have | m ( x ) | ≤ C | x | − M − X | α |≤ M Z R d | D αξ ( ¯ χ ( | ξ | /δ ) D ξ m ( ξ )) | dξ ≤ C | x | − M − δ d + s − − M . We set δ = | x | − and conclude | m ( D )( x ) | ≤ C | x | − d − s . (cid:3) Lemma 4.5.
Let m be as in Lemma 4.4 and < p < ∞ . Moreover, let ≤ N γ >
0. Here we use Lemma 3.7 ( ii ) and Lemma 4.4. Wenote 2 N − s < | x | ) N − d − s ∈ L ( R d ).By the Young inequality, we obtain the desired result. (cid:3) Remark . Replacing the Young inequality by the O’neil theorem (the Younginequality in the Lorentz spaces), we can relax the condition (36) as 2 N ≤ s .We return to the proof of (37) with ψ = χ . We take χ ∈ C ∞ ( R d ) such that χ = 1 on supp χ . We learnΛ s/ µ N,γ =(Λ s/ µ N,γ Λ − s/ µ − N,γ ) × ( µ N,γ Λ s/ χ ( D )( T ( D ) − λ ) − Λ s/ µ − N,γ ) × ( µ N,γ Λ − s/ µ − N,γ Λ s/ ) × Λ − s/ µ N,γ χ ( D )( T ( D ) − λ ) . We set m ( D ) = µ N,γ Λ s/ χ ( D )( T ( D ) − λ ) − Λ s/ µ − N,γ , then m satisfies the assump-tion of Lemma 4.4. Thus the inclusions (35), Corollary 3.8 and Lemma 4.4 imply(37) with ψ = χ . This complete the proof of Proposition 4.2 with u ∈ S ( R n ).In order to remove the condition u ∈ S ( R n ), we may use the Friedrichs modifierand a cut-off function as in [13, Proof of Theorem 1.2]. We omit the detail. (cid:3) The next lemma implies that the potential is ”admissible”.
Lemma 4.7.
Suppose V ∈ L p ( R d ) with d/s ≤ p ≤ ( d + 1) / for d/ ( d + 1) ≤ s < d and V ∈ L ( d +1) / ( R d ) ∩ L d/s ( R d ) for < s < d/ ( d + 1) . Then we have V ∈ B ( X ∗ s , X s ) . Moreover, for each ε > and N ≥ there exists A N,ε , R
N,ε ≥ such that for γ ∈ (0 , , we have k µ N,γ
V u k X s ≤ ε k µ N,γ u k X ∗ s + A N,ε k u k L ( | x |≤ R N,ε ) . (38) Proof.
First, we prove k V u k X s ≤ k V k Y s k u k X ∗ s , (39)where Y s ∈ { L p ( R d ) } d/s ≤ p ≤ ( d +1) / for 2 d/ ( d + 1) ≤ s < d and Y s = L ( d +1) / ( R d ) ∩ L d/s ( R d ). By the Sobolev embedding theorem, we have W l d ,p ∗ d ֒ → L q ∗ ( R d ) , L q ( R d ) ֒ → W − l d ,p d for 2 d/ ( d + s ) ≤ q ≤ p d . For 2 d/ ( d + 1) ≤ s < d and d/s ≤ p ≤ ( d + 1) /
2, we set q p = 2 p/ ( p + 1). We note 2 d/ ( d + s ) ≤ q p ≤ p d . By the H¨older inequality, we have k V u k L qp ( R d ) ≤ k V k L p ( R d ) k u k L q ∗ p ( R d ) . We use X ∗ s ֒ → W l d ,p ∗ d and W − l d ,p d ֒ → X s and conclude V ∈ B ( X ∗ s , X s ) and (39)for 2 d/ ( d + 1) ≤ s < d . In order to prove (39) with 0 < s < d/ ( d + 1), it sufficesto prove k V u k L q ( R d ) ≤ k V k Y s k u k L r ( R d ) , where q ∈ { p d , d/ ( d + s ) } and r ∈ { p ∗ d , d/ ( d − s ) } . This inequality follows fromthe fact V ∈ Y s = L ( d +1) / ( R d ) ∩ L d/s ( R d ) and the complex interpolation.Take χ ∈ C ∞ c ( R d ) such that χ = 1 on | x | ≤ / χ = 0 on | x | ≥
1. For R ≥
1, we set V R = V χ ( x/R ). Then we use the inclusion B ֒ → X s and have k µ N,γ
V u k X s ≤ k V − V R k Y s k µ N,γ u k X ∗ s + k µ N,γ V R u k X s ≤ k V − V R k Y s k µ N,γ u k X ∗ s + k µ N,γ V R u k B . For each ε >
0, we take
R > k V − V R k Y s < ε and we obtain(38). (cid:3) Proof of Theorem 1.3 ( iv ) . We recall H = T ( D ) + V . Suppose that σ pp ( H ) \ { } isnot discrete in R \ { } . Then there exist an orthonormal system { u j } ∞ j =1 ⊂ L ( R d ), δ ≥ { λ j } ∞ j =1 ⊂ { λ ∈ R | δ ≤ | λ | ≤ δ − } such that Hu j = λ j u j . We note u j ∈ L ( R d ) ⊂ B ∗ . Let N ≥ u j and Lemma 4.7 with small ε >
0, we have k µ N,γ u j k X ∗ s ≤ C N,ε k u j k L ( R d ) with C N,ε independent of γ ∈ (0 , − s/ (1 + | x | ) / ε L ( R d ) ֒ → X ∗ s for ε > k (1 + | x | ) − / − ε Λ s/ µ N,γ u j k L ( R d ) ≤ C N,ε k u j k L ( R d ) . Taking γ →
0, we have k (1 + | x | ) − / − ε Λ s/ (1 + | x | ) N u j k L ( R d ) ≤ C N,ε k u j k L ( R d ) = C N,ε . (40)We take ε small enough and N ≥ N > / ε when T ( D ) = ( − ∆) s/ with 2 s / ∈ N . Then (40) implies that u j is bounded in (1 + | x | ) / ε − N Λ − s/ L ( R d ). Since the inclusion (1 + | x | ) / ε − N Λ − s/ L ( R d ) ֒ → IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 23 L ( R d ) is compact, there exists a subsequence { u j k } k such that u j k → u in L ( R d )for some u ∈ L ( R d ). On the other hand, since u j converges to 0 in the weaktopology of L ( R d ), then we have u = 0. This contradicts to k u j k L ( R d ) = 1.The same argument implies that the each eigenspace associated with eigenvalue λ ∈ R \ { } is finite dimensional. (cid:3) Discrete Schr¨odinger operator.
In this subsection, we consider the case of X = Z and consider the discrete Schr¨odinger operators. Proof of Theorem 1.9.
Part ( ii ) directly follows from the following lemma. Lemma 4.8.
Let d ≥ and a signature ± . Then maps z ∈ C ± \ R R ± ( z ) areH¨older continuous in B ( L p ( Z d ) , L p ∗ ( Z d )) for ≤ p < ∗ , where ∗ = 2 d/ ( d + 3) .Proof. We follow the argument in [25, Lemma 4.7]. We prove the lemma in thecase of + only. The case of − is similarly proved. For 1 ≤ p < ∗ , there exists0 < δ ≤ ≤ p < ∗ ,δ , where3 ∗ ,δ = 23 δ/d + (3 + d ) /d . We use the following dispersive estimate ([29]): k e it ∆ d k B ( L p ( Z d ) ,L p ∗ ( Z d )) ≤ C p h t i − d ( p − , ≤ p ≤ . (41)Moreover, | e itz − e itz ′ | ≤ − δ | t | δ | z − z ′ | δ (42)holds for t ≥ z, z ′ ∈ C + since | e itz − e itz ′ | ≤ | e itz − e itz ′ | ≤ | t || z − z ′ | .By (41) and (42), we have k R +0 ( z ) − R +0 ( z ′ ) k B ( L p ( R d ) ,L p ∗ ( R d )) = (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ ( e itz − e itz ′ ) e it ∆ d dt (cid:13)(cid:13)(cid:13)(cid:13) B ( L p ( R d ) ,L p ∗ ( R d )) ≤ C p − δ | z − z ′ | δ Z ∞ | t | δ h t i − d ( p − dt < ∞ for 1 ≤ p < ∗ ,δ . This completes the proof. (cid:3) Now we prove part ( i ). The above lemma implies thatlim ε → ,ε> k R ± ( λ ± iε ) − R ± ( λ ± i k B ( L p ( Z d ) ,L p ∗ ( Z d )) = 0 , λ ∈ R , ≤ p < ∗ , (43)where we recall R ± ( λ ± i
0) are Fourier multipliers of the distributions ( h ( ξ ) − ( λ ± i − . We also use the uniform bounds ([30, Proposition 3.3]):sup z ∈ C ± \ R k R ± ( z ) k B ( L ∗ ( Z d ) ,L ∗ ( Z d )) < ∞ , (44)where 3 ∗ = 2 d/ ( d − z ∈ C ± k R ± ( z ) k B ( L ∗ ( Z d ) ,L ∗ ( Z d )) < ∞ . This proves part ( i ). Note that part ( iii ) with V ∈ L p ( Z d ) for 1 ≤ p < d/ ii )and the H¨older inequality. Part ( iii ) with V ∈ L d/ ( Z d ) follows from the followinglemma. Lemma 4.9.
Let d ≥ and a signature ± . For W , W ∈ L d/ ( Z d ) , a map z ∈ C ± W R ± ( z ) W ∈ B ∞ ( L ( Z d )) is continuous.Proof. Take sequences of finitely supported potentials W ,n , W ,n such that W j,n → W j in L d/ ( Z d ) as n → ∞ for j = 1 ,
2. For z, z ′ ∈ C ± , the H¨older inequality implies k W ( R ± ( z ) − R ± ( z ′ )) W k B ( L ( Z d )) ≤ k W − W ,n k L d/ ( Z d ) k W k L d/ ( Z d ) sup z ∈ C ± k R ± ( z ) k B ( L ∗ ( Z d ) ,L ∗ ( Z d )) +2 k W − W ,n k L d/ ( Z d ) sup n ( k W ,n k L d/ ( Z d ) ) sup z ∈ C ± k R ± ( z ) k B ( L ∗ ( Z d ) ,L ∗ ( Z d )) + k W ,n ( R ± ( z ) − R ± ( z ′ )) W ,n k B ( L ( Z d )) =: I + I + I . Now we let ε >
0. We fix a large n such that I + I is smaller than 2 ε/
3. Since W ,n and W ,n are finitely supported, the previous lemma implies that W ,n ( R ± ( z ) − R ± ( z ′ )) W ,n is H¨older continuous in B ( L ( Z d )). Thus there exists δ > | z − z ′ | < δ implies I = k W ,n ( R ± ( z ) − R ± ( z ′ )) W ,n k B ( L ( Z d )) < ε/ . Thus we conclude that maps z ∈ C ± W R ± ( z ) W are continuous. (cid:3) It remains to prove ( iv ). We follow the argument as in [18] and [21]. Let V ∈ L d/ ( Z d ) be a real-valued function. Set W = (sgn V ) | V | / ∈ L d/ ( Z d ), W = | V | / ∈ L d/ ( Z d ), H = H + V and R ( z ) = ( H − z ) − for z ∈ C \ R . Wenote that for ± Im z > W R ± ( z ) W − W R ( z ) W = W R ( z ) W W R ± ( z ) W . (45)By part ( iii ), it follows that W R ± ( z ) W is continuous in z ∈ I ± and hence is acompact operator . In addition, I + W R ± ( z ) W is invertible in B ( L ( Z d )) for z ∈ C \ R due to the Birman-Schwinger principle. In fact, if I + W R ± ( z ) W isnot invertible at z ∈ C \ R , then the compactness of W R ± ( z ) W implies that I + W R ± ( z ) W has a non-trivial kernel. Then it follows that R ( z ) has a non-trivial kernel by the Birman-Schwinger principle. However, this contradicts to theself-adjointness of H + V . Moreover, if we set σ BS ( H ) = σ ± BS ( H ) = { λ ∈ R | Ker L ( Z d ) ( I + W R ± ( z ) W ) = 0 } , we see that σ BS ( H ) is a closed set with Lebesgue measure zero by PropositionB.3. Since W R ± ( z ) W ∈ B ∞ ( L ( Z d )) for z ∈ I ± , I + W R ± ( z ) W is a Fredholmoperator with index 0. Thus (45) gives W R ( z ) W = W R ± ( z ) W ( I + W R ± ( z ) W ) − , z ∈ I ± \ σ BS ( H ) . Let [ a, b ] ⊂ I \ σ BS ( H ) with a < b . Since ( I + W R ± ( z ) W ) − is continuous in z ∈ [ a, b ] ± , then sup z ∈ [ a,b ] ± k ( I + W R ± ( z ) W ) − k B ( L ( Z d )) < ∞ . IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 25 Combining this with the part ( i ) and H¨older’s inequality, we obtainsup z ∈ [ a,b ] ± k W R ( z ) W k B ( L ( Z d )) < ∞ . Since | W | = | W | , then sup z ∈ [ a,b ] ± k W i R ( z ) W i k B ( L ( Z d )) < ∞ . for i , i = 1 ,
2. By [24, Theorem XIII. 30, 31], the local wave operators s − lim t →±∞ e itH e − itH E H (( a, b )) exist and are complete, where E H ( J ) is the spectralprojection to the interval J ⊂ R associated with H . Since [0 , d ] \ Λ c ( H ) ∪ σ BS ( H ) is a countable union of such interval ( a, b ), the wave operators W ± = s − lim t →±∞ e itH e − itH exist and are complete. (cid:3) As an application of Theorem 1.2, we prove the further estimates of the uniformresolvent estimates for the discrete Schr¨odinger operators.
Proposition 4.10.
Suppose I ⊂ (0 , ∩ (4( d − , d ) if d = 2 and I ⊂ (0 , ∩ (4 d − , d ) if d ≥ . If supp χ ⊂ h − ( I ) , then sup z ∈ I ± k χ ( D ) R ± ( z ) k B ( L p ( Z d ) ,L q ( Z d )) < ∞ . holds for (1 /p, , q ) ∈ S ( d − / .Proof. Let λ ∈ I . As is proved in [15, Lemma 4.3], all principal curvatures of M λ = { h = λ } are non-vanishing. By Example 1, we obtain the desired result. (cid:3) Appendix A. Some estimates for γ z, ± In this section, we give proofs of the estimates for γ z, ± which is needed for theproof of Theorem 1.2.If necessary we take supp χ small, we may assume X = R . We recall the situationof the proof of Theorem 1.2. Set˜ χ ( ξ ′ , ξ d , λ ) = χ ( ξ ′ , ξ d + h λ ( ξ ′ )) e ( ξ ′ , ξ d + h λ ( ξ ′ )) , b ( ξ ′ , ξ d , λ ) = e ( ξ ′ , ξ d + h λ ( ξ ′ ))) − . Note that b is real-valued and min ( ξ ′ ,ξ d ) ∈ supp χ ( · , · ,λ )˜ χ,λ ∈ I b ( ξ ′ , ξ d , λ ) >
0. Recall that γ z, ± ( ξ ′ , x d ) = Z R e πix d ξ d ˜ χ ( ξ ′ , ξ d , λ ) ξ d − i (Im z ) b ( ξ ′ , ξ d , λ ) dξ d , Re z = λ, ± Im z ≥ . Here if ± Im z = 0, we interpret γ z, ± as γ z, ± ( ξ ′ , x d ) = Z R e πix d ξ d χ ( ξ ′ , ξ d + h λ ( ξ ′ )) e ( ξ ′ , ξ d + h λ ( ξ ′ )) ξ d ∓ i dξ d = Z R e πix d ξ d ˜ χ ( ξ ′ , ξ d , λ )) ξ d ∓ i dξ d , where ( ξ d ∓ i − denote the distributions lim ε> ,ε → ( ξ d ∓ iε ) − . In order to estimate γ z, ± , we need some lemmas. Lemma A.1.
Let ψ, ψ ∈ C ∞ c ( R ) and µ , µ ∈ R \ { } . Then | Z R ψ ( µ y d )p . v . e πiy d ξ d y d dy d | ≤ π k ˆ ψ k L ( R ) , | Z R p . v . e πiy d ξ d y d dy d | = π | Z R ψ ( µ y d ) ψ ( µ y d )p . v . e πiy d ξ d y d dy d | ≤ π k ˆ ψ k L ( R ) k ˆ ψ k L ( R ) , Proof.
We leran | Z R p . v . y d ψ ( y d ) e πiy d ξ d dy d | = π | Z R sgn( ξ d − η d ) ˆ ψ ( − η d ) dη d |≤ π k ˆ ψ k L ( R ) . By scaling, we obtain the first inequality. The second equality follows from F (p . v . y d )( ξ d ) = − iπ sgn( ξ d ). The third inequality follows from the first inequality and the Younginequality: k ˆ ψ ˆ ψ k L ( R ) = k ˆ ψ ∗ ˆ ψ k L ( R ) ≤k ˆ ψ k L ( R ) k ˆ ψ k L ( R ) . (cid:3) Lemma A.2.
Let µ ∈ R \ { } and ϕ, a, a ∈ C ∞ c ( R ) such that a, a are real-valuedand a, a > on supp ϕ . ( i ) There exists
C > independent of x d ∈ R , ϕ, a and µ = 0 such that | Z R e πix d ξ d ϕ ( µξ d ) ξ d − ia ( µξ d ) dξ d | ≤ C ( sup ξ d ∈ R | ϕ ( ξ d ) a ( ξ d ) | + k ˆ ϕ k L ( R ) + sup ξ d ∈ R | ϕ ( ξ d ) a ( ξ d ) | ) . (46)( ii ) Let l ≥ be an integer. Then there exists C ′ > independent of x d ∈ R , ϕ, a , l and µ = 0 such that | Z R e πix d ξ d ϕ ( µξ d )( ξ d − ia ( µξ d )) l dξ d | ≤ C ′ ( sup ξ d ∈ R | | ϕ ( ξ d ) || a ( ξ d ) | l + k ϕ k L ∞ ( R ) ) . (47)( iii ) Let l , l ≥ be an integer. Then there exists C ′′ > independent of x d ∈ R , ϕ, a , l and µ = 0 such that | Z R e πix d ξ d ϕ ( µξ d )( ξ d − ia ( µξ d )) l ( ξ d − ia ( µξ d )) l dξ d | ≤ C ′′ ( sup ξ d ∈ R | | ϕ ( ξ d ) || a ( ξ d ) | l + k ϕ k L ∞ ( R ) ) . (48) Proof. ( i ) Take ψ ∈ C ∞ c ( R , [0 , ψ = 1 on | t | ≤ ψ = 0 on | t | ≥ a is real-valued, then | Z R e πix d ξ d ϕ ( µξ d ) ψ ( ξ d ) ξ d − ia ( µξ d ) dξ d | ≤ Z R | ϕ ( µξ d ) ψ ( ξ d ) || a ( µξ d ) | dξ d ≤ sup ξ d ∈ R | ϕ ( ξ d ) a ( ξ d ) |k ψ k L ( R ) . IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 27 We note that Z R e πix d ξ d ϕ ( µξ d )(1 − ψ ( ξ d )) ξ d − ia ( µξ d ) dξ d = Z R e πix d ξ d ϕ ( µξ d )(1 − ψ ( ξ d )) ξ d dξ d + i Z R e πix d ξ d ϕ ( µξ d ) a ( µξ d )(1 − ψ ( ξ d )) ξ d ( ξ d − ia ( µξ d )) dξ d = : I + I . By Lemma A.1, we have | I | = | Z R p . v . e πix d ξ d ϕ ( µξ d ) ξ d dξ d − Z R p . v . e πix d ξ d ϕ ( µξ d ) ψ ( ξ d ) ξ d dξ d |≤ π k ˆ ϕ k L ( R ) (1 + k ˆ ψ k L ( R ) ) . Moreover, since a is real-valued, we have | I | ≤ sup ξ d ∈ R | ϕ ( ξ d ) a ( ξ d ) | Z R − ψ ( ξ d ) ξ d dξ d . Thus we set C = max( k ψ k L ( R ) , π (1 + k ˆ ψ k L ( R ) ) , Z R − ψ ( ξ d ) ξ d dξ d ) , and obtain (46).( ii ) follows from ( iii ).( iii ) Let ψ be as above. Then | Z R e πix d ξ d ϕ ( µξ d ) ψ ( ξ d )( ξ d − ia ( µξ d )) l ( ξ d − ia ( µξ d )) l dξ d | ≤ sup ξ d ∈ R | | ϕ ( ξ d ) || a ( ξ d ) | l | a ( ξ d ) | l k ψ k L ( R ) . Moreover, since a, a is real-valued and l + l ≥
2, then | Z R e πix d ξ d ϕ ( µξ d )(1 − ψ ( ξ d ))( ξ d − ia ( µξ d )) l ( ξ d − ia ( µξ d )) l dξ d | ≤k ϕ k L ∞ ( R ) Z R − ψ ( ξ d ) | ξ d | l + l dξ d ≤k ϕ k L ∞ ( R ) Z R − ψ ( ξ d ) | ξ d | dξ d . Thus we set C ′′ = max( k ψ k L ( R ) , R R − ψ ( ξ d ) | ξ d | dξ d ) and obtain (48). (cid:3) The main result of this section is the following proposition.
Proposition A.3.
Fix a signature ± . ( i ) For α ∈ N d − , there exists C α > such that | ∂ αξ ′ γ z, ± ( ξ ′ , x d ) | ≤ C α (49) for z ∈ I ± , x d ∈ R and ξ ′ ∈ R d − . ( ii ) For α ∈ N d − , there exists C ′ α > such that | ∂ αξ ′ ( γ z, ± ( ξ ′ , x d ) − γ w, ± ( ξ ′ , x d )) | ≤ C ′ α (1 + | x d | ) | z − w | (50) for z, w ∈ I ± with | z − w | ≤ , x d ∈ R and ξ ′ ∈ R d − .Remark A.4 . Let 0 ≤ δ ≤
1. Combining (49) with (50), we have | ∂ αξ ′ ( γ z, ± ( ξ ′ , x d ) − γ w, ± ( ξ ′ , x d )) | ≤ C − sα ( C ′ α ) s (1 + | x d | ) δ | z − w | δ . (51) Proof. ( i ) We follow the argument of the proof of [4, (3.10)]. We may assume0 ≤ ± Im z ≤
1. First, we consider the case of ± Im z = 0. In this case, the claimfollows from the fact that k Z R e πix d ξ d ξ d ∓ i dξ d k L ∞ ( R xd ) < ∞ and that ˜ χ is smooth with respect to ( ξ, ξ d , λ ) ∈ R d × I and has a compact supportwith respect to ( ξ ′ , ξ d )-variable which is bounded in λ ∈ I .We take ψ ∈ C ∞ c ( R , [0 , ψ ( ξ d ) = 1 on | ξ d | ≤
1. We learn γ z, ± ( ξ ′ , x d ) = Z R e πi (Im z ) x d ξ d ˜ χ ( ξ ′ , (Im z ) ξ d , λ ) ξ d − ib ( ξ ′ , (Im z ) ξ d , λ ) dξ d . We note that ∂ αξ ′ γ ( ξ ′ , x d ) is a linear combination of the form Z R e πi (Im z ) x d ξ d ( ∂ α ξ ′ ˜ χ )( ξ ′ , (Im z ) ξ d , λ ) Q lj =1 ( ∂ α j ξ ′ b )( ξ ′ , (Im z ) ξ d , λ )( ξ d − ib ( ξ ′ , (Im z ) ξ d , λ )) l dξ d , where l ≥ α j ∈ N d − for j = 0 , ..., l . Applying Lemma A.2( i ) if l = 1 and ( ii ) if l > ϕ ( ξ d ) = ( ∂ α ξ ′ ˜ χ )( ξ ′ , ξ d , λ ) Q lj =1 ( ∂ α j ξ ′ b )( ξ ′ , ξ d , λ ), a ( ξ d ) = b ( ξ ′ , ξ d , λ ) and µ = Im z , we obtain (49) with | α | ≥ ii ) We set λ = Re z and σ = Re w . We take 0 < ε such thatmin ( ξ ′ ,ξ d ) ∈ supp χ ( · , · ,λ ) , | z − w |≤ δ | b ( ξ ′ , ξ d , σ ) | > . Then we may assume | z − w | < ε . In fact, in order to prove ( ii ), we use ( i ) if | z − w | ≥ ε . Note that γ z, ± ( ξ ′ , x d ) − γ w, ± ( ξ ′ , x d ) = J ( x d ) + J ( x d ) + J ( x d ) , where we set J ( x d ) = Z R e πix d ξ d ( ˜ χ ( ξ ′ , ξ d , λ ) ξ d − i (Im z ) b ( ξ ′ , ξ d , λ ) − ˜ χ ( ξ ′ , ξ d , λ ) ξ d − i (Im w ) b ( ξ ′ , ξ d , λ ) ) dξ d J ( x d ) = Z R e πix d ξ d ˜ χ ( ξ ′ , ξ d , λ ) − ˜ χ ( ξ ′ , ξ d , σ ) ξ d − i (Im w ) b ( ξ ′ , ξ d , λ ) dξ d = Z R e πi (Im w ) x d ξ d ˜ χ ( ξ ′ , (Im w ) ξ d , λ ) − ˜ χ ( ξ ′ , (Im w ) ξ d , σ ) ξ d − ib ( ξ ′ , (Im w ) ξ d , λ ) dξ d J ( x d ) = Z R e πix d ξ d ˜ χ ( ξ ′ , ξ d , σ )( 1 ξ d − i (Im w ) b ( ξ ′ , ξ d , λ ) − ξ d − i (Im w ) b ( ξ ′ , ξ d , σ ) ) dξ d = Z R e πi (Im w ) x d ξ d i ˜ χ ( ξ ′ , (Im w ) ξ d , σ )( b ( ξ ′ , (Im w ) ξ d , λ ) − b ( ξ ′ , (Im w ) ξ d , σ ))( ξ d − ib ( ξ ′ , (Im w ) ξ d , λ ))( ξ d − ib ( ξ ′ , (Im w ) ξ d , σ )) dξ d . First, we estimate J . Similarly to the proof of ( i ), ∂ αξ ′ J ( ξ ′ ) is a finite sum of theform Z R e πi (Im w ) x d ξ d (( ∂ α ξ ′ ˜ χ )( ξ ′ , (Im w ) ξ d , λ ) − ( ∂ α ξ ′ ˜ χ )( ξ ′ , (Im w ) ξ d , σ ))( ξ d − ib ( ξ ′ , (Im w ) ξ d , λ )) l × l Y j =1 ( ∂ α j ξ ′ b )( ξ ′ , (Im w ) ξ d , λ ) dξ d , IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 29 where l ≥ α j ∈ N d − for j = 0 , ..., l . We apply Lemma A.2 ( i )if l = 1 and ( ii ) l ≥ | ∂ αξ ′ J ( ξ ′ ) | ≤ C ′ α | z − w | (52)with C ′ α > x d ∈ R , ξ d ∈ R d − and z, w ∈ I ± with | z − w | ≤ δ .Next, we estimate J . ∂ αξ ′ J is a linear combination of the form Z R e πi (Im w ) x d ξ d ( ∂ α ξ ′ ˜ χ )( ξ ′ , (Im w ) ξ d , σ ) ∂ α ξ ′ ( b ( ξ ′ , (Im w ) ξ d , λ ) − b ( ξ ′ , (Im w ) ξ d , σ ))( ξ d − ib ( ξ ′ , (Im w ) ξ d , λ )) l ( ξ d − ib ( ξ ′ , (Im w ) ξ d , σ )) l × l + l +1 Y j =2 ( ∂ α j ξ ′ b )( ξ ′ , (Im w ) ξ d , λ ) dξ d , where l , l ≥ α j ∈ N d − for j = 0 , ..., l + l + 1. We applyLemma A.2 ( iii ) and obtain | ∂ αξ ′ J ( ξ ′ ) | ≤ C ′ α | z − w | (53)with C ′ α > x d ∈ R , ξ d ∈ R d − and z, w ∈ I ± with | z − w | ≤ ε .Finally, we estimate J . Note that | ∂ αξ ′ J ( x d ) | ≤ C by ( i ). Thus it suffices toprove that | ∂ αξ ′ J ′ ( x d ) | ≤ C ′ α | Im z − Im w | . We learn J ′ ( x d )2 πi = Z R e πix d ξ d ( ξ d ˜ χ ( ξ ′ , ξ d , λ ) ξ d − i (Im z ) b ( ξ ′ , ξ d , λ ) − ξ d ˜ χ ( ξ ′ , ξ d , λ ) ξ d − i (Im w ) b ( ξ ′ , ξ d , λ ) ) dξ d = Z R e πix d ξ d i (Im z − Im w ) ξ d ˜ χ ( ξ ′ , ξ d , λ ) b ( ξ ′ , ξ d , λ )( ξ d − i (Im z ) b ( ξ ′ , ξ d , λ ))( ξ d − i (Im w ) b ( ξ ′ , ξ d , λ )) dξ d = Z R e πi (Im w ) x d ξ d i (Im z − Im w ) ξ d ˜ χ ( ξ ′ , (Im w ) ξ d , λ ) b ( ξ ′ , (Im w ) ξ d , λ )( ξ d − i Im z Im w b ( ξ ′ , (Im w ) ξ d , λ ))( ξ d − ib ( ξ ′ , (Im w ) ξ d , λ )) dξ d . Thus ∂ αξ ′ J ′ ( x ) / ( − π | Im z − Im w | ) is a linear combination of the form( Im z Im w ) l Z R e πi (Im w ) x d ξ d ξ d ∂ α x ′ ˜ χ ( ξ ′ , (Im w ) ξ d , λ ) ∂ α ξ ′ b ( ξ ′ , (Im w ) ξ d , λ )( ξ d − i Im z Im w b ( ξ ′ , (Im w ) ξ d , λ )) l ( ξ d − ib ( ξ ′ , (Im w ) ξ d , λ )) l × l + l +1 Y j =2 ∂ α j ξ ′ b ( ξ ′ , ξ d , λ ) dξ d , where l , l ≥ α j ∈ N d − for j = 1 , ..., l + l + 1. Applying LemmaA.2 ( i ) and ( ii ) with ϕ ( ξ d ) = (Im z ) l ξ d ∂ α x ′ ˜ χ ( ξ ′ , (Im w ) ξ d , λ ) ∂ α ξ ′ b ( ξ ′ , (Im w ) ξ d , λ )( ξ d − i (Im z ) b ( ξ ′ , ξ d , λ )) l ,a ( ξ d ) = b ( ξ ′ , ξ d , λ ), l = l and µ = Im w , we have | ∂ αξ ′ J ′ ( x d ) | ≤ C ′ α | Im z − Im w | .This completes the proof. (cid:3) Appendix B. Complex analysis
We define log + t = log t if 1 ≤ t , log + t = 0 if 0 < t ≤ − t = log t − log + t . Lemma B.1.
Let f : { z ∈ C | | z | ≤ } → C be a continuous function which isholomorphic on {| z | < } and has no zero on {| z | < } . Then f ( e iθ ) = 0 for almosteverywhere θ ∈ [ − π, π ) . Proof.
We follow the argument of [26, Theorem 17.17]. By the mean value proper-ties of the harmonic function, we havelog | f (0) | = 12 π Z π − π log | f ( re iθ ) | dθ (54) = 12 π Z π − π log + | f ( re iθ ) | dθ − π Z π − π log − | f ( re iθ ) | dθ for 0 < r <
1. On the other hand, by using x ≤ e x for x ∈ R and Jensen’sinequality, we have12 π Z π − π log + | f ( re iθ ) | dθ ≤ exp( 12 π Z π − π log + | f ( re iθ ) | dθ ) ≤ π Z π − π | f ( re iθ ) | dθ. By Fatou’s lemma and (54), we obtain log | f ( e iθ ) | ∈ L ([ − π, π )). In particular,log | f ( e iθ ) | < ∞ for almost everywhere θ ∈ [ − π, π ). Thus f ( e iθ ) = 0 for almosteverywhere θ ∈ [ − π, π ). (cid:3) Corollary B.2.
Let J = ( a, b ) be an open interval and r = ( b − a ) / . Let f : { z ∈ C | | z − ( a + b ) / | ≤ r, ± Im z ≥ } → C be a continuous function which isholomorphic and has no zero on {| z − ( a + b ) / | < r, Im z > } . Then f ( λ ) = 0 foralmost everywhere λ ∈ J .Proof. For simplicity, we assume a = − b = 1. Define κ : D = {| z | < , Im z > } → { Im z > } and κ : { Im z > } → {| z | < } by κ ( z ) = (1 + z ) / (1 − z ) and κ ( z ) = ( z − i ) / ( z + i ). Then κ = κ ◦ κ is biholomorphic from {| z | < , Im z > } to {| z | < } and homeomorphic from {| z | ≤ , Im z ≥ } to {| z | ≤ } . Moreover, since κ − ( w ) = q i w − w − q i w − w + 1where we take a branch such that Im √ z >
0, then κ − | | z | =1 : {| z | = 1 } → ¯ D \ D is H¨older continuous. Thus κ − | | z | =1 maps sets of Lebesgue measure zero to sets ofLebesgue measure zero. By Lemma B.1, we obtain the desired result. (cid:3) Next proposition is a variant of [17, Lemma 4.20]. See also [21, Proposition 4.6].
Proposition B.3.
Let Z be a Banach space and fix a sgnature. For J ⊂ R be anopen set, we denote J ± = { z ∈ C | Re z ∈ J, ± Im z ≥ } . Let K : J ± → B ∞ ( Z ) becontinuous and holomorphic on {± Im z > } . If I + K ( z ) has a inverse in B ( Z ) for each z ∈ {± Im z > } , then Γ = { λ ∈ R | I + K ( λ ) is not invertible } is aclosed set with Lebesgue measure zero.Proof. Since the set of all invertible operators in B ( Z ) is open and since K iscontinuous, then Γ is closed. Thus it suffices to prove that the Lebesgue measureof Γ is zero. Note that I + K ( λ ) is not invertible if and only if − K ( λ ) for λ ∈ Γ . Fix λ ∈ Γ . Since K ( λ ) is compact, there exists a circle C λ enclosing − C λ is contained in the resolvent set of K ( λ ). Since K is IMITING ABSORPTION PRINCIPLE ON L p -SPACES AND SCATTERING THEORY 31 continuous, there exists r λ > C λ is contained in the resolvent set of K ( z ) for z ∈ B ± r λ ( λ ) where B ± r λ ( λ ) = { z ∈ C | ± Im z ≥ , | z − λ | < r λ } . We define P z = 12 πi Z C λ ( w − K ( z )) − dw, then z ∈ B ± r λ ( λ ) P z ∈ B ( Z ) is analytic in B ± r λ ( λ ) \ R and continuous in B ± r λ ( λ ).Note that n = dim Ran P z < ∞ is independent of z ∈ B ± r λ ( λ ). Set Z z = Ran P z and fix a linear isomorphism Π λ : C n → Z λ . We choose r λ smaller such that I + P λ ( P z − P λ ) has an inverse in B ( Z λ ). Then Θ z = P z | Z λ : Z λ → Z z is a linearisomorphism with its inverse( I + P λ ( P z − P λ )) − P λ : Z z → Z λ . Now we set X ( z ) = Π − λ Θ − z ( I + K ( z ))Θ z Π λ for z ∈ B ± r λ ( λ ). Then X is continuous on B ± r λ ( λ ) and analytic in B ± r λ ( λ ). More-over, det X ( z ) is also continuous on B ± r λ ( λ ) and analytic in B ± r λ ( λ ). We note thatdet X ( z ) = 0 if and only if − K ( z ). By Corollary B.2 andthe compactness argument, we conclude that the Lebesgue measure of Γ is zero. (cid:3) References [1] J.G. Bak, A. Seeger, Extensions of the Stein-Tomas theorem. Math. Res. 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