The L p -boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities
TTHE L p -BOUNDEDNESS OF WAVE OPERATORS FORTWO DIMENSIONAL SCHR ¨ODINGER OPERATORSWITH THRESHOLD SINGULARITIES KENJI YAJIMA
Abstract.
We generalize the recent result of Erdo˘gan, Goldbergand Green on the L p -boundedness of wave operators for two di-mensional Schr¨odinger operators and prove that they are boundedin L p ( R ) for all 1 < p < ∞ if and only if the Schr¨odinger op-erator possesses no p -wave threshold resonances, viz. Schr¨odingerequation ( − ∆ + V ( x )) u ( x ) = 0 possesses no solutions which sat-isfy u ( x ) = ( a x + a x ) | x | − + o ( | x | − ) as | x | → ∞ for an( a , a ) ∈ R \ { (0 , } and, otherwise, they are bounded in L p ( R )for 1 < p ≤ < p < ∞ . We present also anew proof for the known part of the result. Introduction and main results
Let H = − ∆, D ( H ) = W , ( R ) be the free Schr¨odinger operatoron R d , d ≥ V ( x ) a real measurable function on R d . Supposethat, for some γ > / (cid:104) x (cid:105) γ | V ( x ) | / ( H + 1) − , (cid:104) x (cid:105) = (1 + | x | ) , is acompact operator on L ( R d ). Then, Schr¨odinger operator H = − ∆+ V defined via the quadratic form is selfadjoint ([27]); the spectrum σ ( H )consists of the absolutely continuous part σ ac ( H ) = [0 , ∞ ) and thepoint spectrum σ p ( H ) which can accumulate only to zero ([1]); L ( R d )is the orthogonal sum of the the absolutely continuous subspace L ac ( R d )for H and the space of eigenfunctions of H . The scattering theorycompares the large time behavior of scattering solutions e − itH ϕ , ϕ ∈ L ac ( R d ) of the time dependent Schr¨odinger equation i∂ t u ( t ) = Hu ( t ) , u (0) = ϕ ∈ L ac ( R d ) (1.1)with that of free solutions e − itH ϕ and the wave operators are definedby the following strong limits W ± = lim t →±∞ e itH e − itH . (1.2)It is well known ([1, 23]) that W ± exist, they are complete in the senseImage W ± = L ac ( H ) and they satisfy the intertwining property: e − itH P ac ( H ) = W ± e − itH W ∗± , t ∈ R , (1.3) P ac ( H ) being the orthogonal projection onto L ac ( H ). It follows that allscattering solutions e − itH P ac ( H ) ϕ , become asymptotically free in the a r X i v : . [ m a t h . A P ] A ug K. YAJIMA remote past and far future:lim t →±∞ (cid:107) e − itH P ac ( H ) ϕ − e − itH ϕ ± (cid:107) = 0 , ϕ ± = W ∗± ϕ. Then a simple argument shows that the intertwining property (1.3) canbe extended to Borel functions of H and H : f ( H ) P ac ( H ) = W ± f ( H ) W ∗± , (1.4)which reduces various properties of f ( H ) to those of the Fourier mul-tiplier f ( H ) if W ± satisfy certain corresponding properties.If W ± are bounded in L p ( R d ) for a range of p ∈ I ⊂ [1 , ∞ ], then(1.4) instantly produces a set of estimates that for any { p, q } ∈ I × I ∗ , I ∗ = { q = p/p − p ∈ I } , (cid:107) f ( H ) P ac ( H ) (cid:107) B ( L q ,L p ) ≤ C (cid:107) f ( H ) (cid:107) B ( L q ,L p ) , (cid:107) f ( H ) (cid:107) B ( L p ,L q ) ≤ C − (cid:107) f ( H ) P ac ( H ) (cid:107) B ( L p ,L q ) for a C = C p,q independent of f , where B ( X, Y ) is the Banach space ofbounded operators from X to Y and B ( X ) = B ( X, X ). Such estimatesare often very useful and have many applications. Thus, the problemof whether or not W ± are bounded in L p ( R d ) has attracted interestof many authors and various results have been obtained under variousassumptions: If H has no singularities at zero (see Definition 1.4 be-low), W ± are bounded in L p ( R d ) for all 1 ≤ p ≤ ∞ if d ≥ < p < ∞ if d = 1 , H has singularitiesat zero, a rather complete result is known if d (cid:54) = 2 , p for which W ± are bounded in L p ( R d ) has been determined, whichdepends on the dimensions d , the type of singularities at zero and therate of decay as | x | → ∞ of the corresponding zero energy resonancesor eigenfunctions ([40, 12, 15, 41, 42]); however, only partial resultsare obtained when d = 2 or d = 4. For d = 2, Erdo˘gan, Goldbergand Green [11] have recently shown that W ± are bounded bounded in L p ( R ) for all 1 < p < ∞ if the singularities at zero are associatedeither with s -wave resonances only or with zero energy eigenfunctionsonly. If d = 4 and the singularity is associated with zero eigenvalueonly, then W ± are bounded in L p ( R ) for all 1 ≤ p < ≤ p < ∞ depending on the nature of zero energy eigenfunctions ([13, 19]).The purpose of this paper is to prove the following theorem on the L p -boundedness of wave operators for Schr¨odinger operators in R whichgeneralizes the result of [11]: Theorem 1.1.
Suppose (cid:104) x (cid:105) V ∈ L ( R ) and (cid:104) x (cid:105) γ | V ( x ) | ∈ L ( R ) fora constant γ > . Suppose further that H has no p -wave resonances atthe threshold, viz. Schr¨odinger equation ( − ∆+ V ( x )) u ( x ) = 0 possessesno solutions u ( x ) which satisfy u ( x ) = ( a x + a x ) | x | − + o ( | x | − ) , ( | x | → ∞ ) p -BOUNDEDNESS OF WAVE OPERATORS IN R for an ( a , a ) ∈ R \{ (0 , } . Then, W ± are bounded in L p ( R ) for any < p < ∞ and otherwise, W ± are bounded in L p ( R ) for < p ≤ and unbounded for < p < ∞ . As remarked above the result is known if H is regular ([39]), if ithas only singularities of the first kind at zero or if the singularitiesare associated with zero energy eigenfunctions only ([11]) and we shallgive a new proof for these cases under slightly weaker assumptions that (cid:104) x (cid:105) γ | V ( x ) | ∈ L ( R ) for γ > γ > (cid:104) x (cid:105) V ∈ L ( R ). We also remark that the problem for end points p = 1 and p = ∞ remains open. A similar result has recently beenobtained for Schr¨odinger operators H with point interactions on R :([4], [5] and [43]). Actually the main ideas of the proof is borrowedfrom [4] and [43].The rest of the paper is devoted to the proof of Theorem 1.1. Weprove it only for W + . The result for W − then follows via the complexconjugation C u ( x ) = u ( x ): W − = C W + C − . We briefly explain here thebasic strategy for the proof introducing some notation and displayingthe plan of the paper. Various constants are denoted by C which maydiffer at each appearance and the notation C α is used for emphasizingthat the constant depends on α . We shall use the function space D ∗ = { u ∈ S ( R ) | ˆ u ∈ C ∞ ( R \ { } ) } . (1.5) D ∗ is a dense subspace of L p ( R ) for any 1 < p < ∞ . (cid:107) u (cid:107) p = (cid:107) u (cid:107) L p ( R ) for 1 ≤ p ≤ ∞ , (cid:107) u (cid:107) = (cid:107) u (cid:107) . H is the Hilbert space of operators ofHilbert-Schmidt type on L ( R ).The proof relies on the stationary representation of W + (see e.g.[23]): W + u ( x ) = u ( x ) − (cid:90) ∞ ( G ( − λ ) vM ( λ ) − v Π( λ ) u )( x ) λdλ, u ∈ D ∗ . (1.6)As we shall see the integral (1.6) converges absolutely for almost all x ∈ R for u ∈ D ∗ . Here G ( λ ) = ( H − λ ) − for λ ∈ R \ { } isthe boundary value of the free resolvent defined for λ ∈ C + = { λ ∈ C : (cid:61) λ > } and it is the convolution operator with G λ ( x ) = 1(2 π ) (cid:90) R e ixξ dξξ − λ = i H (1)0 ( λ | x | ) ; (1.7) H (1)0 ( z ) is the Hankel function of the first kind; we often write H ( λ )for i H (1)0 ( λ ); for V ( x ) of the theorem we define U ( x ) = sign V ( x ) , v ( x ) = | V ( x ) | / , w ( x ) = U ( x ) v ( x ) ;then, vG ( λ ) v is an H -valued C function of λ ∈ C + \ { } , C + = { λ ∈ C : (cid:61) λ ≥ } , and we define M ( λ ) = U + vG ( λ ) v, λ > K. YAJIMA M ( λ ) − ∈ B ( L ( R )) exists for any λ > λ ) u ( x ) = 1 iπ ( G ( λ ) − G ( − λ )) u ( x ) (1.8)is the spectral projection of H ; we haveΠ( λ ) u ( x ) = 12 π (cid:90) S e iλωx ˆ u ( λω ) dω = 12 π (cid:90) S ( F τ − x u )( λω ) dω , (1.9)where F u ( ξ ) = ˆ u ( ξ ) is the Fourier transform: F u ( ξ ) = 12 π (cid:90) R e − ixξ u ( x ) dx ; τ y is the translation by y ∈ R : τ y u ( x ) = u ( x − y ).For continuous functions f ( λ ) on (0 , ∞ ) we have f ( λ )Π( λ ) u = Π( λ ) f ( | D | ) u, λ > , (1.10)where f ( | D | ) u ( x ) = F ∗ f ( | ξ | ) F u ( x ) is the Fourier multiplier.From the series expansion of the Hankel function (2.1), we have (seeLemma 5.3) M ( λ ) = U + g ( λ ) v ⊗ v + vN v + λ g ( λ ) G + λ G + O ( gλ ) , (1.11)where with Euler’s constant γ and with principal branch of log g ( λ ) = − π log (cid:16) λ (cid:17) + i − γ π ; (1.12) N is the Newton potential; G is rank 4 operator: N u ( x ) = − π (cid:90) R log | x − y | u ( y ) dy ; G u ( x ) = 14 (cid:90) R | x − y | u ( y ) dy . (1.13) G u ( x ) = 18 π (cid:90) R | x − y | log (cid:18) e | x − y | (cid:19) u ( y ) dy . (1.14)We indiscriminately denote by v ⊗ w or | v (cid:105)(cid:104) w | the one dimensionaloperator f (cid:55)→ v ( x ) (cid:90) R w ( y ) f ( y ) dy without complex conjugate .( u, v ) = (cid:90) R u ( x ) v ( x ) dx, (cid:104) u, v (cid:105) = (cid:90) R u ( x ) v ( x ) dx. These notation will be used whenever the coupling make sense.The proof of Theorem 1.1 will be split into several parts. We takeand fix χ ∈ C ∞ ( R ) such that χ ( λ ) = 1 for | λ | ≤ / χ ( λ ) = 0 for | λ | ≥
1; (1.15) p -BOUNDEDNESS OF WAVE OPERATORS IN R and define for a > χ ≤ a ( λ ) = χ ( λ/a ) , χ ≥ a ( λ ) = 1 − χ ≤ a ( λ ) . (1.16)Then W + = W + χ ≥ a ( | D | ) + W + χ ≤ a ( | D | ) is the decomposition of W + into the high and the low energy parts. They have stationary represen-tations W + χ ≥ a ( | D | ) u = χ ≥ a ( | D | ) u + Ω high u and the likewise, whereΩ high u = (cid:90) ∞ G ( − λ ) vM ( λ ) − v Π( λ ) uχ ≥ a ( λ ) λdλ , (1.17)Ω low u = (cid:90) ∞ G ( − λ ) vM ( λ ) − v Π( λ ) uχ ≤ a ( λ ) λdλ . (1.18)After introducing some more notation in section 2, we record insection 3 estimates on some integral operators which will be repeatedlyused in the paper. A very import role will be played by the singularintegral operator K defined by Ku ( x ) = (cid:90) + ∞ G − λ ( x ) λ (cid:18)(cid:90) S ( F u )( λω ) dω (cid:19) dλ (1.19)which is closely related the representation formula (1.6) (see (3.5)). K is bounded in L p ( R ) for all 1 < p < ∞ (cf. Lemma 3.2). Definition 1.2. (1) We say an operator is a good operator if it isbounded in L p ( R ) for any 1 < p < ∞ .(2) We say an operator T or an operator valued function T ( λ ) is a good producer if the operator defined by (1.6) with T or T ( λ )in place of M ( λ ) − is a good operator.(3) We say a function f ( λ ) is a good multiplier if f ( | D | ) is a goodoperator. M ( R ) denotes the space of good multipliers.(4) For a function f defined near 0 (or near infinity) we say T λ = O ( f ) as λ → λ → ∞ ) if T λ is an H valued C function near 0 (resp. near ∞ ) and satisfies (cid:107) ∂ jλ T ( λ ) (cid:107) H ≤ Cf ( λ ) λ − j there.The following proposition will be proved in Lemmas 3.3, 3.6 and 3.7by the help of the operator K : Proposition 1.3. (A)
Let ε > and a > . The followings are goodproducers: (1) Good multiplier. (2)
Finite rank operators (cid:80) nj,k =1 a jk v j ⊗ w j where u j , v j ∈ L ( R ) . (3) (cid:80) nj,k =1 a jk ( λ ) v j ⊗ w j where v j , w j ∈ L ( R ) and a jk ∈ M ( R ) . (4) T ∈ B (see Definition 3.4 below). (5) T λ = O ( λ ε ) as λ → for an ε > . (6) T λ = O ( λ − ε ) as λ → ∞ for an ε > . (B) Let Z F be defined by (1.6) with multiplication by F ∈ L ( R ) inplace of vM ( λ ) − v . Then, (cid:107) Z F (cid:107) B ( L p ) ≤ C p (cid:107) F (cid:107) for any < p < ∞ . K. YAJIMA
We give a new proof of the known result that W + χ ≥ a ( | D | ) is agood operator ([39]) in section 4. In (1.17) we expand M ( λ ) − = U (1 + vG ( λ ) w ) − , w = U v into the sum (cid:88) j =0 ( − j U ( vG ( λ ) w ) j − U ( vG ( λ ) w ) (1 + vG ( λ ) w ) − , (1.20)which produces Ω high = (cid:80) j =0 Ω h,j . By virtue of Proposition 1.3 (B),Ω h, = Z V is a good operator. For proving that Ω h,j , 1 ≤ j ≤ V G ( λ ) V = (cid:90) R M y ( x ) H ( | y | λ ) τ y dy, (1.21)where M y ( x ) = V ( x ) V ( x − y ) and H ( λ ) = ( i/ H (1)0 ( λ ). Then (1.17)with V G ( λ ) V in place of vM ( λ ) − v is equal by virtue of (1.10) toΩ h, u = (cid:90) R Z M y ( H ( | y || D | ) τ y χ ≥ a ( | D | ) u ) dy. We show (cid:107)H ( | y || D | ) χ ≥ a ( | D | ) (cid:107) B ( L p ) ≤ C p (1 + | log | y || ) in Lemma 4.7.Then, (B) of Proposition 1.3 implies (cid:107) Ω h, (cid:107) B ( L p ) ≤ C p (cid:90) R | V ( x ) V ( y ) | (1 + | log | x − y || ) dxdy < ∞ . For j = 2 , , U ( vG ( λ ) w ) j have expressions similar to (1.21) and Ω h,j become good operators (cf. Proposition 4.8); under the smoothnessassumption (cid:104) x (cid:105) V ∈ L , we have the decay of the resolvent for large λ > (cid:107) vG ( l )0 ( λ ) w (cid:107) H ≤ Cλ − / , j = 0 , , O ( λ − ) as λ → ∞ and Ω h, is also a goodoperator by virtue of Proposition 1.3 (6).We prove that the low energy part W + χ ≤ a ( | D | ) or equivalently Ω low satisfies Theorem 1.1 in section 5, which will be divided into six sub-sections. We define P = ( v/ (cid:107) v (cid:107) ) ⊗ ( v/ (cid:107) v (cid:107) ) , Q = 1 − P, T = U + vN v. We first recall in subsections 5.1 Feshbach formula and Jensen-Nenciu’slemma ([17]) which will be used for analysing M ( λ ) − for λ → M ( λ ) = M ( λ ) − g ( λ ) v ⊗ v − T , M ( λ ) = M ( λ ) − vλ ( gG + G ) v We start studying Ω low in subsection 5.3. We recall the following defi-nition of the type of sigularities of H at zero ([17], see also [30, 10, 11]). Definition 1.4. H is said to be regular at zero if QT Q is invertiblein QL ( R ). Otherwise, H is singular at zero. If H is singular at zero,let S be the projection in QL ( R ) onto Ker QL ( R ) QT Q.(1) We say H has singularities of the first kind at zero, if T = S QT P T QS | S L : S L ( R ) → S L ( R ) is non-singular. p -BOUNDEDNESS OF WAVE OPERATORS IN R (2) If T is singular, let S be the projection in S L ( R ) ontoKer T . We say H has singularities of the second kind atzero, if T = S ( vG v ) S | S L ( R ) : S L ( R ) → S L ( R ) is non-singular.(3) We say H has singularities of the third kind at zero if T issingular. Let S be the projection in S L ( R ) onto Ker T .In subsection 5.3 we prove Ω low is a good operator if H is regular atzero by showing that, with rank 2 operator L , M ( λ ) − = ( g ( λ )+ c ) − L + B + O ( gλ ) as λ → low in the case when H is singular at zero insubsection 5.4. We first collect some preliminaries. We shall takeadvantage here of the fact that rank S is finite and all operators as-sociated with the singularities of M ( λ ) − act in S L ( R ) and that ζ ∈ S L ( R ) satisfies the vanishing moment condition (cid:90) R v ( x ) ζ ( x ) = 0 . (1.22)In subsection 5.5 we prove that W low is a good operator if H has asingularities of the first kind at zero (cf. [11]). In this case rank S = 1and, if S = ζ ⊗ ζ , ζ gives rise to the s -wave resonance via (5.33) and H has no other resonances nor zero energy eigenvalue ([17]); except theleading singular term − c (log λ + c ) ζ ⊗ ζ M ( λ ) − has good producersof Proposition 1.3 (cf. [11]) and, modulo a good operatorΩ low u ≡ − c (cid:90) ∞ (log λ + c ) G ( − λ ) vζ (cid:105)(cid:104) ζv, Π( λ ) u (cid:105) χ ≤ a ( λ ) λdλ . (1.23)By virtue of (1.22), we may replace Π( λ ) u ( x ) in (1.23) by Π( λ ) u ( x ) − Π( λ ) u (0), which we write in the form12 π (cid:90) (cid:18)(cid:90) S iλxωe iλxωθ ( F u )( λω ) dω (cid:19) dθ (1.24)= (cid:88) j =1 iλx j π (cid:90) (cid:18)(cid:90) S F ( τ − θx R j u )( λω ) dω (cid:19) dθ, (1.25)where R j = D j / | D | , j = 1 , λ and µ ( λ ) def = − ic χ ≤ a ( λ )(log λ + c ) λ is a good multi-plier. Then, we apply the following operation to (1.25) consecutively:Multiply − ic χ ≤ a ( λ )(log λ + c ), apply (1.10) for µ ( λ ), take the in-ner product of the result with ζv , multiply by G − λ ( x − z )( ζv )( z ) λ =( ζv )( z )( τ z G − λ )( x ) λ , integrate by dλ and express the result via the op-erator K and, then integrate by dz . Result is that Ω low of (1.23) is equal K. YAJIMA to the sum over j = 1 , y, z ∈ R of good op-erators τ z Kτ − θy µ ( | D | ) R j with the integrable weight ( ζv )( z ) y j ( ζv )( y ):Ω low u = (cid:88) j =1 (cid:90) (cid:90) R × R ( ζv )( z ) y j ( ζv )( y )( τ z Kτ − θy µ ( | D | ) R j u )( x ) dydzdθ and, hence, it itself is a good operator.In subsection 5.6 we study Ω low in the case that H has singularitiesof the second kind. Then, rank S ≤
2. We assume here rank S = 2and let { ζ , ζ } be the basis of S L ( R ). Resonance ϕ produced by ζ ∈ S L ( R ) via (5.33) is a p -wave resonance and the one by ζ ∈ S L ( R ) \ S L ( R ) is a s -wave resonance ([17], see also Remark 5.18).By using also results of previous subsections, we show in Lemma 5.21that Ω low u is equal modulo a good operator to − (cid:90) ∞ g − λ − G ( − λ ) S ( S R S ) − S ( v Π( λ ) u ) χ ≤ a ( λ ) λdλ (1.26)which contains a very singular factor g − λ − in front, where R = v ( G + g − G ) v and ( S R S ) − exists since T is non-singular in S L ( R ). In (1.26) S ( v Π( λ ) u ) = (cid:80) j =1 | ζ j (cid:105)(cid:104) vζ j , Π( λ ) u (cid:105) and we mayagain replace Π( λ ) u ( x ) by (1.24). Here we further expand e iλωxθ of(1.24) and express (cid:104) ζ j v, Π( λ ) u (cid:105) as a sum of good part g j ( λ ) and bad part b j ( λ ): g j ( λ ) = − λ π (cid:90) R (cid:90) S v ( z ) ζ j ( z ) (cid:18)(cid:90) (1 − θ )( izω ) e iλzωθ dθ (cid:19) ˆ u ( λω ) dωdz,b j ( λ ) = iλ π (cid:90) R (cid:90) S v ( z ) ζ j ( z )( zω )ˆ u ( λω ) dωdz. (1.27)The good part g j ( λ ) has the ample factor λ which cancels λ − in away similar as in the previous subsection and we can show (1.26) with (cid:80) j =1 g j ( λ ) ζ j ( x ) in place of S ( v Π( λ ) u ) defines a good operator (cf.Lemma 5.22). The bad part b j ( λ ) has only the factor λ and the oper-ator Ω low , b produced by (cid:80) j =1 b j ( λ ) ζ j ( x ) is bounded in L p ( R ) for only1 < p ≤ < p < ∞ . We avoid outliningthe proof of the boundedness part for not making the introduction toolong. For proving the unboundedness, we express its high energy part χ ≥ a ( | D | )Ω low , b u in the form (cid:80) j =1 a j ( x ) (cid:96) j ( u ) by using linearly inde-pendent a , a ∈ L p ( R ), 1 < p < ∞ and linear functionals (cid:96) ( u ) , (cid:96) ( u ).If Ω low , b ∈ B ( L p ) then, the Hahn-Banch theorem implies that (cid:96) and (cid:96) must be bounded on L p ( R ) and we show by the absurdity that thiscannot happen if 2 < p < ∞ (cf. Lemma 5.25). This is sufficient toconclude that Ω low is unbounded in L p ( R ) for 2 < p < ∞ .In the final subsection 5.7 we assume that H has singularities of thethird kind at zero. Then, T = S G S is necessarily non-singular, ϕ produced by ζ ∈ S L ( R ) via (5.33) is a zero energy eigenfunction of H p -BOUNDEDNESS OF WAVE OPERATORS IN R and by ζ ∈ S L ( R ) \ S L ( R ) p -wave resonance ([17]). The Feshbachformula implies ( S R S ) − = gS T − S + L , L being an operator in S L ( R ); ζ ∈ S L ( R ) satisfies the extra moment condition (cid:90) R x ζ k ( x ) v ( x ) dx = (cid:90) R x ζ k ( x ) v ( x ) dx = 0 . (1.28)Ω low is still given modulo a good operator by (1.26) and, if ( S R S ) − is replaced by gS T − S then, it produces a good operator. This isbecause for z ∈ S L ( R ) the identity (cid:104) ζv, Π( λ ) u (cid:105) = g j ( λ ) is satisfiedwithout the bad part b by virtue of (1.28). If S = S , in which case p -wave resonances are absent, then L = 0 and Ω low becomes a goodoperator. If S (cid:54) = S , however, L (cid:54) = 0 and L has the structure similarto that of ( S R S ) − of the the previous subsection, and, hence Ω low is bounded in L p only for 1 < p ≤ < p < ∞ .In the rest of the paper we shall give the details of the proof outlinedabove. 2. Notation
We introduce here some more notation and immediate consequencesof the definitions. For σ ∈ R , L σ ( R ) = (cid:104) x (cid:105) − σ L ( R ) are weighted L -spaces. For σ, τ > / G ( z ) is a B ( L σ , L − τ )-valued analytic functionof z ∈ C + = { z ∈ C : (cid:61) z > } and it can be continuously extended to C + \ { } . We often write λ for z when we want to emphasize z canalso be real.For the Hankel function H (1)0 ( λ ) we have i H (1)0 ( λ ) = g ( λ ) ∞ (cid:88) k =0 ( − k ( k !) (cid:16) λ (cid:17) k + 12 π (cid:18) λ (1!) − (cid:16) (cid:17) ( λ ) (2!) + (cid:16) (cid:17) ( λ ) (3!) − · · · (cid:19) , (2.1)where g ( λ ) is defined by (1.12). It has also an integral representation([35]): i H (1)0 ( λ ) = e iλ π (cid:90) ∞ e − t t − (cid:18) t − iλ (cid:19) − dt, (2.2)where the square root z is positive for positive z . The followingsare immediate consequences of (2.1) and (2.2). Recall that H ( λ ) =( i/ H (1)0 ( λ ) so that G λ ( x ) = H ( λ | x | ). Lemma 2.1. (1)
The function H ( λ ) satisfies H ( λ ) = e iλ ω ( λ ) , | ω ( j ) ( λ ) | ≤ C j λ − − j , j = 0 , . . . , λ ≥ , (2.3) H ( λ ) = g ( λ ) + λ (cid:16) − g ( λ )4 + 18 π (cid:17) + O ( gλ ) , λ → . (2.4) (2) For any δ > , G λ ( x ) satisfies G λ ( x ) = (cid:26) e iλ | x | ω ( λ | x | ) , λ | x | ≥ g ( λ ) + N ( x ) + O (( λ | x | ) − δ ) , λ | x | < ≤ | · | C (1 + | g ( λ ) | + | log | x || ) . (2.6) Lemma 2.2.
Let ≤ α < β < ∞ . Then: (cid:90) βα |G λ ( x ) | dλ ≤ C α,β ( | x | − + | x | − ) , x ∈ R . (2.7) Proof.
By virtue of Lemma 2.1, the left side is bounded by C (cid:90) α<λ< | x | − ( | log λ | x || + 1) dλ + C (cid:90) | x | − <λ<β ( λ | x | ) − / dλ. (2.8)For | x | < β − the second integral vanishes and for | x | > α − the firstdoes. Thus, when | x | < β − via change of variable (cid:90) βα |G λ ( x ) | dλ ≤ C | x | (cid:90) α | x | ( − log s + 1) ds ≤ C | x | (cid:90) ( − log s + 1) ds = C | x | and, when | x | > α − (cid:90) βα |G λ ( x ) | dλ ≤ C | x | ( β − | x | − ) ≤ Cβ | x | . Since the left hand side is a continuous function of | x | , (2.7) follows. (cid:3) Integral estimates
We collect here some L p estimates on operators involving G ( λ ).We often identify integral operators with their integral kernels. Thefollowing lemma is obvious or well-known (cf. [33]). Lemma 3.1.
For u ∈ D ∗ , Π( λ ) u ( x ) ∈ C ∞ ( R λ × R x ) and for some < α < β < ∞ supp λ Π( λ ) u ( x ) ⊂ ( α, β ) . (3.1) | Π( λ ) u ( x ) | ≤ min( (cid:107) u (cid:107) , C u (cid:104) x (cid:105) − / ) . (3.2)The following lemma is proved in [4]. K is defined by (1.19). Lemma 3.2. (1) Ku ( x ) is a rotationary invariant and Ku ( x ) = lim ε ↓ π ) (cid:90) R u ( y ) dyx − y − iε . (3.3)(2) For any < p < ∞ there exists a constant C p > such that (cid:107) Ku (cid:107) p ≤ C p (cid:107) u (cid:107) p , u ∈ D ∗ . (3.4)(3) We have the identity (cid:90) ∞ G − λ ( x − y )Π( λ ) u ( z ) λdλ = ( τ y Kτ − z u )( x ) . (3.5) p -BOUNDEDNESS OF WAVE OPERATORS IN R Proof.
We give another and simpler proof of (3.3) for readers’ conve-nience. Since dη = λdλdω in the polar coordinates η = λω , we havefor u, v ∈ D ∗ that (cid:104) Ku, v (cid:105) = 12 π (cid:90) + ∞ λ (cid:18)(cid:90) S ˆ u ( λω ) dω (cid:19) (cid:104)G − λ , v (cid:105) dλ = lim ε ↓ π ) (cid:90) + ∞ λ (cid:18)(cid:90) S ˆ u ( λω ) dω (cid:19) (cid:18)(cid:90) R ˆ v ( ξ ) ξ − λ + iε dξ (cid:19) dλ . = lim ε ↓ π ) (cid:90) (cid:90) R ˆ u ( η )ˆ v ( ξ ) ξ − η + iε dξdη = lim ε ↓ − i (cid:90) ∞ e − tε (cid:18) π (cid:90) R e − itη / ˆ u ( η ) dη (cid:19) (cid:18) π (cid:90) R e itξ / ˆ v ( ξ ) dξ (cid:19) dt = − i (cid:90) ∞ ( e − itH / u )(0)( e itH / v )(0) dt Here, in the last step we used dominated convergence theorem and thewell known fact that for u ∈ D ∗ ( e − itH / u )( x ) = e ∓ i/ (2 | t | π ) (cid:90) R e i ( x − y ) / t u ( y ) dy, t ∈ R is rapidly decreasing smooth function of t ∈ R for every x ∈ R . Thus,inserting once more but another vanishing factor, we see that the righthand side is equal tolim ε ↓ − i π ) (cid:90) ∞ e − ε/ t (cid:18)(cid:90) (cid:90) R e i ( x − y ) / t u ( x ) v ( y ) dxdy (cid:19) dtt = lim ε ↓ − i (2 π ) (cid:90) ∞ (cid:18)(cid:90) (cid:90) R e is ( x − y + iε ) u ( x ) v ( y ) dxdy (cid:19) ds = lim ε ↓ π ) (cid:90) (cid:90) R u ( x ) v ( y ) x − y + iε dxdy. This proves (3.3). Change of variable shows that Ku ( √ rω ) = lim ε ↓ π (cid:90) ∞ r − ρ − iε (cid:18) π (cid:90) S u ( √ ρµ ) dµ (cid:19) dρ. This is essentially the Hilbert transform and (cid:107) Ku (cid:107) pp = 2 π (cid:90) ∞ | Ku ( √ rω ) | p dr ≤ C (cid:90) ∞ (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) S u ( √ ρµ ) dµ (cid:12)(cid:12)(cid:12)(cid:12) p dρ ≤ C π (cid:90) ∞ (cid:90) S | u ( √ ρµ ) | p dµdρ = C π (cid:107) u (cid:107) pp , where H¨older’s inequality is used in the third step.(3) The second of (1.9) and (1.19) imply the LHS of (3.5) is equal to12 π (cid:90) ∞ λ G − λ ( x − y ) (cid:18)(cid:90) S ( F τ − z u )( λω ) dω (cid:19) dλ = ( τ y Kτ − z u )( x ) . This completes the proof. (cid:3)
The following lemma proves statements (A2) and (B) of Proposition1.3. Then, by virtue of (1.10) which proves (A1), it also proves (A3).
Lemma 3.3.
Let < p < ∞ and u ∈ D ∗ . (1) Suppose that v, w ∈ L ( R ) . Let W v,w u = (cid:90) ∞ ( G ( − λ ) v )( x ) (cid:104) w | Π( λ ) u (cid:105) λdλ. (3.6) Then, for a constant C p independent of v, w and u , (cid:107) W v,w u (cid:107) p ≤ C p (cid:107) v (cid:107) (cid:107) w (cid:107) (cid:107) u (cid:107) p . (3.7)(2) For a, b ∈ R , define Ω a,b u ( x ) = (cid:90) ∞ G − λ ( x − a )(Π( λ ) u )( b ) λdλ. (3.8) Then, for a constant C p independent of a, b and u , (cid:107) Ω a,b u (cid:107) p ≤ C p (cid:107) u (cid:107) p . (3.9)(3) Suppose F ∈ L ( R ) . Let Z F u ( x ) = (cid:90) ∞ ( G ( − λ ) F Π( λ ) u )( x ) λdλ. (3.10) Then, for a constant C p independent of F and u , (cid:107) Z V u (cid:107) p ≤ C p (cid:107) F (cid:107) (cid:107) u (cid:107) p . (3.11) Proof.
We apply (3.4) after changing the order of integrations.(1) Let D (cid:98) R x . Since | Π( λ ) u ( z ) | ≤ C u (cid:104) z (cid:105) − / and Π( λ ) u ( z ) = 0 for λ (cid:54)∈ ( α, β ) by virtue of (3.1) and (3.2), (2.7) implies (cid:90) R × R | v ( y ) || w ( z ) | (cid:18)(cid:90) D × [0 , ∞ ) λ |G − λ ( x − y ) || Π( λ ) u ( z ) | dxdλ (cid:19) dydz ≤ C (cid:90) R × R (cid:104) z (cid:105) − | v ( y ) || w ( z ) | (cid:18)(cid:90) D × ( α,β ) λ |G − λ ( x − y ) | dxdλ (cid:19) dydz ≤ C (cid:90) R × R (cid:104) z (cid:105) − | v ( y ) || w ( z ) | dydz ≤ C (cid:107) v (cid:107) (cid:107) w (cid:107) < ∞ . (3.12)Fubini’s theorem and (3.5) then imply that W u,v u ( x ) is equal to (cid:90) R × R w ( z ) v ( y ) (cid:26)(cid:90) ∞ G − λ ( x − y )Π( λ ) u ( z ) λdλ (cid:27) dydz = (cid:90) R × R w ( z ) v ( y )( τ y Kτ − z u )( x ) dydz (3.13) p -BOUNDEDNESS OF WAVE OPERATORS IN R for almost every x ∈ R . Apply Minkowski’s inequality and thenLemma 3.2 to (3.13). Since translations are isometries of L p , we have (cid:107) W v,w u (cid:107) p ≤ (cid:90) (cid:90) R × R | w ( z ) v ( y ) |(cid:107) τ y Kτ − z u (cid:107) p dydz ≤ C (cid:107) w (cid:107) (cid:107) v (cid:107) (cid:107) u (cid:107) p . (2) (3.5) implies Ω a,b u ( x ) = ( τ a Kτ − b u )( x ). and (3.9) follows from (3.4).(3) For almost all x ∈ R the order of the integration by dy and dλ may be changed as in the proof of (1) and Z F u ( x ) = (cid:90) ∞ (cid:18)(cid:90) R G − λ ( x − y ) F ( y )Π( λ ) u ( y ) dy (cid:19) λdλ = (cid:90) R F ( y ) (cid:18)(cid:90) ∞ G − λ ( x − y )Π( λ ) u ( y ) λdλ (cid:19) dy = (cid:90) R F ( y )Ω y,y u ( x ) dy . (3.14)Apply Minkowski’s inequality and (3.9) to (3.14). (3.11) follows. (cid:3) Following is a substitute of absolutely integrable operators in [30].
Definition 3.4. B is the space of T ∈ B ( L ( R )) which is (uniquely)expressed in the form T u ( x ) = m ( x ) u ( x ) + (cid:90) R K T ( x, y ) u ( y ) dy (3.15)with m ∈ L ∞ ( R ) and K T ∈ L ( R × R ). We define (cid:107) T (cid:107) B = (cid:107) m (cid:107) ∞ + (cid:107) K (cid:107) H . (3.16) Lemma 3.5. B is a Banach algebra with the unit.Proof. B is obviously algebra with unit T = 1, (cid:107) T (cid:107) B is a norm of B and (cid:107) T S (cid:107) B ≤ (cid:107) T (cid:107) B (cid:107) S (cid:107) B . Since L ∞ ( R ) and H are both Banach spaces, B is a Banach space. (cid:3) The following two lemmas respectively prove statements (A4) and(A5,A6) of Proposition 1.3.
Lemma 3.6.
Let T ∈ B and v, w ∈ L ( R ) . Define for u ∈ D ∗ W u,v ( T ) u ( x ) = (cid:90) ∞ ( G ( − λ ) vT w Π( λ ) u )( x ) λdλ. (3.17) Then, for a constant C p independent of v, w, T and u , (cid:107) W u,v ( T ) u (cid:107) p ≤ C p (cid:107) T (cid:107) B (cid:107) v (cid:107) (cid:107) w (cid:107) (cid:107) u (cid:107) p , < p < ∞ . (3.18) Proof.
Let T = m + K T . Then, (3.11) implies (3.18) is satisfied if T = m . We prove it for T ∈ H . Let D (cid:98) R Since Π( λ ) u ( z ) = 0 for λ (cid:54)∈ [ α, β ] by virtue of (3.1), (3.2) and (2.7) implysup z,y ∈ R (cid:90) D (cid:90) ∞ |G − λ ( x − y )Π( λ ) u ( z ) | λdλdx ≤ C sup y ∈ R (cid:90) D ( | x − y | − + | x − y | − ) dx ≤ C It follows that (cid:90) D × R (cid:90) ∞ | v ( y ) G − λ ( x − y ) T ( y, z ) w ( z )Π( λ ) u ( z ) λ | dλdxdydz (3.19) ≤ C (cid:90) R | v ( y ) || T ( y, z ) || w ( z ) | dydz ≤ (cid:107) T (cid:107) H (cid:107) v (cid:107) (cid:107) w (cid:107) (3.20)Thus, for almost all x ∈ R , we may apply Fubini’s theorem and in-tegrate by dλ first in the following integral. Then, (3.5) implies that W u,v ( T ) u ( x ) is equal to (cid:90) ∞ (cid:18)(cid:90) R v ( y ) G − λ ( x − y ) T ( y, z ) w ( z )Π( λ ) u ( z ) dydz (cid:19) λdλ = (cid:90) (cid:90) R v ( y ) T ( y, z ) w ( z )( τ y Kτ − z u )( x ) dydz and, by virtue of Minkowski’s inequality and (3.4), (cid:107) W u,v ( T ) u (cid:107) p isbounded by C (cid:90) R | v ( y ) T ( y, z ) w ( z ) |(cid:107) u (cid:107) p dydz ≤ C (cid:107) u (cid:107) p (cid:107) v (cid:107) (cid:107) w (cid:107) (cid:107) T (cid:107) H . The lemma follows. (cid:3)
Lemma 3.7.
Let T λ ∈ C ((0 , ∞ ); H ) , a > and v, w ∈ L ( R ) . (1) Let σ > . Suppose that T λ ∈ O ( λ σ ) as λ → . Then, W v,w ( T λ ) u = (cid:90) ∞ G ( − λ ) vT λ w Π( λ ) uλχ ≤ a ( λ ) dλ , u ∈ D ∗ (3.21) satisfies for a constant C p > independent of u, v, w (cid:107)W w,v ( T λ ) u (cid:107) p ≤ C p (cid:107) u (cid:107) p (cid:107) v (cid:107) (cid:107) w (cid:107) , < p < ∞ . (3.22)(2) Let σ < . Suppose that T λ ∈ O ( λ − σ ) as λ → ∞ . Then ˜ W v,w ( T λ ) u = (cid:90) ∞ G ( − λ ) vT λ w Π( λ ) uλχ ≥ a ( λ ) dλ , u ∈ D ∗ (3.23) satisfies for a constant C p > independent of u, v, w (cid:107) ˜ W w,v ( T λ ) u (cid:107) p ≤ C p (cid:107) u (cid:107) p (cid:107) v (cid:107) (cid:107) w (cid:107) , < p < ∞ . (3.24) Proof. (1) By integration by parts we obtain T λ u = (cid:90) ∞ ( λ − µ ) + T (cid:48)(cid:48) µ udµ, (3.25) p -BOUNDEDNESS OF WAVE OPERATORS IN R which we substitute in (3.21). Explicitly W w,v ( T λ ) u ( x ) is equal to (cid:90) ∞ λχ ≤ a ( λ ) (cid:20)(cid:90) R G − λ ( x − y ) v ( y ) (cid:26)(cid:90) ∞ ( λ − µ ) + × (cid:18)(cid:90) R T (cid:48)(cid:48) µ ( y, z ) w ( z )Π( λ ) u ( z ) dz (cid:19) χ ≤ a ( µ ) dµ (cid:27) dy (cid:21) dλ (3.26)Here we are allowed to insert χ ≤ a ( µ ) because χ ≤ a ( µ ) (cid:54) = 1 implies( λ − µ ) + = 0 for λ ∈ supp χ ≤ a . We show that (3.26) is absolutelyintegrable for almost all x ∈ R as in the proof of Lemma 3.6 by usingLemma 3.1, (2.7) and obvious estimate χ ≤ a ( λ ) λ ( λ − µ ) + ≤ a χ ≤ a ( λ ).Indeed, for D (cid:98) R , we have (cid:90) D (cid:18)(cid:90) (cid:90) R × [0 , ∞ ) ] | integrand of (3.26) | dydzdλdµ (cid:19) dx ≤ Ca (cid:90) (cid:90) R × [0 , ∞ )] χ ≤ a ( µ ) | v ( y ) T (cid:48)(cid:48) µ ( y, z ) w ( z ) | dydzdµ ≤ Ca (cid:90) [0 , ∞ )] χ ≤ a ( µ ) (cid:107) T (cid:48)(cid:48) ( µ ) (cid:107) H (cid:107) v (cid:107) (cid:107) w (cid:107) dµ < ∞ . Thus, by virtue of Fubini’s theorem, the order of integration is arbitraryin (3.26) for almost every x ∈ R . Then, the obvious identity( λ − µ ) + = λ − µ + ( µ − λ ) + , (3.27)the spectral theorem (1.10) and the definition (3.17) imply that W w,v ( T λ ) u ( x ) = W (1) u ( x ) + W (2) u ( x ) + W (3) u ( x ) a.e. x ∈ R where W (1) u = (cid:90) ∞ (cid:18)(cid:90) ∞ G ( − λ ) vT (cid:48)(cid:48) µ w Π( λ ) uλ χ ≤ a ( λ ) dλ (cid:19) χ ≤ a ( µ ) dµ = (cid:90) ∞ χ ≤ a ( µ ) W u,v ( T (cid:48)(cid:48) µ )( χ ≤ a ( D ) D u ) dµ. W (2) u = − (cid:90) ∞ (cid:18)(cid:90) ∞ G ( − λ ) vT (cid:48)(cid:48) µ w Π( λ ) uλχ ≤ a ( λ ) dλ (cid:19) µχ ≤ a ( µ ) dµ. = − (cid:90) ∞ µχ ≤ a ( µ ) W u,v ( T (cid:48)(cid:48) µ )( χ ≤ a ( D ) | D | u ) dµ. W (3) u = (cid:90) ∞ (cid:18)(cid:90) ∞ ( µ − λ ) + G ( − λ ) vT (cid:48)(cid:48) µ w Π( λ ) uλχ ≤ a ( λ ) dλ (cid:19) χ ≤ a ( µ ) dµ = (cid:90) ∞ µχ ≤ a ( µ ) W u,v ( T (cid:48)(cid:48) µ )(1 − | D | /µ ) + χ ≤ a ( | D | ) udµ. Since λχ ≤ a ( λ ) and χ ≤ a ( λ ) are good multipliers, Minkowski’s inequalityand (3.18) evidently imply for j = 1 , (cid:107)W ( j ) u (cid:107) p ≤ C p (cid:107) v (cid:107) (cid:107) w (cid:107) (cid:107) u (cid:107) p (cid:90) a (cid:107) T (cid:48)(cid:48) µ (cid:107) H µ j − dµ and, since (cid:107) T (cid:48)(cid:48) µ (cid:107) H ≤ Cµ − ε as µ →
0, we obtain (cid:107)W ( j ) u (cid:107) p ≤ C p (cid:107) v (cid:107) (cid:107) w (cid:107) (cid:107) u (cid:107) p , j = 1 , . (3.28)It is well known that Fourier transform of (1 − | ξ | /µ ) + of ξ ∈ R is integrable with µ -indendent L ( R )-norm (see p. 426 of [33]). Itfollows that (cid:107)W (3) u (cid:107) p ≤ C (cid:107) u (cid:107) (cid:107) v (cid:107) (cid:107) u (cid:107) p (cid:90) ∞ µχ ≤ a ( µ ) (cid:107) T (cid:48)(cid:48) µ (cid:107) H dµ (3.29)Combining (3.28) and (3.29), we obtain the first statement.(2) By integration by parts we have T ( λ ) = (cid:90) ∞ λ (cid:18)(cid:90) ∞ ρ T (cid:48)(cid:48) ( µ ) dµ (cid:19) dρ = (cid:90) ∞ ( µ − λ ) + T (cid:48)(cid:48) ( µ ) dµ, (3.30)which we substitute in (3.23). We see that ˜ W w,v ( T λ ) u ( x ) is expressedagain by (3.26) but with χ ≥ a ( λ ) and χ ≥ a/ ( µ ) in place of χ ≤ a ( λ ) and χ ≥ a ( µ ) respectively. We then repeat the argument for the proof of (1).Actually the argument here is simpler as we need not decomposition(3.27) now. We omit the repetitious details. (cid:3) High energy estimate
In this section, we give a simpler proof of the following theorem whichhas been known for some years now ([39]). The new proof replaces theargument by integration by parts in [39] by the use of Fourier multi-pliers and the singular integral operator K defined by (1.19). Recall χ ≥ a ( λ ) = 0 for λ ≤ a . In what follows in this section we write χ ≥ for χ ≥ a if no confusions are feared. Theorem 4.1.
Suppose that (cid:104) x (cid:105) V ∈ L ( R ) . Then, for any a > , W + χ ≥ a ( | D | ) is a good operator. It suffices to show the theorem for Ω high defined by (1.17). In thissection we always assume that (cid:104) x (cid:105) V ∈ L ( R ), which may not bementioned explicitly.Expanding M ( λ ) − = U (1 + vG ( λ ) w ) − as in (1.20), we expressΩ high u as the sum: (cid:88) j =0 Ω h ,j u = (cid:88) j =0 (cid:90) ∞ G ( λ ) w ( − vG ( λ ) w ) j v Π( λ ) uλχ ≥ ( λ ) dλ + (cid:90) ∞ G ( λ ) w ( vG ( λ ) w ) (1 + vG ( λ ) w ) − v Π( λ ) uλχ ≥ ( λ ) dλ. (4.1)4.1. Estimate of Ω h , . The following is evident from (3.11).
Proposition 4.2.
Suppose V ∈ L ( R ) . Then, there exists a constant C p independent of u ∈ D ∗ and V such that (cid:107) Ω h , u (cid:107) p ≤ C p (cid:107) V (cid:107) (cid:107) u (cid:107) p , < p < ∞ . (4.2) p -BOUNDEDNESS OF WAVE OPERATORS IN R Estimate of Ω h ,n , n = 1 , . . . , .Lemma 4.3. (1) Let v, w ∈ L ( R ) . Then, vG ( λ ) w ∈ H for any λ > and for ε > there exists a C ε > such that for λ ≥ ε we have (cid:107) vG ( λ ) w (cid:107) H ≤ C ε λ − / (cid:107) v (cid:107) / (cid:107) w (cid:107) / . (4.3)(2) Let j = 1 , . Suppose that (cid:104) x (cid:105) j v, (cid:104) x (cid:105) j w ∈ L ( R ) . Then, vG ( λ ) w is an H -valued C j function of λ ∈ (0 , ∞ ) . For any ε > , there existsa constant C ε such that for λ ≥ ε we have (cid:107) ( d j /dλ j ) vG ( λ ) w (cid:107) H ≤ C ε λ − / (cid:107)(cid:104) x (cid:105) j v (cid:107) / (cid:107)(cid:104) x (cid:105) j w (cid:107) / . (4.4) Proof. (1) By virtue of (2.3) and (2.4), (cid:107) vG ( λ ) w (cid:107) is bounded by aconstant times the sum F + F where F = (cid:90) | x − y | >ε/λ | v ( x ) | | w ( y ) | λ | x − y | dxdy,F = (cid:90) | x − y | <ε/λ ( | log | λ | x − y || + C ) | v ( x ) | | w ( y ) | dxdy. The generalized Young’s inequality (e.g. [24]) then implies F ≤ Cλ − (cid:107) v (cid:107) / (cid:107) w (cid:107) / ,F ≤ C (cid:107) v (cid:107) / (cid:107) w (cid:107) / (cid:90) | z | <ε/λ ( | log | ( λz ) || + C ) dz ≤ C ε λ − (cid:107)(cid:107) v (cid:107) / (cid:107) w (cid:107) / and we obtain the estimate (4.3).(2) From formulas 6.6, 2.5 and 7.7 of chapter 10 of [7], we learn that( d/dλ ) G λ ( x ) and ( d /dλ ) G λ ( x ) are respectively given by | x | (cid:16) ddz H (1)0 (cid:17) ( λ | x | ) = −| x | H (1) − ( λ | x | ) ≤ | · | C (cid:26) λ − | x | , | x | λ ≥ .λ − , | x | λ < | x | (cid:16) d dz H (1)0 (cid:17) ( λ | x | ) = | x | (cid:18) H (1) − ( λ | x | ) + 1 λ | x | H (1) − ( λ | x | ) (cid:19) ≤ | · | C (cid:26) λ − | x | , | x | λ ≥ .λ − , | x | λ < . (4.6)Here a ≤ | · | b means | a | ≤ | b | . It follows as in the proof of (1), if (cid:104) x (cid:105) v ( x ) , (cid:104) x (cid:105) w ( x ) ∈ L ( R ), that vG (cid:48) (0) ( λ ) w ∈ H for 0 < λ < ∞ and (4.4) is satisfied for j = 1, where with a slight abuse of nota-tion we wrote G (cid:48) (0) ( λ ) for the convolution operator with ( d/dλ ) G λ ( x ).Moreover, by virtue of (4.5) and the smoothness propery of H − ( λ | x | )with respect to λ >
0, the dominated convergence theorem implies that vG (cid:48) (0) ( λ ) w is an H -valued continuous function. It follows that vG ( λ ) w − vG ( µ ) w = (cid:90) λµ vG (cid:48) (0) ( ρ ) wdρ as the Riemann integral of an H -valued continuous function. Thisproves vG ( λ ) w is an H -valued C function of λ ∈ (0 , ∞ ) and for thederivative vG (cid:48) ( λ ) w = vG (cid:48) (0) ( λ ) w .Repeating the argument above by using (4.6) instead of (4.5), wemay be able to show that vG ( λ ) w is an H -valued C -function of λ ∈ (0 , ∞ ) if x v ( x ) , x w ( x ) ∈ L ( R ) and second derivative satisfies(4.4) for j = 2. We, however, omit the repetitious details. (cid:3) As a warm up we prove the following proposition.
Proposition 4.4.
For any < p < ∞ , there exists a constant C p independent of u and V such that (cid:107) Ω h , u (cid:107) p ≤ C p (cid:107) u (cid:107) p (cid:90) (cid:90) R | V ( x ) || V ( y ) | (1 + | log | x − y || ) dxdy. (4.7)For the proof we need lemmas. Lemma 4.5.
Let m y ( x ) = v ( x ) w ( x − y ) and u ∈ L ( R ) . Then for λ (cid:54) = 0 vG ( λ ) wu ( x ) = (cid:90) R m y ( x ) H ( | y | λ )( τ y u )( x ) dy, a.e. x ∈ R . (4.8) Proof.
The change of variables in vG ( λ ) wu ( x ) = (cid:90) R v ( x ) H ( λ | x − y | ) w ( y ) u ( y ) dy implies the lemma. (cid:3) Lemma 4.6.
Let M y ( x ) = V ( x ) V ( x − y ) and Z M y be the operatordefined by (3.10) with M y in place of V . Then, Ω h , u ( x ) = − (cid:90) R Z M y ( H ( | y || D | ) χ ≥ a ( | D | ) τ y u )( x ) dy, u ∈ D ∗ . (4.9) Proof.
Substitute (4.8) for
V G ( λ ) V inΩ h , u = − (cid:90) ∞ G ( − λ ) V G ( λ ) V Π( λ ) uλχ ≥ ( λ ) dλ. (4.10)Explicitly the integral on the right side becomes (cid:90) ∞ (cid:18)(cid:90) (cid:90) R G − λ ( x − y ) M z ( y ) H ( λ | z | )(Π( λ ) τ z u )( y ) dzdy (cid:19) λχ ≥ ( λ ) dλ. (4.11)Again, we want to change the order of integrations in (4.11) by showingthat it is absolutely integrable for almost all x ∈ R . We repeat the p -BOUNDEDNESS OF WAVE OPERATORS IN R argument used for (3.26): We let D (cid:98) R x and, by applying Lemma3.1, (2.6), (2.7) and obvious | g ( λ ) | ≤ C for λ ∈ ( α, β ), we obtain (cid:90) ∞ (cid:90) D × R |G − λ ( x − y ) M z ( y ) H ( | z | λ )Π( λ )( τ z u )( y ) | λχ ≥ ( λ ) dxdzdydλ ≤ C (cid:90) βα (cid:90) D × R |G − λ ( x − y ) | ( | log | z || + C ) | M z ( y ) | dxdzdydλ ≤ C (cid:90) R | V ( y ) V ( z ) | (1 + | log | y − z || ) dzdy < ∞ . Thus, integrating by dλ first in (4.11) and applying the spectral theo-rem (1.10) for f ( λ ) = χ ≥ ( λ ) H ( | z | λ ), we obtainΩ h , u = − (cid:90) R (cid:18)(cid:90) ∞ G ( − λ ) M y Π( λ ) H ( | y || D | ) χ ≥ ( | D | ) τ y uλdλ (cid:19) dy. Recalling the definition (3.11) of Z F , we obtain the identity (4.9). (cid:3) In view of the different behavior of H ( λ ) for small and large λ , wesplit it as H ( λ ) = H low ( λ ) + H high ( λ ) by defining for a a > H low ( λ ) = χ ≤ a ( λ ) H ( λ ) , H high ( λ ) = χ >a ( λ ) H ( λ ) . Lemma 4.7.
For y ∈ R , H high ( | y || D | ) and H low ( | y || D | ) χ ≥ a ( | D | ) aregood operators. (cid:107)H high ( | y || D | ) (cid:107) B ( L p ) is y -independent and (cid:107)H low ( | y || D | ) χ ≥ a ( | D | ) u (cid:107) p ≤ C a,p (1 + | log | y || ) (cid:107) u (cid:107) p , u ∈ D ∗ . (4.12) It follows that H ( | y || D | ) χ ≥ a ( | D | ) is a good operator and (cid:107)H ( | y || D | ) χ ≥ a ( | D | ) u (cid:107) p ≤ C a,p (1 + | log | y || ) (cid:107) u (cid:107) p , u ∈ D ∗ . (4.13) Proof. (1) Since H high ( λ ) satisfies (2.3), the theory of spatially homoge-nous Fourier integral operators (see [26, 32, 34]) implies that H high ( | D | )is a good operator. Then, the scaling argument implies the same for H high ( | y || D | ) and (cid:107)H high ( | y || D | ) (cid:107) B ( L p ) is y -independent.(2) By virtue of (2.4) ( H ( λ ) − g ( λ )) χ ≤ a ( λ ) is a good multiplier and (cid:107)H low ( | y || D | ) − g ( | y || D | ) χ ≤ a ( | y || D | ) (cid:107) B ( L p ) is y -independent. Thus, itsuffices to prove (4.12) for g ( | y || D | ) χ ≤ a ( | y || D | ) χ ≥ a ( | D | ) or for( g ( | y | ) − (2 π ) − log | D | ) χ ≤ a ( | y || D | ) χ ≥ a ( | D | ) . However, (cid:107) g ( | y | ) χ ≤ a ( | y || D | ) χ ≥ a ( | D | ) (cid:107) B ( L p ) ≤ C (1 + | log | y || ) is evi-dent and we need consider only − (1 / π )(log | D | ) χ ≤ a ( | y || D | ) χ ≥ a ( | D | ).Define F ( λ ) = (1 / π )(log λ ) χ ≤ a ( | y | λ ) χ ≥ a ( λ ). Then, we have | F ( j ) ( λ ) | ≤ Cλ − j (1 + | log | y || ) . j = 0 , , . (4.14)Indeed F ( λ ) (cid:54) = 0 only if | y | < a < λ < a/ | y | and, | F ( λ ) | ≤ (2 π ) − max( | log a | , | log a/ | y || ) ≤ (2 π ) − ( | log | y || + | log a | ) , which implies (4.14) for j = 0. The proof of j = 1 and j = 2 is similar.Then, Mikhlin’s theorem implies that (cid:107) F ( | D | ) (cid:107) B ( L p ) ≤ C (1 + | log | y || )and (4.12) follows. (4.13) is then obvious. (cid:3) Proof of Proposition 4.4.
We apply Minkowski’s inequality, (3.11)and (4.13) to (4.9). Then, (cid:107) Ω h , u (cid:107) p ≤ (cid:90) R (cid:107) Ω M y ( H ( | y || D | ) χ ≥ a ( | D | ) τ y u ) (cid:107) p dy ≤ C p (cid:90) R (cid:107) M y (cid:107) (1 + | log | y | ) (cid:107) u (cid:107) p dy. This implies the proposition. (cid:3)
Estimate of Ω h ,n , n = 2 , , . . Results for Ω h ,n , n = 2 , , Proposition 4.8.
Let Ω ( n ) for n = 0 , , . . . be the operators define byreplacing M ( λ ) − by U ( − vG ( − λ ) w ) n . Then, for any < p < ∞ thereexists a constant C p independent of V and u ∈ D ∗ such that (cid:107) Ω ( n ) u (cid:107) p ≤ C p C ( n ) (cid:107) u (cid:107) p , u ∈ D ∗ , (4.15) where C ( n ) ( V ) is given by the integral (cid:90) R n +1) (cid:32) n +1 (cid:89) i =1 | V ( y i ) | (cid:33) (cid:32) n (cid:89) i =1 (1 + | log | y i − y i +1 || ) (cid:33) dy · · · dy n +1 . which is bounded by ( C (cid:107)(cid:104) x (cid:105) V (cid:107) ) n +1 .Proof. We have already proved the cases n = 0 and n = 1 and welet n ≥
2. Define M (1) y ( x ) = V ( x ) w ( x − y ), M (2) y ( x ) = v ( x ) V ( x − y )and m y ( x ) = v ( x ) w ( x − y ). Since vG ( λ ) w is the operator of Hilbert-Schmidt class and it can be expressed by (4.8), we have vU ( − vG ( λ ) w ) n vu ( x ) = ( − n ( V G ( λ ) w )( vG ( λ ) w ) n − ( vG ( λ ) V ) u = ( − n (cid:90) (cid:90) R n M (1) y ( x ) H ( λ | y | ) τ y × (cid:32) n − (cid:89) j =2 m y j ( x ) H ( λ | y j | ) τ y j (cid:33) M (2) y n ( x ) H ( λ | y n | ) τ y n u ( x ) dy . . . dy n = ( − n (cid:90) (cid:90) R n M y ,...,y n ( x ) (cid:32) n (cid:89) j =1 H ( λ | y j | ) (cid:33) τ y + ··· + y n u ( x ) dy . . . dy n , (4.16)where M y ,...,y n ( x ) = V ( x ) V ( x − y ) · · · V ( x − y − · · · − y n ). We sub-stitute (4.16) for vM ( λ ) − v in (1.17). The result is thatΩ ( n ) u ( x ) = ( − n (cid:90) (cid:90) R n (cid:90) ∞ G − λ ( x − y ) M z ,...,z n ( y ) × (cid:32) n (cid:89) j =1 H ( | y j | λ ) (cid:33) Π( λ ) τ y + ··· + y n u ( x ) λχ ≥ ( λ ) dy · · · dy n dλ (4.17) p -BOUNDEDNESS OF WAVE OPERATORS IN R and (4.17) is absolutely integrable for almost all x ∈ R . We omit therepetitious proof of the integrability of (4.17) which is similar to thatfor (4.11) and which uses Lemma 3.1, (2.6) and (2.7).Integrating (4.17) with respect to λ first after applying (1.10) and re-calling the definition of Z F in Proposition 1.3 (B), we see that Ω ( n ) u ( x )is equal to (cid:90) (cid:90) R n Z M y ,...,yn χ ≥ a ( | D | ) H ( | y || D | ) · · · H ( | y n || D | ) τ y + ··· + y n udy . . . dy n . Minkowski’s inequality, Proposition 1.3 (B) and Lemma 4.7 then imply (cid:107) Ω ( n ) u (cid:107) p ≤ C p (cid:90) R n (cid:107) M y ,...,y n (cid:107) n (cid:89) j =1 (1 + | log | y j || ) (cid:107) u (cid:107) p dy . . . dy n which is equivalent to (4.15). Since (1 + | log | x − y || ) ≤ C ε (cid:104) x (cid:105) ε (cid:104) y (cid:105) ε forany ε > C ( n ) ( V ) is evidently bounded by ( C (cid:107)(cid:104) x (cid:105) V (cid:107) ) n +1 .. (cid:3) Estimate of Ω h , . Estimate (4.15) implies that, if (cid:107)(cid:104) x (cid:105) V (cid:107) issmall, the expansionΩ high = ∞ (cid:88) j =0 (cid:90) ∞ G ( λ ) w ( − vG ( λ ) w ) j Π( λ ) λχ ≥ ( λ ) dλ converges in B ( L p ) for any 1 < p < ∞ and Ω high becomes a goodoperator. However, we want to avoid such an assumption which makes H regular at zero. We shall instead exploit the decay property (4.3)of (cid:107) vG ( λ ) w (cid:107) H and prove that Ω h , of (4.1), hence Ω high is a goodoperator. The following proposition completes the proof of Theorem4.1. Proposition 4.9. If (cid:104) x (cid:105) V ∈ L ( R ) then, Ω h , is a good operator.Proof. Evidently (cid:104) x (cid:105) v, (cid:104) x (cid:105) w ∈ L ( R ). Define T ( λ )= − U ( vG ( λ ) w ) (1 + vG ( λ ) w ) − χ ≥ ( λ ) (4.18)so that W h , u = ˜ W v,v ( T ( λ )) u . Lemma 4.3 implies that(1 + vG ( λ ) w ) − − vG ( λ ) w (1 + vG ( λ ) w ) − is an H -valued C -function of λ , hence so is T ( λ ) and that (cid:107) T ( j ) ( λ ) (cid:107) H ≤ C (cid:104) λ (cid:105) − , j = 0 , , . (4.19)Then, Lemma 3.7 (2) implies that ˜ W v,v ( T ( λ )) u is a good operator. (cid:3) Low energy estimates , threshold analysis
We now study the low energy part W low = W + χ ≤ a ( | D | ) of the waveoperator or equivalently Ω low defined by (1.18):Ω low u = (cid:90) ∞ G ( − λ ) vM ( λ ) − v Π( λ ) uλχ ≤ a ( λ ) dλ. (5.1)In this section we write χ ≤ for χ ≤ a when no confusions are feared; weshall always assume u ∈ D ∗ unless otherwise stated.5.1. Feshbach formula and Jensen-Nenciu Lemma.
For studyingthe behavior of M ( λ ) − as λ →
0, we repeatedly apply the Feshbachformula and the lemma due to Jensen and Nencie ([17]) which we recallhere. Feshbach formula is for the operator matrix A = (cid:18) a a a a (cid:19) :on the direct sum of Banach spaces Y = Y ⊕ Y . Lemma 5.1.
Suppose a , a are closed and a , a are boundedoperators. Suppose that a − exists. Then A − exists if and only if d = ( a − a a − a ) − exists. In this case we have A − = (cid:18) d − da a − − a − a d a − a da a − + a − (cid:19) . (5.2)Following is the lemma due to Jensen and Nenciu ([17]). Lemma 5.2.
Let A be a closed operator in a Hilbert space X and S a projection. Suppose A + S has a bounded inverse. Then, A has abounded inverse if and only if B = S − S ( A + S ) − S has a bounded inverse in S X . In this case, A − = ( A + S ) − + ( A + S ) − SB − S ( A + S ) − . (5.3)5.2. Preliminaries.
We recall that g ( λ | x | ) = N ( x ) + g ( λ ). In whatfollows we normalize g ( λ ) and define g ( λ ) def = g ( λ ) (cid:107) V (cid:107) . (5.4)Recall that the derivatives G ( j ) λ ( x ) = ( ∂/∂λ ) j G λ ( x ), j = 0 , , G ( j ) λ ( x ) satisfies the estimate (4.5) and (4.6). Moreover the expansion(2.1) implies that that for λ | x | ≤ ∂ jλ ( G λ ( x ) − g ( λ | x | )) ≤ | · | ∂ jλ ( (cid:104) g ( λ ) (cid:105) λ ) O ( | x | (cid:104) log | x |(cid:105) ) (5.5) ∂ jλ (cid:18) G λ ( x ) − g ( λ | x | ) (cid:18) − ( λ | x | ) (cid:19) + ( λ | x | ) π (cid:19) ≤ | · | ∂ jλ ( (cid:104) g ( λ ) (cid:105) λ ) O ( | x | (cid:104) log | x |(cid:105) ) (5.6) p -BOUNDEDNESS OF WAVE OPERATORS IN R The following lemma has been proved in [10] (see also [11]) underslightly different assumptions.
Lemma 5.3. (1)
Suppose (cid:104) x (cid:105) γ V ∈ L ( R ) , γ > . Then, as λ → , M ( λ ) = g ( λ ) P + T + M ( λ ) , M ( λ ) = O ( g ( λ ) λ ) . (5.7)(2) Suppose (cid:104) x (cid:105) σ V ∈ L ( R ) for some σ > . Then, as λ → , M ( λ ) = − g ( λ ) λ vG v − λ vG v + O ( λ (cid:104) log λ (cid:105) ) . (5.8) Proof.
We may assume 0 < λ <
1. (5.5) implies that for j = 0 , , (cid:18)(cid:90) | x − y | λ< | ( d/dλ ) j M ( λ, x, y ) | dxdy (cid:19) ≤ C | g ( λ ) λ − j |(cid:107)(cid:104) x (cid:105) γ V (cid:107) . For λ | x − y | ≥
1, (4.5) and (4.6) imply d j dλ j ( M ( λ ) + v ( x ) g ( λ | x − y | ) v ( y )) = O ( λ − / | x − y | j − / ) v ( x ) v ( y )for j = 0 , ,
2. Since (cid:104) x (cid:105) − (cid:104) y (cid:105) − ≤ Cλ , we have (cid:90) | x − y | >λ − | g ( λ ) | | v ( x ) v ( y ) | dxdy ≤ Cλ | g ( λ ) | (cid:107)(cid:104) x (cid:105) V (cid:107) , (cid:90) | x − y | >λ − | log | x − y || | v ( x ) v ( y ) | dxdy ≤ Cλ | g ( λ ) | (cid:107)(cid:104) x (cid:105) γ V (cid:107) , (cid:90) | x − y | >λ − λ − | x − y | j − | v ( x ) v ( y ) | dxdy ≤ Cλ γ − j (cid:107)(cid:104) x (cid:105) γ V (cid:107) . These estimates yield the first statement of the lemma. Proof of (2) issimilar and we omit the repetitious details. (cid:3)
We often use the following trivial but important identity:(1 + X ) − = 1 − X (1 + X ) − = 1 − X + X (1 + X ) − X. (5.9)5.3. Regular case.
In this subsection we prove the following theorem.
Theorem 5.4.
Suppose that (cid:104) x (cid:105) γ V ∈ L ( R ) for a γ > and that H is regular at zero, viz. QT Q is invertible in QL ( R ) . Then, Ω low is agood operator. For the proof we use the following lemma. We define v ∗ = v/ (cid:107) v (cid:107) . Lemma 5.5.
Under the assumption of
Theorem 5.4 we have M ( λ ) − = h ( λ ) L + Q ( QT Q ) − Q + O ( gλ ) (5.10) where h ( λ ) is the good multiplier defined by h ( λ )=( g ( λ ) + c ) − , c = (cid:104) v ∗ | T − T Q ( QT Q ) − QT | v ∗ (cid:105) , (5.11)rank L = 2 and Q ( QT Q ) − Q ∈ B . We first prove Theorem 5.4 admitting Lemma 5.5 is true. We substitute(5.10) for M ( λ ) − in (5.1). Then, h ( λ ) L produces1 iπ (cid:90) ∞ G ( − λ ) vLv Π( λ ) h ( | D | ) uλχ ≤ ( λ ) dλ . (5.12)Since h ( λ ) ∈ M ( R ) and rank L ≤
2, Proposition 1.3 (3) implies that(5.12) is a good operator; Q ( QT Q ) − Q ∈ B and O ( gλ ) are goodproducers by virtue of Lemma 3.6 and Lemma 3.7 respectively. Thus,Ω low is a good operator.@ (cid:3) We prove Lemma 5.5. Recalling (5.7), we first show g P + T isinvertible. In the matrix form in L ( R ) = P L ⊕ QL , g P + T = (cid:18) g P + P T P P T QQT P QT Q (cid:19) . (5.13)Here QT Q is invertible by the assumption; for small λ > g P + P T P − P T Q ( QT Q ) − QT P = ( g ( λ ) + c ) P is invertible in P L and ( g P + P T P − P T Q ( QT Q ) − QT P ) − = h ( λ ) P . Then, Lemma 5.1 implies that ( g P + T ) − exists and it isgiven by h ( λ ) (cid:18) P − P T Q ( QT Q ) − − ( QT Q ) − QT P ( QT Q ) − QT P T Q ( QT Q ) − + ( QT Q ) − (cid:19) def = h ( λ ) L + Q ( QT Q ) − Q, (5.14)where it is obvious that rank L ≤ Q ( QT Q ) − Q ∈B ([30]). It follows that M ( λ ) = (1 + M ( λ )( g P + T ) − )( g P + T )and (5.7) implies ( g P + T ) − M ( λ ) = O ( λ ). Hence, for small λ > H and M ( λ ) − = ( g P + T ) − (1 + M ( λ )( g P + T ) − ) − = ( g P + T ) − + ∞ (cid:88) j =1 ( g P + T ) − ( M ( λ )( g P + T ) − ) j , which implies (5.10) by virtue of (5.14). (cid:3) The singular case. Preliminaries.
We next consider the casethat H is singular at zero and let S be the orthogonal projection in QL ( R ) onto Ker QL QT Q | QL (recall Definition 1.4). In this subsec-tion, we collect some preliminary results and definitions. We assume (cid:104) x (cid:105) γ V ∈ L ( R ) for γ > (cid:104) x (cid:105) γ V ∈ L for a γ > Lemma 5.6 ([17, 11]) . The spectrum of QT Q in L ( R ) is discreteoutside {− , } and hence so is in QL ( R ) . (1) The projection S is of finite rank. Let n = rank S . p -BOUNDEDNESS OF WAVE OPERATORS IN R (2) QT Q + S has a bounded inverse in QL ( R ) and D = ( QT Q + S ) − ∈ B . (5.15) Proof. (1) is proved in [17]. Schlag([30]) actually proves D ∈ B . (cid:3) Definition 5.7.
We take and fix an orthonormal basis { ζ , . . . , ζ n } of S L ( R ) ⊂ QL ( R ). Define S = QS Q and write A for M ( λ ).We apply Lemma 5.2 to the pair ( A, S ) for studying A − for smallas λ >
0. We first show that A + S = g ( λ ) P + T + S + M ( λ ) (5.16)is invertible. We define h ( λ ) = ( g ( λ ) + c ) − , c = (cid:104) v ∗ | T − T QD QT | v ∗ (cid:105) . (5.17)It is evident that h is a good multiplier. Lemma 5.8. ( g P + T + S ) − exists in L ( R ) and in the decomposition L ( R ) = P L ⊕ QL it is given by h ( λ ) (cid:18) P − P T QD − D QT P D QT P T QD (cid:19) + QD Q = h ( λ ) L + QD Q. (5.18) Here L is λ -independent, rank L ≤ and QD Q ∈ B .Proof. In the decomposition L ( R ) = P L ⊕ QL , g P + T + S = (cid:18) g P + P T P P T QQT P QT Q + S (cid:19) . (5.19)By virtue of Lemma 5.6, D = ( QT Q + S ) − ∈ B . It is obvious thatfor small λ > g P + P T P − P T QD QT P = ( g + c ) P has the inverse h ( λ ) P . It follows by Lemma 5.1 that (5.19) is invertiblefor small λ > L ≤ (cid:3) Lemma 5.9. (1) M ( λ ) + S is invertible for small λ and ( M ( λ ) + S ) − = h ( λ ) L + QD Q + O ( gλ ) . (5.20) The following operator Ω (1)low is a good operator: Ω (1)low u = (cid:90) ∞ G ( − λ ) v ( M ( λ ) + S ) − v Π( λ ) uλχ ≤ a ( λ ) dλ. (5.21)(2) If (cid:104) x (cid:105) γ V ∈ L for a γ > then, O ( gλ ) in (5.20) is equal to λ ( h L + QD Q ) v ( gG + G ) v ( h L + QD Q ) + O ( λ g ) . (5.22) Proof.
Denote ˜ L =( g P + T + S ) − . Then, (5.18) and (5.7) imply A + S = (1 + M ˜ L )( g P + T + S ) is invertible for small λ > A + S ) − = ˜ L − ˜ L M ˜ L + ˜ L M ˜ L (1 + M ˜ L ) − M ˜ L (5.23)which implies (5.20). Since h ( λ ) ∈ M ( R ), rank L ≤ QD Q ∈B , the argument of the proof of Theorem 5.4 implies that Ω (1)low is a goodoperator.(2) The last term on the right of (5.23) is O ( λ g ). Equation (5.8)implies M ( λ ) = − v ( gλ G + λ G ) v + O ( gλ ) under the conditionand (5.22) follows from (5.23). (cid:3) We define B and T on S L ( R ) by B = S − S ( A + S ) − S , T = S QT P T QS (5.24)and R on L ( R ) by R = v ( G + g − G ) v. (5.25) Lemma 5.10. On S L ( R ) , we have as λ → B = − h ( λ )( T − λ X ) , X = O ( g ) . (5.26) If (cid:104) x (cid:105) γ V ∈ L ( R ) for a γ > , then X is given more precisely by − S h − g (cid:0) R + h ( L R + R L ) + h L R L + O ( gλ ) (cid:1) S . (5.27) Proof.
We substitute (5.20) for ( A + S ) − in (5.24). By the definitionof D , we have S QD QS = S and (5.18) implies S L S = (cid:18) S (cid:19) (cid:18) P − P T QD − D QT P D QT P T QD (cid:19) (cid:18) S (cid:19) = T . Then, (5.26) and (5.27) follow immediately by using (5.20) and (5.22)respectively. (cid:3) If B − ∈ B ( S L ( R )) exists, we have by virtue of Lemma 5.2, M ( λ ) − = ( A + S ) − + ( A + S ) − S B − S ( A + S ) − = A + A (5.28)where the definition of A and A should be obvious.5.5. The case with singularities of the first kind. . We first studythe case that H has singularities of the first kind at zero, viz. when T is nonsingular in S L ( R ), and prove the following theorem. Theorem 5.11.
Suppose (cid:104) x (cid:105) γ V ∈ L ( R ) for a γ > and H hassingularities of the first kind at zero. Then, Ω low is a good operator. Since rank T = 1, T is non-singular on S L ( R ) if and only ifrank S = 1. Take and fix a normalized basis ζ of S L ( R ). Then, c = (cid:104) T ζ, ζ (cid:105) = (cid:107) P T ζ (cid:107) > T = c ζ ⊗ ζ . Since S L ( R ) ⊂ QL ( R ), we have P ζ = 0 and ζ satisfies (cid:90) R v ( x ) ζ ( x ) dx = 0 (5.29) p -BOUNDEDNESS OF WAVE OPERATORS IN R Lemma 5.12. B ( λ ) − exists and B − = h ( λ ) − | ζ (cid:105)(cid:104) ζ | , h ( λ ) = − c h (1 + O ( g λ )) . (5.30) Proof.
Since X = O ( g ), B = − h ( λ ) T (1 − λ T − X ) is invertiblefor small λ > (cid:3) We use the notation of (5.28). Then, Lemma 5.9 shows that A is agood producer and we need to prove that the same is true for A . Ifwe insert (5.30) for B − and use QD Qζ = ζ , we have A = h − ( h L + QD Q )( ζ ⊗ ζ )( h L + QD Q ) + O ( g λ )= h ( λ ) − ( ζ ⊗ ζ ) − c − ( L ζ ⊗ ζ + ζ ⊗ L ζ ) − c − h ( λ )( L ζ ) ⊗ ( L ζ ) + O ( g λ ) . (5.31)Here Proposition 1.3 shows that all terms except h ( λ ) − ( ζ ⊗ ζ ) are goodproducers. Thus, the following lemma completes the proof of Theorem5.11 because v ( x ) ζ ( x ) ∈ L ( R ), (cid:104) ζ, v (cid:105) = 0 by virtue of (5.29) and h ( λ ) − λχ ≤ ( λ ) is a good multiplier. Lemma 5.13.
Suppose that (cid:104) x (cid:105) ϕ ( x ) ∈ L and (cid:82) R ϕ ( x ) dx = 0 . Sup-pose that λρ ( λ ) χ ≤ ( λ ) ∈ M ( R ) . Then, the operator W defined by W u = (cid:90) ∞ G ( − λ ) | ϕ (cid:105)(cid:104) ϕ | Π( λ ) u (cid:105) λρ ( λ ) χ ≤ ( λ ) dλ, u ∈ D ∗ is bounded from L p ( R ) to L p ( R ) for any < p < ∞ .Proof. The proof is the duplication of the one of the correspondingtheorem for the point interaction ([43]). Since (cid:104) ϕ, (cid:105) = 0, we have asin (1.25) (cid:104) ϕ, Π( λ ) u (cid:105) = (cid:104) ϕ, Π( λ ) u ( x ) − Π( λ ) u (0) (cid:105) = (cid:28) ϕ, π (cid:90) S ( e iλzω − F u )( λω ) dω (cid:29) = (cid:88) j =1 (cid:28) ϕ, iz j π (cid:90) (cid:18)(cid:90) S λω j ( F τ θz u )( λω ) dω (cid:19) dθ (cid:29) . (5.32)Define µ ( λ ) = − ρ ( λ ) λχ ≤ ( λ ). Then µ ( λ ) ∈ M ( R ) and, as translationsand multiplier commute each other, W u is equal to i π (cid:90)
10 2 (cid:88) j =1 (cid:18)(cid:90) ∞ G ( − λ ) | ϕ (cid:105)(cid:104) z j ϕ | Π( λ )( R j τ θz µ ( | D | )) u (cid:105) λdλ (cid:19) dθ, where R j = D j / | D | , j = 1 , W u ( x ) = (cid:88) j =1 (cid:90) (cid:18)(cid:90) (cid:90) R z j ϕ ( z ) ϕ ( y )( τ y Kτ z R j τ θz µ ( | D | ) u )( x ) dydz (cid:19) dθ. Then, Minkowki’s inequality, Riesz theorem and Mikhlin’s multipliertheorem jointly imply (cid:107) W u (cid:107) p ≤ (cid:88) j =1 (cid:90) (cid:18)(cid:90) (cid:90) R | z j ϕ ( z ) ϕ ( y ) |(cid:107) K (cid:107) B ( L p ) (cid:107) u (cid:107) p dydz (cid:19) dθ Then, Lemma 3.2 implies (cid:107) W u (cid:107) p ≤ C (cid:107) zϕ (cid:107) (cid:107) ϕ (cid:107) (cid:107) u (cid:107) p . (cid:3) Remark 5.14. If H has singularities of the first kind at zero, theleading singularities of M ( λ ) − as λ → is given by ( (cid:107) V (cid:107) / πc )( ζ ⊗ ζ ) log λ . It is shown in [17] that ϕ ( x ) = c + 12 π (cid:90) R log( | x − y | ) v ( y ) ζ ( y ) dy, c = (cid:104) v, T ζ (cid:105) / (cid:107) V (cid:107) (5.33) is a solution of Schr¨odinger equation ( − ∆ + V ) ϕ ( x ) = 0 . Here c (cid:54) = 0 since (cid:54) = (cid:104) T ζ, ζ (cid:105) = (cid:107) P T ζ (cid:107) = |(cid:104) v , T ζ (cid:105)| . Moreover as (cid:104) ζ, v (cid:105) = 0 , ϕ ( x ) satisfies ϕ ( x ) = c + a x + a x | x | + o ( | x | − ) | x | → ∞ (5.34) and ϕ ( x ) is an s -wave resonance by definition( [17] ). The case with singularities of the second kind.
We nextassume T = S QT P T QS is also singular in S L ( R ) and define S =projection in S L ( R ) to Ker S L ( R ) T . (5.35)Since rank T ≤ T | S L ( R ) is singular if rank S ≥ S = 1and T = 0. In what follows we often write S for S Q, QS and S for S S Q, QS S for simplicity. In the rest of the paper we assume (cid:104) x (cid:105) γ V ∈ L ( R ) for γ > . (5.36) Lemma 5.15.
The projection S annihilates T and L : T S = S T = 0 , S L = L S = 0 . (5.37) Proof.
Since QT QS = 0, P T QS = T QS . Then T = S QT P T QS = ( T QS ) ∗ ( T QS )and Ker S L ( R ) T = Ker S L ( R ) T QS . Thus, T QS S = 0 and S S QT = ( T QS S ) ∗ = 0 or T S = 0 and S T = 0. We have P S = P QS = 0 and likewise S P = 0; D S = D S S = S S = S and likewise S D = S . It follows S L = S ( P − P T QD − D QT P + D QT P T QD ) = 0 . The proof for L S = 0 is similar. (cid:3) p -BOUNDEDNESS OF WAVE OPERATORS IN R Recall that M ( λ ) − is given by (5.28) if B = − h ( λ )( T − λ X )is invertible in S L ( R ) where X is defined by (5.27). We define forsimplicity A = T − λ X so that B = − h ( λ ) A (5.38)and study A − by the help of Lemma 5.2. Since rank S = n < ∞ it isobvious that ( T + S ) − exists in S L ( R ). Lemma 5.16. (1)
For small λ > , ( A + S ) − exists and ( A + S ) − = ( T + S ) − + λ ( T + S ) − X ( T + S ) − + O ( g λ ) . (5.39)(2) Define B on S L ( R ) by B = S − S ( A + S ) − S . (5.40) Then, as λ → , B = λ h − g { ( S R S ) + S O ( g λ ) S } . (5.41) Proof. (1) We have A + S = ( T + S )(1 − λ ( T + S ) − X ) and X = O ( g ). It follows that ( A + S ) − exists for small λ > S ( T + S ) − = ( T + S ) − S = S implies S ( A + S ) − S = S + λ S XS + S O ( g λ ) S . (5.27) and Lemma 5.15 imply (5.41) since h − g = O ( g ). (cid:3) In this subsection, we assume that T = S ( vG v ) S | S L ( R ) (5.42)is non-singular, viz. H has singularities of the second kind and provethe following theorem: Theorem 5.17.
Suppose (cid:104) x (cid:105) γ V ∈ L ( R ) for a γ > and H hassingularities of the second kind at zero. Then, Ω low is bounded in L p ( R ) for < p ≤ but is unbounded in L p ( R ) for < p < ∞ . The proof is long and is given by a series of lemmas. The case p = 2is trivial and we assume p (cid:54) = 2. We follow the strategy of the proof ofthe corresponding results for two dimensional Schr¨odingers with pointinteractions (see section 4.3 of [43]). Remark 5.18. (1)
We have rank T = rank S is one or two and T isnegative definite. Indeed, for ζ ∈ S L ( R ) , we have (cid:104) vG vζ, ζ (cid:105) = − (cid:90) R ( x y + x y ) v ( x ) v ( y ) ζ ( x ) ζ ( y ) dxdy (5.43)= − (cid:88) j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R x j v ( x ) ζ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) . (5.44) Hence T = − (1 / x v ⊗ x v + x v ⊗ x v ) and rank T ≤ however x v and x v can be linearly dependent which is in particular the case if rank S = 1 (see Theorem 6.2 of [17] ). (2) If ζ ∈ S L ( R ) , then ϕ defined by (5.33) is a solution of ( − ∆ + V ) ϕ = 0 as in Remark 5.14 . However, c = 0 now since T ζ = 0 but T ζ (cid:54) = 0 implies ( a , a ) (cid:54) = (0 , . Thus, ϕ ( x ) produced by ζ ∈ S L ( R ) are p -wave resonances of − ∆ + V . (3) If T = 0 , then S L ( R ) = S L ( R ) and there is no s -wave reso-nances. If T (cid:54) = 0 , then S L ( R ) (cid:9) S L ( R ) = Image T = { cT v : c ∈ C } and rank S = rank S + 1 ( [17] ). In this case, if H has singularitiesof second kind, it has both s - and p -wave resonances but no zero energyeigenvalues. Reduction. If H has singularities of the second kind at zero,then S R S is also non-singular in S L ( R ) and we prove here thatit suffices for the proof of Theorem 5.17 to show that the operator ˜Ω low defined by (cid:90) ∞ g − λ − G ( − λ ) vS ( S R S ) − S ( v Π( λ ) u ) χ ≤ a ( λ ) dλ (5.45)satisfies the property stated in the theorem.Since S R S is nonsingular, B is invertible in S L ( R ) by virtueof (5.41). Then, Lemma 5.2 implies A − = ( A + S ) − + ( A + S ) − B − ( A + S ) − ; (5.46)Recall (5.28) that M ( λ ) − = A + A and A is a good producerby virtue of Lemma 5.9. We substitute B − = − h ( λ ) − A − in theexpression for A in (5.28). This produces A = − ( A + A ) where A = h − ( A + S ) − S ( A + S ) − S ( A + S ) − , (5.47) A = h − ( A + S ) − S ( A + S ) − S B − S ( A + S ) − S ( A + S ) − . (5.48)Remark 5.18 implies rank T = rank S is equal to one or two and T is negative selfadjoint. In what follows we assume rank T = 2.Modification for the case rank T = 1 is obvious and we omit it. Wetake the orthonormal basis { ζ , . . . , ζ n } of S L ( R ) such that ζ , ζ are(real) eigenfunctions of T ζ j = − κ j ζ j , j = 1 ,
2. Here n = 2 or n = 3according to the absence or presence of the s -wave resonance. We usethe matrix notation in the space S L ( R ) and define: Z u = (cid:18) (cid:104) ζ , u (cid:105)(cid:104) ζ , u (cid:105) (cid:19) , C ( λ ) = (cid:18) c ( λ ) c ( λ ) c ( λ ) c ( λ ) (cid:19) (5.49)where C ( λ ) is the matrix of S R S in the basis { ζ , ζ } : c jk ( λ ) = (cid:104) S R S ζ j , ζ k (cid:105) = − κ j δ jk + g − (cid:104) G vζ j , vζ k (cid:105) ; (5.50) C ( λ ) is non-singular and entries of C ( λ ) − = ( d jk ) are good multipliers. Lemma 5.19. (1)
We have (cid:26) ( A + S ) − S = S + h L S + O ( gλ ) S ,S ( A + S ) − = S + h S L + S O ( gλ ) (5.51) p -BOUNDEDNESS OF WAVE OPERATORS IN R and modulo O ( λ g ) (cid:26) ( A + S ) − S S = S + gλ ( h L + QD Q ) R S ,S S ( A + S ) − = S + gλ S R ( h L + QD Q ) (5.52)(2) The operator valued function −A is a good producer.Proof. (1) Multiplying (5.20) by S from the right or the left yields(5.51). Insert (5.22) for O ( gλ ) in (5.51), multiply the resulting equa-tion by S from the right or from the left and apply (5.37). We obtain(5.52).(2) (5.39) shows that S ( A + S ) − S = S ( T + S ) − S + O ( g λ ) . It follows by using also (5.51) that, with λ -independent finite rankoperators F and F , −A is equal to − h − ( S + h L S )( T + S ) − ( S + h S L ) + O ( g λ )= − h ( λ ) − S ( T + S ) − S + F + h ( λ ) F + O ( g λ ) . (5.53)In (5.53), O ( g λ ) is a good producer by virtue of Lemma 3.7 and,Proposition 1.3 (2) and (3) imply F and h ( λ ) F are good producers. h ( λ ) − S ( T + S ) − S is also a good producer by virtue of Lemma5.13 since, in the basis { ζ , . . . , ζ n } , S ( T + S ) − S = (cid:80) e jk | ζ (cid:105)(cid:104) ζ k | withconstants { e jk } and (cid:104) z j , v (cid:105) = 0 , j = 1 , . . . , n . The lemma follows. (cid:3) Lemma 5.19 in particular implies that A is a good producer and,hence, may be ignored in what follows. In (5.41) S R S is invertiblein S L ( R ) for small λ >
0, it follows that B − = λ − h g − ( S R S ) − ( S + S O ( g λ ) S ( S R S ) − ) − = λ − h g − ( S R S ) − + S O (1) S + S O ( g λ ) S , (5.54)which we substitue for B − in the expression (5.48) for A . Then,modulo a term O ( g λ ) which is a good producer, A ( λ ) becomes g − λ − ( A + S ) − S ( A + S ) − S ( S R S ) − S ( A + S ) − S ( A + S ) − + ( A + S ) − S ( A + S ) − S O ( g ) S ( A + S ) − S ( A + S ) − . (5.55) Lemma 5.20. (1)
There exist λ -independent operators F (1) j and F (2) j on S L ( R ) , j = 0 , , with rank at most two such that ( A + S ) − S = S + λ h − g (cid:88) j =0 O ( h j ) F (1) j + O ( g λ ) (5.56) S ( A + S ) − = S + λ h − g (cid:88) j =0 O ( h j ) F (2) j + O ( g λ ) . (5.57)(2) The second line of (5.55) is a good producer.
Proof. (1) is evident from (5.39) and (5.27) since L S = S L = 0 by(5.37) and rank S ≤
2. Note that g − = O ( h ).(2) Combining (5.56), (5.57), (5.51) and (5.37), we see that the secondline of (5.55) is equal to( A + S ) − S S O ( g ) S S ( A + S ) − + O ( λ g )= S O ( g ) S + O ( λ g ) . (5.58)Here O ( λ g ) is a good producer by virtrue of Lemma 3.7; in the basis { ζ , ζ } of S L ( R ) which satisfy (cid:104) ζ j , v (cid:105) = 0, j = 1 , S O ( g ) S = g ( λ ) (cid:88) b jk ( λ ) ζ j ⊗ ζ k , (5.59) b jk ( λ ) = g − (cid:104) z j , O ( g ) ζ k (cid:105) ∈ M ( R ) . Then, Lemma 5.13 implies that S O ( g ) S is a good producer. (cid:3) Lemma 5.21.
Define operator A = − g ( λ ) − λ − S ( S R S ) − S . (5.60) Then, modulo a good producer M ( λ ) − ≡ A ( λ ) . (5.61) Proof.
It suffices to show that the first line of (5.55) is equal to −A modulo good producers. By virtue of (5.56) and (5.57), we have λ − g − ( A + S ) − S ( S R S ) − S ( A + S ) − = λ − g − (cid:0) S + λ g (cid:88) h k − F (1) k (cid:1) ( S R S ) − (cid:0) S + λ g (cid:88) h k − F (2) k (cid:1) = λ − g − S ( S R S ) − S + h − L ( λ ) + O ( g λ ) , (5.62)where the sums are taken over k = 0 , , F (1) k and F (2) k are operatorsin S L ( R ); and L ( λ ) = (cid:16) (cid:88) h k F (1) k (cid:17) ( S R S ) − + ( S R S ) − (cid:16) (cid:88) h k F (2) k (cid:17) = n (cid:88) j,k γ jk ( λ ) ζ j ⊗ ζ k , γ jk ∈ M ( R ) . (5.63)We multiply (5.62) by ( A + S ) − S from the left and by S ( A + S ) − from the right. ( A + S ) − S O ( g λ ) S ( A + S ) − is again of class O ( g λ ) and is a good producer; (5.63) and (5.51) imply( A + S ) − S ( h − L ( λ )) S ( A + S ) − S = h − L ( λ ) + L S L ( λ ) + L S L ( λ ) + O ( g λ ) . Here h − L ( λ ) is a good producer by Lemma 5.13 as ζ j , ζ k ∈ S L ( R ); L S L ( λ ) + L S L ( λ ) is also a good producer as it is a finite sum ofrank one operators multiplied by a good multiplier; and O ( g λ ) is agood producer by Lemma 3.7 p -BOUNDEDNESS OF WAVE OPERATORS IN R Thus, applying also (5.52), we see that modulo a good producer − M ( λ ) − = g − λ − ( A + S ) − S S ( S R S ) − S S ( A + S ) − . = g − λ − S ( S R S ) − S − ( h L + QD Q ) R S ( S R S ) − S − S ( S R S ) − S R ( h L + QD Q ) + O ( gλ ) . Recall that S ( S R S ) − S = (cid:80) j,k =1 d jk ζ j ⊗ ζ k with d jk ∈ M ( R ).Note also that ( h L + QD Q ) R S and S R ( h L S + QD Q ) arepolynomials of order two of h and g − whose coefficients are finiterank operators. It follows that M ( λ ) + g − λ − S ( S R S ) − S is agood producer. This proves the lemma. (cid:3) Lemma 5.21 reduces the proof of Theorem 5.17 to studying ˜Ω low of(5.45). In the matrix notation defined by (5.49), S ( S R S ) − S = Z ∗ C ( λ ) − Z , and ˜Ω low u is expressed in the form − (cid:90) ∞ g − λ − G ( − λ )( Z v )( x ) ∗ C ( λ ) − Z ( v Π( λ ) u ) χ ≤ a ( λ ) dλ . (5.64)Recall that χ ≤ a ( λ ) d ij ( λ ) are good multipliers for 1 ≤ i, j ≤ u ∈ D ∗ , that Π( λ ) u = 0 for λ (cid:54)∈ ( α, β ), see Lemma 3.1.5.6.2. Decomposition of ˜Ω low u into good and bad parts. Since (cid:104) v, ζ j (cid:105) =0, j = 1 ,
2, we may express (cid:104) vζ j , Π( λ ) u (cid:105) as in (5.32) (see also (1.24)and (1.25)): (cid:104) vζ j , Π( λ ) u (cid:105) = 12 π (cid:28) vζ j , (cid:90) (cid:18)(cid:90) S iλzωe iλωzθ ˆ u ( λω ) dω (cid:19) dθ (cid:29) . (5.65)Then, defining the good part g j ( λ ) and the bad part b j ( λ ) by (1.27) byfurther expanding e iλωθ , we write (cid:104) vζ j , Π( λ ) u (cid:105) = g j ( λ ) + b j ( λ ). Defineˆ g ( λ ) = (cid:18) g ( λ ) g ( λ ) (cid:19) , ˆ b ( λ ) = (cid:18) b ( λ ) b ( λ ) (cid:19) (5.66)so that Z v Π( λ ) u = ˆ g ( λ ) + ˆ b ( λ ) and, correspondingly˜Ω low u = ˜Ω low , g u ( x ) + ˜Ω low , b u ( x ) , ˜Ω low , g u ( x ) = − (cid:90) ∞ λ − g ( λ ) − G ( − λ ) v Z ∗ C ( λ ) − ˆ g ( λ ) χ ≤ a ( λ ) dλ , ˜Ω low , b u ( x ) = − (cid:90) ∞ λ − g ( λ ) − G ( − λ ) v Z ∗ C ( λ ) − ˆ b ( λ ) χ ≤ a ( λ ) dλ . Lemma 5.22.
The good part ˜Ω low , g is a good operator. Proof.
By applying the routine change of order of integrations, we mayexpress ˜Ω low , g u ( x ) as the sum over 1 ≤ j, k ≤ ≤ i, l ≤ − (cid:90) (1 − θ ) dθ (cid:90) R dydz v ( y ) ζ j ( y ) v ( z ) ζ k ( z ) z l z i × (cid:26) π (cid:90) ∞ d jk g − χ ≤ a G − λ ( x − y ) (cid:18)(cid:90) S F ( τ θz R l R i u )( λω ) dω (cid:19) λdλ (cid:27) . Define µ jk ( λ ) = d jk ( λ ) g ( λ ) − χ ≤ a ( λ ). Then, µ jk ∈ M ( R ) and, in viewof the definition (1.19), the function inside the brace {· · · } is equal to τ y π (cid:90) ∞ G − λ ( x ) (cid:18)(cid:90) S F ( µ jk ( | D | ) τ θz R l R i u )( λω ) dω (cid:19) λdλ = ( τ y Kµ jk ( | D | ) τ θz R l R i u )( x ) . (5.67)Since (cid:107) τ y Kµ jk ( | D | ) τ θz R l R i u (cid:107) p ≤ C (cid:107) u (cid:107) p with constant independent of y, z ∈ R , 0 < θ < u ∈ D ∗ , Minkowki’s inequality implies (cid:107) ˜Ω low , g u (cid:107) p ≤ C p (cid:32) (cid:88) j,k,i,l =1 (cid:107) z l vζ j (cid:107) (cid:33) (cid:107) u (cid:107) p . (5.68)Since p -wave resonances ζ j satisfies (cid:104) x (cid:105) − δ ζ ∈ L ( R ) for any 0 < δ < (cid:107) z l ζ j v (cid:107) ≤ C (cid:107)(cid:104) x (cid:105) V (cid:107) . Thus, the good part˜Ω low , g is a good operator. (cid:3) Estimate of bad part 1, Positive result.
For the bad part, we firstshow that ˜Ω low , b is bounded in L p ( R ) for 1 < p ≤
2. We decompose˜Ω low , b u = χ ≥ a ( | D | ) ˜Ω low , b u + χ ≤ a ( | D | ) ˜Ω low , b u where a is the constant which appears in the definition of W + χ ≤ a ( | D | ). Lemma 5.23.
For < p ≤ , χ ≥ a ( | D | ) ˜Ω low , b ∈ B ( L p ( L ( R ))) .Proof. Via components we express in the form˜Ω low ,b u ( x ) = (cid:88) j,k i π (cid:90) ∞ λ − g − G ( − λ )( vζ j )( x ) d jk ( λ ) b k ( λ ) χ ≤ a dλ.. As previously µ jk ( λ ) = g ( λ ) − d jk ( λ ) χ ≤ a ( λ ) for j, k = 1 , µ jk ( λ ) = 0 for λ > a evidently. Explicitly b k ( λ ) isgiven by (see (1.27)) b k ( λ ) = (cid:88) l =1 iλ π (cid:104) ζ k , z l v (cid:105) (cid:90) S ( F R l u )( λω ) dω. Thus, χ ≥ a ( | D | ) ˜Ω low , b u ( x ) = (cid:88) j,k,l =1 X jkl u ( x ) p -BOUNDEDNESS OF WAVE OPERATORS IN R where for j, k, l = 1 , X jkl u ( x ) = i (cid:104) z l v | ζ k (cid:105) π (cid:90) ∞ χ ≥ a ( | D | ) G ( − λ )( vζ j )( x ) × (cid:18)(cid:90) S ( F µ jk ( | D | ) R l u )( λω ) dω (cid:19) dλ. (5.69)We decompose the convolution kernel of χ ≥ a ( | D | ) G ( − λ ). Define µ ( ξ ) = χ ≥ a ( ξ ) | ξ | − and split as χ ≥ a ( | ξ | )( ξ − λ ) − = µ ( ξ ) + λ µ ( ξ )( ξ − λ ) − . (5.70)Note that for λ < a for which µ jk ( λ ) (cid:54) = 0 functions in (5.70) aresmooth functions of ξ ∈ R . Via Fourier transform χ ≥ a ( | D | ) G − λ ( x − y ) = ˆ µ ( x − y ) + λ µ ( | D | ) G − λ ( x − y ) (5.71)and χ ≥ a ( | D | ) G ( − λ ) vζ j ( x ) becomes the sumˆ µ ∗ ( vζ j )( x ) + λ (cid:90) R µ ( | D | ) G − λ ( x − y )( vζ j )( y ) dy. (5.72)We substitute (5.72) for χ ≥ a ( | D | ) G ( − λ )( vζ j )( x ) in (5.69). Then, X jkl u ( x ) = X ( b ) jkl u ( x ) + X ( g ) jkl u ( x ) . where X ( b ) jkl u ( x ) and X ( g ) jkl u ( x ) are the functions produced by the firstand the second terms of (5.72) respectively.Using the fact that second factor contains the ample factor λ , wefirst show that X ( g ) jkl u ( x ) is a good operator. Denote ρ jk ( λ ) = λµ jk ( λ ).Then, by changing the order of integration and by integrating withrespect to dλ first as usual, we obtain X ( g ) jkl u ( x ) = i (cid:104) z l v | ζ k (cid:105) π (cid:90) R ( vζ j )( y ) × (cid:26)(cid:90) ∞ µ ( | D | ) τ y G − λ ( x ) (cid:18)(cid:90) S ( F ρ jk ( | D | ) R l u )( λ ) dω (cid:19) λdλ (cid:27) dy = (cid:104) z l v | ζ k (cid:105) · (cid:90) R ( vζ j )( y )( µ ( | D | ) Kρ jk ( | D | ) R l u )( x − y ) dy. (Here we have omitted the repetitious argument for the changing theorder of integration.) Then, Minkowski’s inequality and (3.4) imply (cid:107) X ( g ) jkl u (cid:107) p ≤ C p |(cid:104) z l v | ζ k (cid:105)|(cid:107) vζ j (cid:107) (cid:107) u (cid:107) p (5.73)for any 1 < p < ∞ . Since ˆ µ ∗ ( vζ j )( x ) is independent of λ , we may express X ( b ) jkl u ( x ) inthe form X ( b ) jkl u ( x ) = i (cid:104) z l v | ζ k (cid:105) π ( µ ∗ vζ j )( x ) (cid:90) ∞ (cid:18)(cid:90) S ( F µ jk ( | D | ) R l u )( λ ) dω (cid:19) dλ = i (cid:104) z l v | ζ k (cid:105) π ( µ ∗ vζ j )( x ) (cid:90) R ( F µ jk ( | D | ) R l u )( ξ ) | ξ | − dξ. (5.74)Here µ ∗ vζ j ∈ L p ( R ) for any 1 < p < ∞ since µ ∈ L p ( R ) and vζ j ∈ L ( R ). Since µ jk ( λ ) | ξ | − = ( g ( | ξ | ) | ξ | ) − d jk ( | ξ | ) χ ≤ a ( | ξ | ) ∈ L p ( R )as long as 1 ≤ p ≤
2, the linear functional (cid:90) R ( F µ jk ( | D | ) R l u )( ξ ) | ξ | − dξ = (cid:90) R u ( x ) F ( µ jk ( | ξ | ) ξ l | ξ | − )( x ) dx is bounded on L p ( R ) for 1 ≤ p ≤ X ( b ) jkl is bounded in L p ( R ) for 1 < p ≤ (cid:3) The next lemma together with previous Lemma 5.23 proves that˜Ω low , b is bounded in L p ( R ) if 1 < p < Lemma 5.24.
Let < p < . The low energy part χ ≤ a ( | D | ) ˜Ω low , b isbounded from L p ( R ) to itself.Proof. The argument patterns after the proof of Lemma 4.4 of [43] forthe corresponding result for the point interactions. As in the proof ofLemma 5.23 it suffices to show the lemma for ˜ X jkl u ( x ), j, k, l = 1 , (cid:90) ∞ χ ≤ a ( | D | ) G ( − λ ) vζ j ( x ) (cid:18)(cid:90) S ( F µ jk ( | D | ) R l u )( λω ) dω (cid:19) dλ . (5.75)Recall that µ jk ( λ ) = d jk ( λ ) g ( λ ) − χ ≤ a ( λ ). Write u jkl = µ jk ( | D | ) R l u .We have for λ < aχ ≤ a ( | D | ) G ( − λ ) vζ j ( x ) = 12 π (cid:90) R e ixξ χ ≤ a ( | ξ | ) F ( vζ j )( ξ ) ξ − λ + i dξ (5.76) (cid:18) def = lim s ↓ π (cid:90) R e ixξ χ ≤ a ( | ξ | ) F ( vζ j )( ξ ) ξ − λ + iσ dξ (cid:19) . Since (cid:104) vζ j , (cid:105) = 0, we have as previously F ( vζ j )( ξ ) = 12 π (cid:90) (cid:18)(cid:90) R ( − izξ ) e − iθzξ ( vζ j )( z ) dz (cid:19) dθ, which we substitute in (5.76). Then χ ≤ a ( | D | ) G ( − λ ) vζ j ( x ) becomesthe sum over m = 1 , ,
1] with respect p -BOUNDEDNESS OF WAVE OPERATORS IN R θ of − i (2 π ) (cid:90) R (cid:18)(cid:90) R ξ m χ ≤ a ( | ξ | ) e i ( x − θz ) ξ z m ( vζ j )( z ) ξ − λ + i dz (cid:19) dξ = − i π (cid:90) R z m ( vζ j )( z ) τ θz (cid:18) π (cid:90) R e ixξ ξ m χ ≤ a ( | ξ | ) ξ − λ + i dξ (cid:19) dz. (5.77)Here the change of order of integral is trivially justified if + i iσ ; for the change of the order of the limit σ → u ∈ D ∗ , ˆ u ( ξ ) = 0 for | ξ | < α and for λ > α ,12 π (cid:90) R e ixξ ξ m χ ≤ a ( | ξ | ) ξ − λ + iσ dξ = G ( √ λ − iσ ) F ( ξ m χ ≤ a )( x )has a limit as σ → +0 uniformly with respect to x ∈ R and thedominated convergence theorem applies. Define ω m = ξ m | ξ | − andwrite ξ m ξ − λ + i λω m ξ − λ + i ω m | ξ | + λ . Then the inner integral in the right of (5.77) becomes R m λχ ≤ a ( | D | ) G − λ ( x ) + R m (cid:18) π (cid:90) R e ixξ χ ≤ a ( | ξ | ) | ξ | + λ dξ (cid:19) (5.78)where ( R , R ) = ( D / | D | , D / | D | ) are Riesz transforms. Combining(5.75), (5.77) and (5.78) together and changing the order of integra-tions, we see that ˜ X jkl u ( x ) is equal to the sum over m = 1 , − iR m π (cid:90) R z m ( vζ j )( z ) τ θz χ ≤ a ( | D | ) (cid:18)(cid:90) ∞ (cid:90) S G − λ ( x ) (cid:100) u jkl ( λω ) dωλdλ (cid:19) dz (5.79)+ − iR m (2 π ) (cid:90) R z m ( vζ j )( z ) τ θz (cid:18)(cid:90) R e ixξ χ ≤ a ( | ξ | )( | ξ | + | η | ) | η | (cid:100) u jkl ( η ) dξdη (cid:19) dz (5.80)By virtue of the definition (1.19), (5.79) is equal to − i π (cid:90) R z m ( vζ j )( z ) τ θz R m χ ≤ a ( | D | ) Ku jkl ( x ) dz. Thus, Minkowski’s inequality, (3.4) and the multiplier theory implythat for δ > δ < γ/ (cid:107) (5 . (cid:107) p ≤ C (cid:90) R | z m ( vζ j )( z ) |(cid:107) u jkl (cid:107) p ≤ C (cid:107)(cid:104) z (cid:105) − δ ζ j (cid:107) (cid:107)(cid:104) x (cid:105) δ v (cid:107) (cid:107) u (cid:107) p for 1 < p < ∞ , where in the last step we used he fact that µ jk ( | ξ | ) ∈M ( R ). The inner integral of (5.80) may be written in the form (cid:90) R (cid:18)(cid:90) R e ixξ − iyη χ ≤ a ( | ξ | ) χ ≤ a ( | η | )( | ξ | + | η | ) | η | g ( | η | ) dξdη (cid:19) ( d jk ( | D | ) R m u )( y ) dy (5.81) It is known that (Lemma 4.7 of [43]) for 1 < p < L ( x, y ) = (cid:90) R e ixξ − iyη χ ≤ a ( | ξ | ) χ ≤ a ( | η | )( | ξ | + | η | ) | η | g ( | η | ) dξdη is bounded in L p ( R ) and (cid:107) (5 . (cid:107) p ≤ C (cid:107) u (cid:107) p . Lemma 5.24 is proved. (cid:3) Estimate of bad part 2, Negative result.
The following lemmaimplies that W + is unbounded in L p ( R ) when 2 < p < ∞ . Lemma 5.25.
Let < p < ∞ . Then, χ ≥ a ( | D | ) ˜Ω low , b is unbounded in L p ( R ) .Proof. By virtue of Lemma 5.22 and (5.73), it suffices to show that thelemma for the operator ˜Ω low ∗ defined by˜Ω low ∗ u = (cid:88) j,k,l =1 X ( b ) jkl u. By virtue of (5.74), we have˜Ω low ∗ u ( x ) = (cid:88) j =1 (cid:96) j ( u ) µ ∗ ( vζ j )( x ) , (5.82) (cid:96) j ( u ) = (cid:88) k,l =1 i (cid:104) z l v | ζ k (cid:105) (cid:90) R u ( x ) F ( µ jk ( | ξ | ) ξ l | ξ | − )( x ) dx. (5.83)We have seen that µ ∗ ( vζ j ) ∈ L p ( R ), j = 1 , ≤ p < ∞ .Actually they are linearly independent for, suppose c µ ∗ ( vζ ) + c µ ∗ ( vζ ) = µ ∗ ( c vζ + c vζ ) = 0 for constants c , c ∈ C ; since µ ( x ) isentire, this implies v ( x )( c ζ + c ζ )( x ) = 0 and ( c ζ + c ζ )( x ) = 0 for x such that v ( x ) (cid:54) = 0. But ζ j ( x ) satisfies − κ j ζ j ( x ) = v ( x )( G vζ j )( x )and ζ j ( x ) = 0 whenever v ( x ) = 0. Thus, ( c ζ + c ζ )( x ) = 0 and c = c = 0 because (cid:104) ζ j , ζ k (cid:105) = δ jk .Suppose that ˜Ω low ∗ is bounded in L p ( R ) for a 2 < p < ∞ . Then,since µ ∗ ( vζ ) and µ ∗ ( vζ ) are linearly independent, the Hahn-Banachtheorem implies both (cid:96) and (cid:96) must be continuous functionals on L p ( R ); then, by virtue of the Riesz representation theorem, it mustbe that (cid:88) k,l =1 e kl F ( µ jk ( | ξ | ) ξ l | ξ | − ) ∈ L q ( R ) , e kl = (cid:104) z l v | ζ k (cid:105) p -BOUNDEDNESS OF WAVE OPERATORS IN R for the dual exponent q of p and hence, for j = 1 , (cid:88) k =1 d jk ( | ξ | ) (cid:88) l =1 e kl χ ≤ a ( ξ ) ξ l g ( | ξ | ) − | ξ | − ∈ L p ( R )by Haudorff-Young’s inequality. This means in vector notation that χ ≤ a ( | ξ | ) g ( | ξ | ) | ξ | C ( | ξ | ) − N ( ξ ) ∈ L p ( R , C ) , N ( ξ ) = (cid:18) e ξ + e ξ e ξ + e ξ (cid:19) . (5.84)But C ( | ξ | ) − has bounded inverse for | ξ | ≤ a , it must be that χ ≤ a ( | ξ | ) g ( | ξ | ) | ξ | ( e k ξ + e k ξ ) ∈ L p ( R ) , k = 1 , . (5.85)Since p > e k = e k = 0, k = 1 , (cid:104) z v | ζ k (cid:105) = (cid:104) z v | ζ k (cid:105) =0. This contradicts with (5.44). Thus, ˜Ω low ∗ mustbe unbounded in L p ( R ) for any 2 < p < ∞ . (cid:3) Sigularities of the third kind.
Finally we consider the casethat T = S vG vS | S L ( R ) is singular in S L ( R ) and let S be theprojection in S L ( R ) onto Ker T . It is well known ([17]) in thiscase that T = S vG vS is necessarily non-singular in S L . We firstassume that S (cid:9) S (cid:54) = 0. But it will become evident from the proof ofLemma 5.27 that W + becomes a good operator if S = S (see Theorem5.28). Theorem 5.26.
Suppose that H has singularities of the third kind atzero and S (cid:40) S . Then W + is bounded in L p ( R ) for < p ≤ andis unbounded in L p ( R ) for < p < ∞ . By virtue of Lemma 5.21 modulo a good producer vM ( λ ) − v ≡ − g ( λ ) − λ − vS ( S R S ) − S v. (5.86)and we need only study the operator ˜Ω low defined by (5.45). For short-ening formulas, define˜ R = S R S , T = S vG vS (= T ) , ˜ T = S vG vS , so that ˜ R = T + g − ˜ T and T = S ˜ T S . Decompose S L ( R ) intothe orthogonal sum S L ( R ) = X ⊕ X , X = ( S (cid:9) S ) L ( R ) , X = S L ( R )and express ˜ R in S L ( R ) as the operator matrix in this decomposi-tion: ˜ R = (cid:18) T + g − ˜ T g − ˜ T g − ˜ T g − ˜ T (cid:19) . (5.87)Here T jk is the X k → X j part of T for j, k = 2 , T and we have used that T = 0 , T = 0 and T = 0. Note that˜ T = T . We apply Feshbach Lemma 5.1 to (5.87). ( g − T ) − = gT − existsby the assumption and T is non-singular in X by the definition. Itfollows that T + g − ( ˜ T − ˜ T ˜ T − ˜ T )is invertible in X for small λ > d def = T − (1 X + g − ( ˜ T − ˜ T ˜ T − ˜ T ) T − ) − = T − ∞ (cid:88) n =0 g ( λ ) − E n , E = 1 X , (5.88)Then, Lemma 5.1 implies˜ R − = gS T − S + L , L = (cid:18) d − d ˜ T ˜ T − − ˜ T − ˜ T d ˜ T − ˜ T d ˜ T ˜ T − (cid:19) . (5.89)We now take the basis { ζ , . . . , ζ m } of S L ( R ) such that { ζ , . . . , ζ l } ,1 ≤ l ≤ T = S vG vS with strictlynegative eigenvalues − κ , . . . , − κ l and { ζ l +1 , . . . , ζ m } is the basis of S L ( R ) = Ker S L ( R ) T which consists of eigenfunctions of T | X with non-zero eigenvalues α l +1 , . . . , α m . Since T ζ k = 0 for k = l +1 , . . . , m , (5.44) implies for these k ’s (cid:90) R x ζ k ( x ) v ( x ) dx = (cid:90) R x ζ k ( x ) v ( x ) dx = 0 . (5.90) Lemma 5.27.
Suppose that H has singularities of the third kind atzero and S (cid:40) S . Then W + is bounded in L p ( R ) for < p ≤ .Proof. We represent ˜ R − = gS T − S + L by the matrix in the or-thonormal basis { ζ , . . . , ζ m } and substitute it for C ( λ ) − in (5.64) withthe obvious modification for Z v ( x ) which is now m -component vectordefined via this basis. The matrix is the sum for the ones for gS T − S and for L . Let B ( λ ) = β . . . β m ... ... ... β m . . . β mm (5.91)be the one for L . Observe that, by virtue of (5.89) and (5.88), β jk ( λ )are analytic functions of g ( λ ) − for small λ and, hence, β jk ( λ ) χ ≤ ( λ )are good multipliers. and the moment property that (cid:104) ζ j , v (cid:105) = 0 holdsfor all j = 1 , . . . , m . Then, the proof of the positive part of Theorem5.17 implies that the operator produced by B ( λ ): (cid:90) ∞ g − λ − (cid:104) G ( − λ )( Z v )( x ) ∗ | B ( λ ) |Z ( v Π( λ ) u ) (cid:105) χ ≤ ( λ ) dλ . (5.92)is bounded in L p ( R ) for 1 < p < d jk ( λ ) and ζ j which are used there). p -BOUNDEDNESS OF WAVE OPERATORS IN R The matrix for gS T − S is the diagonal matrix which we denote˜ D = diag( α , . . . , α m ) whose first l elements vanish. Hence, the oper-ator (5.64) with ˜ D in place of C ( λ ) − is equal to m (cid:88) j = l +1 α j (cid:90) ∞ λ − ( G ( − λ ) ζ j v )( x ) (cid:104) ζ j v, Π( λ ) u (cid:105) χ ≤ ( λ ) dλ. (5.93)Note that these are operators of the form (3.6) but with the singularfactor λ − in place of λ . However, we can get around this difficulty byusing the extra cancellation property (cid:104) x l ζ j , v (cid:105) = 0, l = 1 , (cid:104) ζ j , v (cid:105) = 0 for j = l + 1 , . . . , m which follows from (5.44) with ζ = ζ j ∈ S L ( R ) for which the left side vanishes. Then, this allowsthe decomposition (cid:104) ζ j v | Π( λ ) u (cid:105) = g j ( λ ) + b j ( λ ) of (1.27) with b j = 0and, as in Lemma 5.22,(5 .
93) = − m (cid:88) k = l +1 2 (cid:88) l,m =1 α k (cid:90) (1 − θ ) × (cid:18)(cid:90) R z l z m ( ζ k v )( y )( ζ k v )( z )( τ y Kχ ≤ a ( | D | ) R m R l τ θz u )( x ) dydz (cid:19) dθ. Then by using Mikowski’s inequality, (3.4) and the multiplier theoremwe obtain (cid:107) (5 . (cid:107) p ≤ C m (cid:88) j = l +1 (cid:107) z ζ j v (cid:107) (cid:107) ζ j v (cid:107) (cid:107) u (cid:107) p ≤ C (cid:107)(cid:104) z (cid:105) γ V (cid:107) (cid:107) u (cid:107) p , γ > . Thus, (5.93) is a good operator and the proof of Lemma 5.27 is com-pleted. (cid:3)
The proof of Lemma 5.27 produces the following theorem for thecase S = S . Theorem 5.28.
Suppose that H has singularities of the third kind atzero and T = 0 , viz. S = S . Then W + is bounded in L p ( R ) for all < p < ∞ .Proof. Under the condition we have L = 0 and ˜ R − = gT − . Then,Theorem 5.28 is evident by the second part of the proof of the previouslemma. (cid:3) The following lemma completes the proof of Theorem 5.26
Lemma 5.29.
Suppose that H has singularities of the third kind atzero and S (cid:40) S . Then W + is unbounded in L p ( R ) if < p < ∞ .Proof. Let B ( λ ) be the matrix of (5.91). In view of the proof of Lemma5.27, it suffices to prove that the operator ˜Ω B u ( x ) defined by˜Ω B u ( x ) = (cid:90) ∞ g − λ − ( G ( − λ )( Z v )( x ) ∗ B ( λ ) Z v Π( λ ) u )( x ) χ ≤ ( λ ) dλ is unbounded in L p ( R ) if 2 < p < ∞ . We define ˆ g ( λ ) and ˆ b ( λ ) by(1.27) so that Z ( v Π( λ ) u ) = ˆ g ( λ ) + ˆ b ( λ ) and ˜Ω B u ( x ) = ˜Ω B,g u + ˜Ω B,b u where˜Ω B,g u ( x ) = (cid:90) ∞ λ − g ( λ ) − G ( − λ ) v Z ∗ B ( λ )ˆ g ( λ ) χ ≤ ε ( λ ) dλ , (5.94)˜Ω B,b u ( x ) = (cid:90) ∞ λ − g ( λ ) − G ( − λ ) v Z ∗ B ( λ )ˆ b ( λ ) χ ≤ ε ( λ ) dλ . (5.95)Then, the proof of Lemma 5.22 with B ( λ ) in place of C ( λ ) implies ˜Ω B,g is a good operator.Then, it suffices to prove χ ≥ a ( | D | ) ˜Ω B,b is unbounded in L p ( R ) if2 < p < ∞ . We prove this by slightly modifying the argument of theproof of Lemma 5.25. χ ≥ a ( | D | ) ˜Ω B,b u ( x ) is given by the right of (5.82)and (5.83) with β jk in place of d jk which is hidden in the definition of µ jk ( λ ). Here again we can show that µ ∗ ( vζ ) , . . . , µ ∗ ( vζ m ) are linearlyindependent by modifying the argument there: for concluding ζ j ( x ) = 0when v ( x ) = 0 we now use that − κ j ζ j ( x ) = v ( x )( G vζ j )( x ) only for j = 1 , . . . , l and, for j = l + 1 , . . . , m , we use ρ j ζ j ( x ) = v ( x )( G vζ j )( x )for ρ j (cid:54) = 0. Then as previously, Hahn-Banach theorem implies that if˜Ω B,b is bounded in L p ( R ) for a 2 < p < ∞ , then we must have (5.84)with B ( | ξ | ) in place of C ( | ξ | ) − , viz χ ≤ ( | ξ | ) g ( | ξ | ) | ξ | B ( | ξ | ) N ( ξ ) ∈ L p ( R ) , N ( ξ ) = e ξ + e ξ ... e m ξ + e m ξ . (5.96)Here, for k = l + 1 , . . . , m , vG vζ k = 0 implies e kh = i (cid:104) z h v | ζ k (cid:105) / π = 0, h = 1 , e k = e k = 0 by virtue of (5.90). Thus, consideringonly the first l components of (5.96), we must have χ ≤ ( | ξ | ) g ( | ξ | ) | ξ | ˜ B ( | ξ | ) ˜ N ( ξ ) ∈ L p ( R ) , ˜ N ( ξ ) = e ξ + e ξ ... e l ξ + e l ξ (5.97)where ˜ B ( | ξ | ) is the first l × l small matrix of B ( | ξ | ), viz. the represen-tation matrix of d = T − (1 X + g − ( ˜ T − ˜ T T − ˜ T ) T − ) − which is non-singular with bounded inverse for small 0 < λ < a . Thus, e k = e k = 0 must be true also for k = 1 , . . . , l which again contradictsto (5.44). This completes the proof. (cid:3) References [1]
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Department of Mathematics, Gakushuin University, 1-5-1 Mejiro,Toshima-ku, Tokyo 171-8588 (Japan).
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