The Calder′o n Projection:
New De?nition and Applications
Bernhelm Booss-Bavnbek,Matthias Lesch,and Chaofeng Zhu
Abstract.We consider an arbitrary linear elliptic ?rst–order di?erential op-
erator A with smooth coe?cients acting on sections of a complex vector bundle
E over a compact smooth manifold M with smooth boundary Σ.We describe
the analytic and topological properties of A in a collar neighborhood U of Σ
and analyze various ways of writing A U in product form;we discuss the sec-
torial projections of the corresponding tangential operator;construct various
invertible doubles of A by suitable local boundary conditions;obtain Poisson
type operators with di?erent mapping properties;and provide a canonical con-
struction of the Calder′o n projection.We apply our construction to generalize
the Cobordism Theorem and to determine su?cient conditions for continuous
variation of the Calder′o n projection and of well–posed self-adjoint Fredholm
extensions under continuous variation of the data.
Contents
List of Figures 21.Introduction 21.1.Dirac operator folklore 21.2.In search of generalization 3
1.3.Our present approach 4
2.Elliptic di?erential operators of ?rst order on manifolds with boundary 6
2.1.Product form and metric structures near the boundary 6
2.2.The general set–up 9
2.3.Regular boundary conditions 10
3.Sectorial projections of an elliptic operator 12
3.1.Parameter dependent ellipticity 12
3.2.Sectorial operators:abstract Hilbert space framework 14
3.3.Sectorial operators:parametric elliptic di?erential operators 18
4.The invertible double 202000Mathematics Subject Classi?cation.Primary 58J32;Secondary 35J67,58J50,57Q20.Key words and phrases.Calder′o n projection,Cauchy data spaces,cobordism theorem,con-tinuous variation of operators and boundary conditions,elliptic di?erential operator,ellipticity with parameter,Lagrangian subspaces,regular boundary value problem,sectorial projection,self-adjoint Fredholm extension,Sobolev spaces,symplectic functional analysis.
This work was supported by the network “Mathematical Physics and Partial Di?erential Equations”of the Danish Agency for Science,Technology and Innovation.The second named author was partially supported by Sonderforschungsbereich/Transregio “Symmetries and Univer-sality in Mesoscopic Systems”(Bochum–Duisburg/Essen–K¨o ln–Warszawa)and the Hausdor?Center for Mathematics (Bonn).The third named author was partially supported by FANEDD 200215,973Program of MOST,Fok Ying Tung Edu.Funds 91002,LPMC of MOE of China,and Nankai University.
1a r X i v :0803.4160v 1 [m a t h .D G ] 28 M a r 2008
2BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
4.1.The construction of?A P(T)20
4.2.The local ellipticity of P(T)for?A21
4.3.The solution space ker?A P(T)22
4.4.The main result23
5.Calder′o n projector from the invertible double23
5.1.Sobolev scale23
5.2.Induced Poisson type operators and inverses24
5.3.The Calder′o n projection28
6.The Cobordism Theorem,revisited30
6.1.The General Cobordism Theorem31
6.2.Elements of symplectic functional analysis32
6.3.Proof of Theorem6.137
6.4.Alternative routes to the General Cobordism Theorem41
7.Parameter dependence42
7.1.Some estimates44
7.2.Continuous dependence of?R T(B0)46
7.3.Continuous dependence of the invertible double46
7.4.Continuous dependence of S(A,T)48
7.5.Continuous dependence of P±on input data48
7.6.Continuity of families of well–posed self–adjoint Fredholm extensions50
Appendix A.Smooth symmetric elliptic continuations with constant
coe?cients in normal direction52 References56
List of Figures
1Construction of a closed coneΛsuch that B0?λId is elliptic with parameterλ∈Λ13 2The contoursΓ±in the plane de?ning the sectorial projections Q±15 3A two-component contourΓ,separating an inner sector around the real axis where all eigenvalues of B0show up,from two outer sectors which
totally belong to the resolvent set of B017 4Three contours encircling all eigenvalues in the right half plane,on the imaginary axis,and all eigenvalues in the left half plane,respectively.39
1.Introduction
This paper is about basic analytical properties of elliptic operators on compact manifolds with smooth boundary.Our main achievements are(i)to develop the basic elliptic analysis in full generality,and not only for the generic case of opera-tors of Dirac type in product metrics(i.e.,we assume neither constant coe?cients in normal direction nor symmetry of the tangential operator);(ii)to establish the cobordism invariance of the index in greatest generality;and(iii)to prove the con-tinuity of the Calder′o n projection and of related families of global elliptic boundary value problems under parameter variation.We take our point of departure in the following observations.
1.1.Dirac operator folklore.Most analysis of geometrical and physical problems involving a Dirac operator A on a compact manifold M with smooth boundaryΣacting on sections of a(complex)bundle E seems to rely on quite a few basic facts which are part of the shared folklore of people working in this
THE Calder′o n PROJECTION3?eld of global analysis(e.g.,see Boo?and Wojciechowski[BBW93]for properties (WiUCP),(InvDoub)and(Cob),and Nicolaescu[Nic97,Appendix]and Boo?, Lesch and Phillips[BBLP05]for property(Param)):
WiUCP:the weak inner unique continuation property(also called weak UCP to the boundary),i.e.,there are no nontrivial elements in the null
space ker A vanishing at the boundaryΣof M;
InvDoub:the existence of a suitable elliptic invertible continuation A of A, acting on sections of a vector bundle over the closed double or another
suitable closed manifold M which contains M as submanifold;this yields
a Poisson type operator K+which maps sections over the boundary into
sections over M;and a precise Calder′o n projector C+,i.e.,an idempotent
mapping of sections over the boundary onto the Cauchy data space which
consists of the traces at the boundary of elements in the nullspace of A
(possibly in a scale of Sobolev spaces);
Cob:the existence of a self–adjoint regular Fredholm extension of any total (formally self–adjoint)Dirac operator A in the underlying L2-space with
domain given by a pseudodi?erential boundary condition;that implies the
vanishing of the signature of the associated quadratic form,induced by
the principal symbol in normal direction at the boundary;moreover,that
actually is equivalent to the Cobordism Theorem asserting a canonical
splitting of the tangential operator B=B+⊕B?with ind B+=0;
Param:the continuous dependence of a family of operators,their associated Calder′o n projections,and of any family of well-posed(elliptic)boundary
value problems on continuous or smooth variation of the coe?cients.
1.2.In search of generalization.With the renewed interest in geometri-cally de?ned elliptic operators of?rst order of general type,arising,e.g.,from perturbations of Dirac operators,we ask to what extent the preceding list can be generalized to arbitrary linear elliptic di?erential operators with smooth coe?-cients.It is hoped that the results of this paper can serve as guidelines for similar constructions and results for hypo-and sub-elliptic operators where the symbolic calculus is not fully available.
There are immediate limits for generalization of some of the mentioned features by counter examples:UCP,even weak inner UCP may not hold for arbitrary elliptic systems of?rst order,see indications in that direction in Pli′s[Pli61,Corollary1, p.610]and the?rst-order Alinhac type counterexample to strong UCP[B¨a r99, Example,p.184].Moreover,from just looking at the de?ciency indices,we see
that the formally self-adjoint operator i d
dx on the positive line does not admit a
self-adjoint extension.This example is instructive because,quite opposite to the half-in?nite domain,on a bounded one-dimensional interval any system of?rst-order di?erential equations satis?es property(Cob)by a deformation argument.
We go through the list.
Property(WiUCP):It seems that the precise domain of validity is unknown. The local stability of weak inner UCP has been obtained by Boo?and Zhu in [BBZ05,Lemma3.2].In spite of the local de?nition of UCP,the property(WiUCP) has a threefold global geometric meaning:(i)there are no ghost solutions,i.e., each section u∈C∞(M,E)belonging to the null space ker A over the manifold M has a non-trivial trace u Σat the boundary;(ii)equivalently,the maximal extension A max=(A t min)?is surjective in L2(M,E)for densely de?ned closed minimal A min:D(A min)=H10(M,E)→L2(M,E);and(iii),as noted by Boo?and Furutani in[BBF98,Section3.3]and in various follow-ups,it seems that assuming weak inner UCP of A and A t is mandatory for obtaining the continuity
4BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
of Cauchy data spaces and the continuous change of the Calder′o n projection under variation of the coe?cients,i.e.,property (Param).
Property (InvDoub ):Di?erent approaches are available:one approach ([BBW93,Chapter 9])has been the gluing of A and its formal adjoint A t to an invertible
elliptic operator A
over the closed double M .This construction is explicit,if the metric structures underlying the Dirac operator’s de?nition are product near the boundary.In the self-adjoint case,it yields at once the Lagrangian property of the Cauchy data space in the symplectic Hilbert space L 2(Σ,E Σ)of square integrable sections in E Σ:=E Σ.Then (Cob)follows.
Property (InvDoub)generalizes to the non-product case for operators of Dirac type and,as a matter of fact,for any elliptic operator satisfying weak inner UCP un-der the somewhat restrictive condition that the tangential operator has self-adjoint principal symbol.Here the trick is that this condition permits the prolongation of the given operator to a slightly larger manifold M with boundary reaching con-stant coe?cients in normal direction close to the new boundary and maintaining UCP under the prolongation (as well as formal self-adjointness of the coe?cients,if present at the old boundary).This is explained in the Appendix.
But what can be done for general elliptic operators?A very general and elegant construction of the Calder′o n projection was given by H¨o rmander in [H¨o r85,Theo-rem 20.1.3]on the symbol level.Unfortunately,he obtains only an almost projection (up to smoothing operators)which limits its applicability in our context.
In this paper,we shall exploit another general de?nition of the Calder′o n pro-jection which is due to Seeley [See69,Theorem 1and Appendix,Lemma].Seeley’s construction provides a precise projection,not only an approximate one,and does not require UCP.First he replaces A by an invertible operator A 1by adding the projection onto the ?nite–dimensional space of inner solutions.Then he extends the operator D := 0A t 1A 10
to the closed double M of a slightly extended man-ifold M with boundary.In general,such a prolongation may,however,destroy weak inner UCP even when UCP was established on the original manifold.Seeley constructs on the symbol level (and by adding a suitable correction term)an el-
liptic extension D of D over the whole of the closed manifold M which is always invertible.He shows that D
provides the wanted Poisson operator and a truly pseudodi?erential Calder′o n projection P +along the original boundary Σ.In the formally self-adjoint case,P +has self-adjoint principal symbol and can be replaced by the orthogonal projection which is also pseudodi?erential and has the same sym-bol (and may be denoted by the same letter P +).In this way,the choices in the construction of the invertible double are removed totally,as the operation of the resulting P +is concerned.This makes P +a good candidate for property (Cob).
However,Seeley’s general construction,in di?erence to the simple gluing in the case of Dirac type operators of [BBW93,Chapter 9],does not lead immedi-ately to the Lagrangian property of im P +.Moreover and more seriously,when working with curves of elliptic problems Seeley’s construction does not give a hint under what conditions the Calder′o n projections vary continuously when varying a parameter.There are too many choices involved in Seeley’s construction.
1.3.Our present approach.This motivates our present approach (inspired by Himpel,Kirk and Lesch [HKL04]),namely the construction of the invertible double as a canonically given local boundary problem for the double D exactly on the original manifold M ,without any choices,prolongations etc.This leaves us with full control of the UCP situation;leads directly to the wanted Fredholm Lagrangian
THE Calder′o n PROJECTION5 property(Cob);and,moreover and here most decisively,provides explicit formulas for treating the parameter dependence in property(Param).
This program is opened in Section2by explaining our basic choice of product structures near the boundary for the sake of comprehensible analysis,even if the original geometric structures are non-product;moreover,for the convenience of the reader and for?xing our notation we summarize a few basic facts about regular boundary conditions.
To begin with,we do not assume self–adjoint principal symbol of the tangential operator B0nor constant coe?cients B x=B0along an inward coordinate x.Most of our estimates depend on the single fact that B0?λis parameter dependent elliptic forλin a conic neighborhood of i R in the sense of Shubin[Shu80,Sec-tion II.9].More precisely,we depend on the related concept of sectorial spectral projections introduced in1970by Burak[Bur70]and recently further developed in Ponge[Pon06,Section3]as part of the current upsurge of interest in spectral properties of non-selfadjoint elliptic operators.Because of our interest in the con-tinuous dependence of this kind of generalized positive spectral projections on the input data we found it necessary to develop the concept of sectorial projections once again from scratch.This is done in Section3where we develop an abstract Hilbert space framework for the concept of sectorial projections and apply it to tangential operators perceived as parametric elliptic operators.
In Section4we provide the construction and the relevant properties of the invertible double yielded by a local elliptic boundary value problem.
In Section5we establish suitable Sobolev regularity of the inverse operator which leads to the de?nition and basic properties of Poisson operator and Calder′o n projection.
Property(Cob):In Section6,we give a?rst application of our construction of the Calder′o n projection:we give our reading of Ralston[Ral70]and infer that the arguments of this1970paper establish the following?ndings for any formally self-adjoint di?erential operator A over a compact manifold M with smooth boundary Σ:
?the existence of a self-adjoint Fredholm extension A P given by a pseudo-
di?erential boundary condition P;
?the vanishing of the signature of iωon the space V(B0)of all purely imag-
inary eigensections of the tangential operator B0of A over the boundary
(or on ker B0in the case that B0is formally self-adjoint);here iωdenotes
the form induced by the Green form of A on the symplectic von Neumann
spaceβ(A):=D(A max)/D(A min),i.e.,the principal symbol of A over the
boundaryΣin normal direction;
?and,equivalently,but seemingly never recognized by people working in
global analysis,the General Cobordism Theorem,stating that the index
of any elliptic di?erential operator B+over a closed manifoldΣmust
vanish,if B+can be written as the left lower corner of a formally self-
adjoint tangential operator overΣinduced by an elliptic formally self–
adjoint A on a smooth compact manifold M with?M=Σ.
Property(Param):In Section7,as a second application of our construction of the Calder′o n projection,we establish that property in great generality.Roughly speaking,we let an operator family(A z)and the family(A t z)of formally adjoint operators vary continuously in the operator norm from L21(M)to L2(M)and as-sume that the principal symbol(J0(A z))of(A z)over the boundaryΣin normal direction also varies continuously in the L2
(Σ)operator norm with z running in
1/2
a parameter space Z.We assume for all A z and A t z property(WiUCP)or,almost equivalently,that the dimensions of the spaces of“ghost solutions”without trace at
6BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
the boundary remain constant under the variation.Then in Theorems 7.9and 7.2a we show that the inverse of the “invertible double”and,under slightly sharpened continuity,the Poisson operator in respective operator norms vary continuously;and so does the resolvent of a family (A z P z )of well-posed Fredholm extensions of now formally self–adjoint (A z )with orthogonal pseudodi?erential projections (P z )varying in L 21/2(Σ)operator norm.
Unfortunately,in the same generality we can neither prove nor disprove the continuous variation of the Calder′o n projection.However,if the principal symbol of the tangential operator is self-adjoint,we can prove the continuous variation of the sectorial projection (Proposition 7.15)and so (Theorem 7.4)of the Calder′o n projections by our correction formula (5.30)and Theorem 7.2b.Our Proposition
7.13shows that the di?culties for proving continuous variation of the sectorial projection disappears also for non-selfadjoint principal symbol,if the variation is of order <1.
In Appendix A we discuss various special cases with emphasis on constant coe?cients in normal direction in a collar around the boundary.
2.Elliptic di?erential operators of ?rst order on manifolds with
boundary
2.1.Product form and metric structures near the boundary.We shall begin with a basic observation:Dirac operators emerge from a Riemannian struc-ture on the manifold and a Hermitian metric on the vector bundle (together with Cli?ord multiplication and a connection).Talking about a general di?erential op-erator it is in our view very misleading to pretend that the operator will depend on metrics and such.All we need is the operator and an L 2–structure on the sections.The latter basically only requires a density (take dvol in the Riemannian case)on the manifold and a metric on the bundle.In this paper,we prefer to choose metrics and such as simple as possible and push all complications into the operator.
The message is this:we can always work in the product case and do have to worry only about the operator.In detail:
Let M be a compact manifold (with or without boundary)and π:E →M a vector bundle.Given a Hermitian metric h on E and a Riemannian metric g on M we can form the Hilbert space L 2(M,E ;g,h )which is the completion of C ∞0(M \?M,E )with regard to the scalar product
(2.1) u,v g,h := M
h (u (x ),v (x ))dvol g (x ).
Given another Riemannian metric g 1and another Hermitian metric h 1on E there is a smooth positive function ∈C ∞(M )such that
(2.2)dvol g 1= dvol g ,
and there is a smooth section θ∈C ∞(M,End E )such that for x ∈M,ξ,η∈E x we have
(2.3)h 1,x (ξ,η)=h x (θ(x )ξ,η).
Furthermore,with regard to h the operator θ(x )is self–adjoint and positive de?nite,thus we may form √θwhich is again a smooth self–adjoint and positive de?nite section of End E .It is clear that (2.3)determines θ(x )uniquely and the claimed smoothness of x →θ(x )can be checked easily in local coordinates.In sum,we ?nd
THE Calder′o n PROJECTION7 for u,v∈C∞(M,E)
u,v g
1,h1
=
M
h1(u(x),v(x))dvol g
1
(x)
=
M
h(θ(x)u(x),v(x)) (x)dvol g(x) =
θu,
θv g,h.
(2.4)
Thus we arrive at
Lemma2.1.The map
Ψ:C∞(M,E)?→C∞(M,E),u→
θu
extends to an isometry from L2(M,E;g1,h1)onto L2(M,E;g,h).
Now assume that we are given a di?erential operator A in L2(M,E;g1,h1)of ?rst order.It may be a Dirac operator which is constructed from the metrics g1and h1.g1and h1may be wildly non–product near the boundary.Suppose there are metrics g,h which we like more,e.g.,product near the boundary.Then consider the di?erential operatorΨAΨ?1in L2(M,E;g,h).ΨAΨ?1is still a di?erential operator and sinceΨis unitary all spectral properties are preserved.
Let us be even more speci?c and choose a neighborhood U of?M=:Σand a di?eomorphismφ:U→[0,ε)×Σwithφ Σ=idΣ.Furthermore,we choose a metric g on M such that
φ?g=dx2⊕gΣ,
gΣ:=g1 Σ,
(2.5)
is a product metric which induces the same metric on the boundary as g1.Here x denotes the normal inward coordinate near the boundary in the metric g.
φis covered by a bundle isomorphism F:E U→[0,ε)×EΣ,EΣ:=E Σ,i.e., we have the commutative diagram
(2.6)E U F
π[0,ε)×EΣ
id×π
Uφ[0,ε)×Σ.
Likewise,we may now choose a metric h on E such that h(x):=F?h {x}×Σ=h Σ=h1 Σ=:hΣis independent of x∈[0,ε).The mappings F andφinduce a map
(2.7)Ψ1:C∞(U,E)→C∞([0,ε),C∞(EΣ))
f→
x→
p→F(f(φ?1(x,p)))
which extends to a unitary isomorphism L2(U,E;g,h)→L2([0,ε],L2(Σ,EΣ)).On Σand EΣwe have the?xed metrics gΣrespectively hΣand we suppress the reference to them in the notation.
Together with the unitary isomorphismΨof Lemma2.1we obtain the claimed isometry
(2.8)Φ:=Ψ1?Ψ:L2(U,E;g1,h1)→L2([0,ε],L2(EΣ)).
NowΦAΦ?1is a?rst order di?erential operator in the product Hilbert space L2([0,ε))?L2(Σ,EΣ;gΣ,h)and hence it takes the form
(2.9)D:=ΦAΦ?1=:J d
dx
+B
with a bundle endomorphism J∈C∞([0,ε),C∞(Σ,End EΣ))and a smooth family of?rst order di?erential operators B∈C∞([0,ε),Di?1(Σ,EΣ)).For the moment
8BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
we consider here only the smooth case,but so far one can replace’smooth’by ’continuous’or’Lipschitz’or whatever.
Let us repeat:now all metric structures are product near the boundary and we do not have to worry about them.If we start with a Dirac operator on a Riemannian manifold with non–product metric the’non–product situation’is re?ected in the varying coe?cients of D.From now on we have to worry only about those varying coe?cients and nothing else.We assume that product metrics g,h are?xed and therefore we will suppress them in the notation.
For further reference we record Green’s formula for A:
Lemma2.2.Letν∈C∞(Σ,T M Σ)be the outward normal vector?eld.Then we have for f,g∈C∞(M,E)
Af,g L2(M,E;g
1,h1)
? f,A t g L2(M,E;g
1
,h1)
=1
i
Σ
σ1A(νb)f Σ,g Σ dvol=? J(0)f Σ,g Σ L2(Σ,E
Σ
;gΣ,h1)
,
(2.10)
where i:=√
?1,νb denotes the cotangent vector?eld corresponding toνin the
metric g1,andσ1
A
denotes the leading symbol of A.
Note thatφ?ν=?d
dx .Note also that by construction all transformations are
trivial on the boundary,that is,
φ Σ=id;F EΣ=id,
(Ψ1f) Σ=f Σ;(Ψf) Σ=f Σ;
(Φf) Σ=f Σ.
(2.11)
For a di?erential operator A we denote by A t its formal adjoint.If A is viewed as an unbounded operator in some Hilbert space,we denote by A?its functional an-alytic adjoint.For0th order operators and for elliptic operators on closed manifolds the di?erence between formal adjoint and(true)adjoint does not really matter;so in this case we use both notations interchangeably.
NearΣwe have
A=J x d
dx
+B x
and
(2.12)
?A t=J t x
d
dx
?B t x J t x+(J x)t
(2.13)
=J t x d
dx
?(J t x)?1B t x J t x
+(J x)t.
If A is formally self–adjoint,we have the relations
(2.14)J t=?J,JB=J ?B t J
( denotes di?erentiation by x).The ellipticity of A implies that J(x)is invertible and that B(x)is elliptic for all x.
Alternatively,we may choose the following normal form in a collar of the bound-ary:
A=J(x) d
dx
+B(x)
+
1
2
J (x),
=:J x d
dx
+B x
+
1
2
J x.
(2.15)
To avoid an in?ation of parentheses we will most often use the notation B x,J x instead of B(x),J(x)etc.Only to avoid double subscripts we will write B(x),J(x) in subscripts.
A=A t implies the relations
(2.16)J t=?J,JB=?B t J.
THE Calder′o n PROJECTION9 The normal form(2.15)determines J and B uniquely.
2.2.The general set–up.We are going to?x some notation which will be used throughout the paper.Assume that the following data are given:?a compact smooth Riemannian manifold(M,g)with smooth boundary
Σ:=?M,
?Hermitian vector bundles(E,h E),(F,h F),
?a?rst order elliptic di?erential operator
(2.17)A:C∞(M,E)→C∞(M,F).
We consider A as an unbounded operator between the Sobolev(and Hilbert) spaces
(2.18)L2s(M,E;g,h E),L2s(M,F;g,h F),s≥0.1
The closure of A C∞0(M\Σ,E)in L2is denoted by A min and we put
(2.19)D(A max):=
f∈L2
Af∈L2
.
As explained in Section2.1there exists a collar U≈[0,ε)×Σand linear isomor-phisms
(2.20)ΦG:C∞(U,G)→C∞([0,ε),C∞(GΣ)),G=E,F,
which extend to isometries
(2.21)L2(U,G;g,h G)→L2([0,ε],L2(Σ,GΣ;gΣ,h GΣ),G=E,F, where gΣ=g Σ,GΣ:=G Σand h GΣ=h G GΣ.
Now we consider
(2.22)D:=ΦF A(ΦE)?1:C∞([0,ε),C∞(EΣ))→C∞([0,ε),C∞(FΣ)). Since A is a?rst order elliptic di?erential operator we?nd
(2.23)D=J x d
dx
+B x
,
where J x∈Hom(EΣ,FΣ),0≤x≤ε,is a smooth family of bundle homomorphisms and(B x)0≤x≤εis a smooth family of?rst order elliptic di?erential operators between sections of EΣ.
The ellipticity of D imposes another restriction on B https://www.doczj.com/doc/0e983039.html,ly,ellipticity of
D means that forλ∈R,ξ∈T?pΣ,(λ,ξ)=(0,0)the operator
(2.24)iλ+σ1
B(x)
(p,ξ)
is invertible for all(x,p)∈[0,ε)×Σ.Here,σ1
B(x)
denotes the leading symbol of
B x.In other words,forξ∈T?pΣ\{0}the endomorphismσ1
B(x)(p,ξ)∈End(E p)
has no eigenvalues on the imaginary axis i R.
Furthermore we note that in view of(2.10)J0equals iσ1
A (νb),whereν=?d
dx
is the outward normal vector?eld.
1For simplicity we content ourselves to Sobolev spaces of nonnegative order.On a manifold with boundary Sobolev spaces of negative order are a nuisance,although with some care they could be dealt with here.Cf.[LM72]
10BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
2.3.Regular boundary conditions.For the convenience of the reader and to ?x some notation we brie?y summarize a few basic facts about boundary condi-tions for A .Standard references are [BBW93,H¨o r85,LM72,See69].We will adopt the point of view of the elementary functional analytic presentation [BL01].However,we try to be as self-contained as possible.
It is well-known that the trace map
(2.25)
:C ∞0(M,E )→C ∞(Σ,E ),f →f Σextends by continuity to a bounded linear map between Sobolev spaces (2.26)L 2s (M,E )→L 2s ?1/2(Σ,E Σ),
s >1/2.For the domain of A max this can be pushed a bit https://www.doczj.com/doc/0e983039.html,ly,for s ≥0the trace map extends by continuity to a bounded linear map
(2.27)
D (A max ,s )→L 2s ?1/2(Σ,
E Σ),s ≥0,that is,there is a constant C s ,such that for f ∈L 2s (M,E )with Af ∈L 2s (M,E )(2.28) f s ?1/2≤C s ( f s + Af s )
Here f s denotes the Sobolev norm of order s.Furthermore,norms of operators
from L 2s to L 2s will be denoted by · s,s ,and · ∞denotes the sup-norm of a
function.
The proof (2.27),(2.28)in [BBW93,Theorem 13.8and Corollary 13.9]simpli-?es [LM72]for operators of Dirac type but remains valid for any elliptic di?erential operator of ?rst order.cf.also [BL01,Lemma 6.1].
Definition 2.3.(a)Let CL 0(Σ;E Σ,G )denote the space of classical pseudo-di?erential operators of order 0,acting from sections of E Σto sections of another smooth Hermitian bundle G over Σ.
(b)Let P ∈CL 0(Σ;E Σ,G ).We denote by A P the operator A acting on the domain (2.29)D (A P ):= f ∈L 21
(M,E ) P ( f )=0 ,and by A max ,P the operator A acting on the domain (2.30)D (A max ,P ):= f ∈L 2(M,E ) Af ∈L 2(M,F ),P ( f )=0 .(c)The boundary condition P for A is called regular if A max ,P =A P ,i.e.,if f,Af ∈L 2,P ( f )=0already implies that f ∈L 21(M,E ).
(d)The boundary condition P is called strongly regular if f ∈L 2,Af ∈L 2k ,P ( f )=
0already implies f ∈L 2k +1(M,E ),k ≥0.
(2.28)shows that D (A P )is in any case a closed subspace of L 21(M,E ).
Proposition 2.4.Let P be regular for A .Then A P is a closed semi-Fredholm operator with ?nite–dimensional kernel.
Proof.Let (f n )?D (A P )be a sequence with f n →f and Af n →g ∈L 2
(M,F ).Then Af =g weakly and hence in view of (2.27),(2.28)we have P ( f )=0and the regularity of P implies f ∈L 21(M,E ),thus f ∈D (A P ).
Hence A P is closed and thus D (A P )is complete in the graph norm.The pre-vious argument shows in view of the Closed Graph Theorem that the inclusion ι:D (A P ) →L 21(M,E )is bounded.ιis thus an injective bounded linear map from the Hilbert space D (A P )(equipped with the graph norm)onto a closed subspace of L 21(M,E );the closedness is also a consequence of the argument at the begin-ning of this proof.Consequently,on D (A P )the graph norm and the L 21-norm are equivalent.I.e.,for f ∈D (A P )we have
(2.31)1C f 1≤ f 0+ Af 0≤C f 1.
THE Calder′o n PROJECTION 11
Since the inclusion L 21(M,E )
→L 2(M,E )is compact,the inclusion D (A P ) →L 2(M,E )is compact,too.Consequently,A P is a semi-Fredholm operator with ?nite–dimensional kernel.
Remark 2.5.(1)P =Id is strongly regular and its domain D (A P )=L 21,0(M,E )equals
the closure of C ∞0(M \Σ,E )in L 21(M,E ).This is seen by https://www.doczj.com/doc/0e983039.html,ly,if f ∈L 2,Af =g ∈L 2and f =0we may extend f by 02to obtain a solution of ?f,?A ?f ∈L 2loc and hence f ∈L 21.For the induction we ?rst note that similarly as in [BL01,Cor.2.14]one shows that to the map
(2.32) (k +1):L 2k +1(M,E )?→k
j =0L 2k ?j +1/2(Σ,E Σ),f →( A j f )k j =0
there exists a continuous linear right–inverse
(2.33)e (k +1):k
j =0L 2k ?j +1/2(Σ,E Σ)?
→L 2k +1(M,E ).To complete the induction consider f ∈L 2,Af ∈L 2k , f =0.By induction we
may assume f ∈L 2k .Put
(2.34)f 1:=f ?e (k +1)(0, Af,..., A k f ).
Then f ?f 1∈L 2k +1,f 1∈L 2k and A j f 1=0,j =0,...,k .Hence we may extend f
by 0to obtain ?f ∈L 2with A j ?f ∈L 2,j =0,...,k +1.From local elliptic regularity we infer ?f ∈L 2k +1,loc and thus f ∈L 2k +1.(im A Id )⊥= f ∈L 2(M,F ) A t f =0 which is known to be (or see Section
5.3below)in?nite–dimensional if dim M >1.Hence we cannot expect regularity to imply that A P is Fredholm.However,if P and the dual boundary condition for A t are regular then A P is Fredholm.
(2)Regular boundary conditions are closely related to the well-posed boundary conditions of Seeley [See69,BBW93].One of the main results in [BL01]states that for symmetric Dirac operators and symmetric boundary conditions (given by operators P with closed range)regularity and well–posedness are equivalent.
(3)It is well-known that if P has closed range then P may be replaced by an orthogonal projection with the same kernel.
In this setting the dual boundary condition can easily be computed:
Proposition 2.6.Let P ∈CL 0(Σ,E Σ),P =P 2=P ?.Then
(2.35)(A P )?=A t max ,(Id ?P )J t 0
.Proof.This follows easily from Green’s formula Lemma 2.2. We recall from [H¨o r85,De?nition 20.1.1](see also [BBW93,Remark 18.2d]):Definition 2.7.Let P ∈CL 0(Σ;E Σ,G ).We say that P de?nes a local elliptic boundary condition for our ?xed operator A (or,equivalently,we say P satis?es
the ˇSapiro-Lopatinski ˇi condition for A ),if and only if the principal symbol σ0P of
P maps the space M +y,ζisomorphically onto the ?bre G y for each point y ∈Σ
and each cotangent vector ζ∈T ?y (Σ),ζ=0.Here M +y,ζdenotes the space of
boundary values of bounded solutions u on the positive real line of the linear system
2We think of M as being a subset of an open manifold ?M to which A can be extended to as an elliptic operator.
12BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
d dt u+σ1
B(0)
(y,ζ)u=0of ordinary di?erential equations,whereσ1
B(0)
denotes the
principal symbol of the tangential operator B(0).
Remark2.8.Note that a solution of the ordinary di?erential equation d dt u+σ1
B(0)
(y,ζ)u=0is bounded if and only if the initial value u0belongs to the range of
the positive spectral projection P+(σ1
B(0)
(y,ζ))(cf.Section3below)of the matrix
σ1
B(0)(y,ζ).Hence M+
y,ζ
=im P+(σ1
B(0)
(y,ζ))and local ellipticity means thatσ0
P
maps im P+(σ1
B(0)
(y,ζ))isomorphically onto G y.
We obtain from[H¨o r85,Theorem20.1.2,Theorem20.1.7](di?erently also
along the lines of[BBW93,Theorem19.6]):
Proposition2.9.Any P satisfying theˇSapiro-Lopatinskiˇi condition for A is strongly regular and makes A P a Fredholm operator.
3.Sectorial projections of an elliptic operator
3.1.Parameter dependent ellipticity.Regarding properties of the tan-gential operator B0onΣ,it is natural to distinguish three situations of increasing generality:
(i)B0is formally self-adjoint,
(ii)B0?B t0is an operator of order zero,and
(iii)B0is the tangential operator of an elliptic operator over the whole mani-fold M.
Whereas(i)implies that the spectrum spec(B0)of B0is contained in the real axis
and(ii)that for all p∈Σandξ∈T?pΣthe principal symbolσ1
B(0)
(p,ξ)∈End(E p)
is self-adjoint,the general case(iii)a priori only implies thatσ1
B(0)
(p,ξ)has no eigenvalues on the imaginary axis i R for all p∈Σandξ∈T?pΣ\{0},as explained above after(2.24).
One may ask,what consequences can be drawn from the general property(iii) for the spectrum of B0?A?rst answer is Proposition3.3below.In fact,(iii)
contains more information than just that the principal symbolσ1
B(0)(p,ξ)has no
eigenvalues on i R.
For the convenience of the reader let us brie?y recall the notion of a(pseudo)-di?erential operator with parameter,cf.Shubin[Shu80,Section II.9].
LetΛ?C be an open conic subset,i.e,z∈Λ,r>0?rz∈Λ.For an open subset U?R n let S m(U,R n×Λ)denote the space of smooth functions
a:U×R n×Λ?→C
(x,ξ,λ)→a(x,ξ,λ)
such that for multi-indicesα,β∈Z n+,γ∈Z2+and each compact subset K?U we
have
?α
x ?β
ξ
?γ
λ
a(x,ξ,λ)
≤C K
1+|ξ|+|λ|
m?|β|?|γ|
.
We emphasize that?γ
λdenotes real partial derivatives–we do not require holomor-
phicity inλ.
In other words,S m(U,R n×Λ)are the symbols of H¨o rmander type(1,0).
We shall call a symbol a∈S m(U,R n×Λ)classical if it has an asymptotic expansion
(3.1)a～
∞
j=0
a m?j,
where a m?j∈S m?j(U,R n×Λ)with homogeneity
a m?j(rξ,rλ)=r m?j a m?j(ξ,λ)for r≥1,|ξ|2+|λ|2≥1.
THE Calder′o n PROJECTION
13
Figure 1.Construction of a closed cone Λsuch that B 0?λId is
elliptic with parameter λ∈Λ
We denote the classical symbols by CS m (U,R n ×Λ)?S m (U,R n ×Λ).
Definition 3.1.Let E Σbe a complex vector bundle of ?nite ?bre dimension N over a smooth closed manifold Σand let Λ?C be open and conic.A classical pseudodi?erential operator of order m with parameter λ∈Λis a family B (λ)∈CL m (Σ,E Σ),λ∈Λ,such that locally B (λ)is given by B (λ)u (x )=(2π)?n R n U
e i x ?y,ξ b (x,ξ,λ)u (y )dydξ
with b an N ×N matrix of functions belonging to CS m (U,R n ×Λ).
Remark 3.2.(a)A pseudodi?erential operator with parameter is more than just a map from Λto the space of pseudodi?erential operators.
(b)Our de?nition of a pseudodi?erential operator with parameter is slightly di?erent from that of Shubin,loc.cit.;however,the main results of loc.cit.do also hold for this class of operators.
The principal symbol of a classical pseudodi?erential operator B of order m
with parameter is now a smooth function σm B (x,ξ,λ)on T ?Σ×Λ\{(x,0,0)|x ∈Σ}
which is homogeneous in the following sense
σm B (x,rξ,rλ)=r m σm B (x,ξ,λ)for (ξ,λ)=(0,0),r >0.
Parameter dependent ellipticity is de?ned as invertibility of this homogeneous principal symbol.The basic example of a pseudodi?erential operator with param-eter is the resolvent of an elliptic di?erential operator.
Proposition 3.3.Let Σbe a closed manifold and let B ∈Di?1(Σ,E Σ)be a ?rst order di?erential operator.Let Λ?C be an open conic subset such that B ?λ,λ∈Λ,is parameter dependent elliptic,i.e.,for each (p,ξ)∈T ?Σ,ξ=0,and each λ∈Λthe homomorphism
σ1B (p,ξ)?λ:E p ?→E p
is invertible.Then there exists R >0such that B ?λis invertible for λ∈Λ,|λ|≥R ,and we have
(B ?λ)?1 s,s +α≤C α|λ|?1+α
for such λand 0≤α≤1.
For the proof see [Shu80,Theorem 9.3].In our situation Proposition 3.3has the following consequences:
14BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU Proposition3.4.LetΣbe a closed manifold and let
D=J x d
dx
+B x
be a?rst order elliptic di?erential operator on the collar[0,ε)×Σ.Then
(a)B0?λis parameter dependent elliptic in an open conic neighborhoodΛ
of the imaginary axis i R.
(b)B0is an operator with compact resolvent,spec B0consists of a discrete
set of eigenvalues of?nite multiplicity.At most?nitely many eigenvalues
lie on the imaginary axis i R.
For an eigenvalueλeven the generalized eigenspace
N
ker(B0?λ)N is?nite–
dimensional;note that B0is not necessarily self–adjoint.
Proof.From the ellipticity of D we infer thatσ1B(0)(p,ξ)?it is invertible for (p,ξ,t)∈T?Σ×R,(ξ,t)=(0,0).Since
(p,ξ)∈T?Σ,|ξ|=1specσ1
B(0)
(p,ξ)
is bounded in C and in view of the homogeneity we?nd an angle?>0such that
specσ1
B(0)
(p,ξ)∩Λ=?.
HereΛis as in Figure1.
(a)now follows from the previous proposition.Since B0is elliptic,its spectrum is either discrete or equals C.The previous lemma implies that B0?λis invertible forλ∈Λlarge enough.Hence we conclude that spec B0is discrete and that(b) holds.
3.2.Sectorial operators:abstract Hilbert space framework.We shall now discuss the positive respectively negative sectorial spectral projections of an elliptic di?erential operator B of?rst order on a closed manifoldΣ.We start with
a purely functional analytic discussion.
3.2.1.Idempotents in a Hilbert space.Let us brie?y summarize some facts about(not necessarily bounded)idempotents in a Hilbert space.A densely de-?ned operator P in the Hilbert space H is an idempotent if im P?D(P)and P?P=P.
Given subspaces U,V?H with
(1)U∩V={0},
(2)U+V dense in H,
the projection P U,V along U onto V is a(not necessarily bounded)idempotent and every idempotent P in H is of this form with D(P U,V)=U+V,U=ker P and V=im P.
It is easy to see that P?
U,V
=P V⊥,U⊥is also a(not necessarily densely de?ned) idempotent.Thus P U,V is closable if U⊥+V⊥is dense or,equivalently U∩V={0}.
In that case,the closure of P U,V is P U,V=P
U,V .Consequently,P U,V is closed if
and only if U,V are closed subspaces of H.
Lemma3.5.(a)Let P U,V be an idempotent in the Hilbert space H,where U,V are closed subspaces satisfying(1),(2)above.Then P U,V is bounded if and only if U+V=H.
(b)Let P=P U,V be a bounded idempotent in the Hilbert space H.Then P+Id?P?is an invertible operator.
THE Calder′o n PROJECTION15
Figure2.The contoursΓ±in the plane de?ning the sectorial
projections Q±
Denote by P ort the orthogonalization of P,i.e.,P ort=P V⊥,V is the orthogonal projection onto im P.Then we have
(3.2)
P ort=P(P+Id?P?)?1,
(3.3)
(P?)ort=(P+Id?P?)?1P.
Proof.(a)is a consequence of the Closed Graph Theorem.
(b)By(a)U,V are closed subspaces of H satisfying U∩V={0},U⊕V=H.Then Id?P?=P U⊥,V⊥.Since bounded idempotents are bounded below P U⊥maps U⊥=ker P⊥bijectively onto V and Id?P?maps U=ker(Id?P?)⊥bijectively onto V⊥.Hence P+Id?P?is invertible.Moreover,this description gives(P+ Id?P?)?1explicitly:given v∈V=im P then(P+Id?P?)?1v is the unique elementξ∈U⊥with P u=v and thus P(P+Id?P?)?1v=v.Furthermore, if v∈V⊥then(P+Id?P?)?1v is the unique elementη∈U=ker P with (Id?P?)η=v.This proves P(P+Id?P?)?1=P V⊥,V=P ort.The equality(3.3) is proved similarly.
Alternatively,one may apply(3.2)to P?to?nd(P?)ort=P?(P?+Id?P)?1. Then the claim follows from(P+Id?P?)P?=P P?=P(P?+Id?P).
Our construction of P ort is a slight modi?cation of the construction given by M.Birman and A.Solomyak and disseminated in[BBW93,Lemma12.8].
Lemma3.5(a)shows that unbounded idempotents in a Hilbert space are abun-dant.See also Example3.13below.
3.2.2.The semigroup Q±(x)of a sectorial operator.In this subsection let H be a separable Hilbert space and B a closed operator in H.
Definition3.6.We call B a weakly sectorial operator if
(1)B has compact resolvent.
(2)There exists a closed conic neighborhoodΛof i R such that spec B∩Λis
?nite and
(3.4) (B?λ)?1 =O(|λ|?α),|λ|→∞,λ∈Λ,
for some0<α≤1.
Ifα=1then we call B a sectorial operator.
16BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
We ?x a weakly sectorial operator B in the sense of De?nition 3.6and choose contours Γ±as in Figure 2.
Convention 3.7.
(a)We ?x a c >0large enough such that (3.5)spec B ∩ z ∈C |z |=c =?,and such that z ∈C |z |=c contains all eigenvalues on the imaginary
axis.
(b)We specify two complementary contours Γ±in the plane as sketched in
Figure 2with Γ+encircling,up to ?nitely many exceptions,the eigen-
values of B with imaginary part ≥0and Γ?encircling the remaining
eigenvalues.Of course,for this to be possible c has to be large enough.
Definition 3.8.
Q +(x ):=12πi Γ+e ?λx (λ?B )?1dλ,x >0,(3.6)
=Id +12πi Γ+e ?λx λ?1B (λ?B )?1dλ(3.7)
Q ?(x ):=12πi Γ?e ?λx (λ?B )?1dλ,x <0,(3.8)
=12πi Γ+e ?λx λ?1B (λ?B )?1dλ.(3.9)When the dependence on B matters we will write Q ±(x,B ).
Formulas (3.7),(3.9)are obtained by adding and subtracting λ?1inside the integral and taking into account that 0lies inside Γ+but outside Γ?.
Q ±(x )are certainly bounded operators for x >0(x <0).Heuristically,Q ±(0)should be the positive/negative sectorial spectral projection of B ,obtained from holomorphic functional calculus.However,Q ±(0)is not de?ned everywhere.To avoid ambiguities,we shall keep to the following two rigorous de?nitions instead of dealing directly with Q ±(0).Definition 3.9.We put D (P +,0):= ξ∈H lim x →0+
Q +(x )ξexists and P +,0ξ:=lim x →0+
Q +(x )ξfor ξ∈D (P +,0).P ?,0is de?ned analogously using Q ?(x ).(3.7),(3.9)and the estimate (3.4)imply that D (B )?D (P +,0)and for ξ∈D (B )we have P +,0ξ=ξ+12πi Γ+λ?1(λ?B )?1dλ(Bξ)(3.10)P ?,0ξ=12πi Γ?
λ?1(λ?B )?1dλ(Bξ),(3.11)thus P ±,0is densely de?ned (D (B )is indeed a core for P ±,0).Note that Q ±(x,B )?=Q ±(x,B ?)(cf.Prop. 3.11),hence the densely de?ned operator P ±,0(B ?)is con-tained in P ±,0(B )?.Thus P ±,0is closable.Its closure will be denoted by P ±.
Definition 3.10.The closure of P ±,0will be called the positive/negative sec-torial spectral projection P ±of B .
Proposition 3.11.For x,y >0we have
(a)Q +(x,B )?=Q +(x,B ?),Q ?(?x,B )?=Q ?(?x,B ?).
(b)Q +(x )Q +(y )=Q +(x +y ).
(c)Q +is di?erentiable and Q +(x )=?BQ +(x ).
(d)Q +(x )Q ?(?y )=Q ?(?y )Q +(x )=0.
THE Calder′o n PROJECTION
17
Figure 3.A two-component contour Γ,separating an inner sec-tor around the real axis where all eigenvalues of B 0show up,from
two outer sectors which totally belong to the resolvent set of B 0
(e)P +Q +(x )?Q +(x )P +,P +Q ?(?x )=0.
Proof.The proof is straightforward and analogous to the proof in [Pon06]of the fact that P +is an idempotent. Corollary 3.12.P ±are complementary,i.e.,P +=Id ?P ?,(possibly un-bounded)idempotents in H .
Proof.Since D (B )is a core for P ±it su?ces to check that for ξ∈D (B )we
have P 2±ξ=P ±ξand (P ++P ?)ξ=ξ.
If ξ∈D (B )then using Proposition 3.11we ?nd
Q +(x )P +ξ=lim y →0+Q +(x )Q +(y )ξ=lim y →0+
Q +(x +y )ξ=Q +(x )ξ,(3.12)hence P +ξ∈D (P +,0)?D (P +)and P 2+ξ=P +ξ.To prove P +=Id ?P ?we take a ξ∈D (B )and ?nd (P ++P ?)ξ=ξ+12πi Γ
λ?1(λ?B )?1dλ(Bξ),where Γis chosen as in Figure 3.Pushing the radius of the circle arches to ∞shows (P ++P ?)ξ=ξ.
The fact that the sectorial projections are a priori unbounded operators may seem strange.The following example shows that the phenomenon really occurs:
Example 3.13.Let D be a discrete self–adjoint positive de?nite operator in H .I.e.,there is an orthonormal basis (e n )n ∈N of H such that De n =λn e n ,where 0<λ1≤λ2≤...→∞.
Pick a parameter 0≤α≤1and de?ne the operator B in H ⊕H as follows:D (B ):= (u,v )∈H ⊕H v ∈D (D ),Du ?D 2?αv ∈H ,B (u,v ):=(Du ?D 2?αv,?Dv ).
(3.13)One immediately checks that for λ∈spec D ∪?spec D the resolvent of B is given by
(3.14)(B ?λ)?1(ξ,η)=((D ?λ)?1ξ?2(D ?λ)?1D 2?α(D +λ)?1η,?(D +λ)?1η).
18BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
Because of 0≤α≤1the resolvent is indeed bounded.Furthermore,(3.14)shows that outside a conic neighborhood of the real axis,equivalently in a conic neigh-borhood of the i R ,we have an estimate
(3.15) (B ?λ)?1 =O (|λ|?α),|λ|→∞.
Hence,if 0<α≤1then B is a weakly sectorial operator in the sense of De?nition
3.6,spec B =spec D ∪?spec D and the positive/negative spectral subspaces of B are given by
im P +(B )=H ⊕0,ker P +(B )= (u,D α?1u ) u ∈H =Graph(D α?1).
(3.16)Consequently,if 0<α<1then im P +(B )⊕ker P +(B )is not closed and hence the positive sectorial projection P +(B )is not bounded.
We leave it as an intriguing problem to ?nd an example of a sectorial operator with decay rate α=1in (3.4)such that P +is unbounded.
3.3.Sectorial operators:parametric elliptic di?erential operators.
3.3.1.The geometric situation.We return to our geometric situation and con-sider the tangential operator B (previously denoted by B (0)or B 0,for convenience we omit (0)as long as we do not need B (x ))of an elliptic di?erential operator A on a compact manifold with boundary,cf.Sec.2,in particular (2.23).
Then it is known that the positive sectorial projection is bounded:
Theorem 3.14.Let B be a ?rst order elliptic di?erential operator on the closed manifold Σ.Furthermore,assume that B ?λis parametric elliptic in an open conic neighborhood Λof i R .Then the positive/negative sectorial projections P ±of B are pseudodi?erential operators of order 0.In particular P ±acts as bounded operator in each Sobolev space L 2s (Σ,E Σ).
The proof is an adaption of the classical complex power construction of See-ley [See67].See Burak [Bur70],Wojciechowski [Woj85],and recently Ponge
[Pon06].
We also note that it follows from Proposition 3.3that B is a sectorial operator in the sense of De?nition 3.6.Also,recall from loc.cit.the resolvent estimate:
For all s ∈R ,0≤α≤1,we have
(3.17)
sup λ∈Γ±|λ|1?α (λ?B )?1 s,s +α≤C (s,α),
where · s,s +αdenotes the operator norm between the Sobolev spaces L 2s (Σ,E Σ)
and L 2s +α(Σ,E Σ),see also the following remark.
Here and in the following we shall denote the closed interval [0,∞)by R +.Similarly Z +:={0,1,2,3,...}.
Remark 3.15.(a)We recall that (cf.e.g.[BL01,Cor.2.20])
(3.18)L 2s (R +×Σ,E Σ)=L 2s (R +,L 2(Σ,E Σ))∩L 2(R +,L 2s (Σ,E Σ)),s ≥0.
In particular,if s ∈Z +then a Sobolev norm for L 2s (R +×Σ,E Σ)is given by (3.19) f 2L 2s (R +×Σ,E Σ)= ∞0
?s x f (x ) 2L 2(Σ,E Σ)+ (Id +|B |)s f (x ) 2L 2(Σ,E Σ)dx.Since the spaces L 2s (...)
have the interpolation property ([Tay96,Sec.4.2],[BL01,Sec.2])for s ≥0,it will be su?cient in most cases to deal with integer s ∈Z +.(b)Note that since B is elliptic,the Sobolev norms on sections of E Σcan be de?ned using B ,i.e.,
(3.20) ξ 2L 2s (Σ,E Σ)= (Id +|B |)s ξ 2L 2(Σ,E Σ).
THE Calder′o n PROJECTION 19
(c)Whenever it is clear from the context whether we are taking norms of sections over R +×Σor over Σwe will denote Sobolev norms of order s by a subscript s .
Furthermore,norms of operators from L 2s to L 2s will be denoted by · s,s ,and
· ∞denotes the sup-norm of a function.
3.3.2.Mapping properties of Q +.The following Proposition will be useful for the study the mapping properties of the invertible double and of the remainder terms in the construction of the Poisson operator and the Calder′o n projector,see Subsections 5.2and 5.3.
The following Proposition establishes a weak convergence of Q +(x )→P +,x →0+,in compensation for the generally not valid convergence in the operator norm.
Proposition 3.16.Let ?∈C ∞0(R +),m ∈Z +.
(a)For s ∈R the operator (3.21)id m R +
?Q +:ξ→ x →x m ?(x )Q +(x )ξ maps L 2s (Σ,E Σ)continuously to L 2comp (R +,L 2s +m +1/2(Σ,E Σ)).
(b)For s ≥?1/2it maps continuously to L 2s +m +1/2,comp (R +×Σ,E Σ).
Proof.Let us ?rst prove the claim (b).It is fairly easy to see that id m R +
?Q +maps L 2s (Σ,E Σ)continuously to L 2comp (R +,L 2s (Σ,E Σ))for some s .Thus once we have proved that then range of id m R +?Q +is contained in the space
L 2comp (R +,L 2s +m +1/2(Σ,E Σ))the continuity will follow from the Closed Graph The-
orem.
Furthermore,since id m R +?Q +commutes with B it su?ces to prove the claim for s large enough:namely,we pick a λ0in the resolvent set of B .Then for arbitrary s ≥?1/2we choose k large enough such that the claim holds for s +k .The claim for s now follows from the identity (3.22)id m R +?Q +|L 2s =(λ0?B )k id m R +?Q +|L 2s +k (λ0?B )?k |L 2s .Finally,by complex interpolation (cf. e.g.[Tay96,Sec. 4.2])it su?ces therefore to consider s =n +1/2,n ∈Z +.Now pick ξ∈L 2n +1/2(Σ,E Σ)and
put f (x ):=x m ?(x )Q +(x )ξ.It is straightforward to check that f is smooth on
(0,∞)×Σ.From
(3.23)(d dx +B )j f (x )= d dx
j x m ?(x ) Q +(x )ξwe infer by the boundedness of P +(B )(according to Theorem 3.14)that
(d dx +B )j f x =0= 0,j =0,...,m ?1, d dx j |x =0x m ?(x )P +(B )ξ∈L 2n +1/2(Σ,E Σ),j ≥m,
∈L 2s +m +1/2?j ?1/2(Σ,E Σ).
(3.24)From (an obvious adaption of)[BL01,Cor. 2.17](cf.also Remark 2.5)we infer that f ∈L 2s +m +1/2(R +×Σ,E Σ).Hence (b)is proved.
For s ≥?1/2the claim (a)follows from (b).For arbitrary s we again conjugate by (λ0?B )k as above and we reach the conclusion.
Remark 3.17.The claim of the previous proposition also follows by applying more sophisticated pseudodi?erential techniques (cf.Grubb [Gru96,Thm.2.5.7]).Our proof only uses the basic trace results for Sobolev spaces,the ellipticity of B ,and the boundedness of the positive sectorial projection on Sobolev spaces.The
20BERNHELM BOOSS-BA VNBEK,MATTHIAS LESCH,AND CHAOFENG ZHU
previous proposition can therefore be generalized to situations where pseudodi?er-ential techniques are not necessarily available anymore.An abstract version is as follows (see also Subsection 5.1where scales of Hilbert spaces are recalled to some extent):
Proposition 3.18.Let B be a sectorial operator in a Hilbert space.Let H s :=D ((B ?B )s/2),s ≥0,be the scale of Hilbert spaces of B ?B and H
s :=D ((BB ?)s/2)be the scale of Hilbert spaces of BB ?.For negative s the spaces H s and H
s are de?ned by duality (cf.[BL01,Sec.2.A]).Furthermore,put for s ≥0
H s (R +,H ?):= 0≤t ≤s
L 2s (R +,H s ?t )
(cf.[BL01,Sec.2,Prop.2.10]for other descriptions).
Assume that the positive sectorial projection P +of B maps H s continuously to H
s for all s .Let ?∈C ∞0(R +),m ∈Z +.Then
(a)For s ∈R the operator
(3.25)id m R +
?Q +:ξ→ x →x m ?(x )Q +(x )ξ maps H s (R +,H ?)continuously to L 2comp (R +, H s +m +1/2).
(b)For s ≥?1/2it maps continuously to H s +m +1/2,comp (R +, H
?).For elliptic pseudodi?erential operators the distinction between H s and H
s is,of course,unnecessary.For general unbounded operators,however,we cannot expect H s = H
s . 4.The invertible double
We return to the set–up described in Subsection 2.2and give a construction of the invertible double of a general ?rst order elliptic di?erential operator.
4.1.The construction of ?A
P (T ).We introduce the operator (4.1)?A
:=A ⊕(?A t ):C ∞(M,E ⊕F )→C ∞(M,F ⊕E ).We are going to consider a special class of boundary conditions for ?A
:Definition 4.1.Let T ∈CL 0(Σ;E Σ,F Σ)be a classical pseudodi?erential op-erator of order 0,acting from sections of E Σto sections of F Σ.We put (4.2)P (T )= ?T Id ∈CL 0(Σ;E Σ⊕F Σ,F Σ).
Viewed as an operator in L 2s (Σ,E Σ⊕F Σ)the operator P (T )has closed range which equals
(4.3)im P (T )=L 2s (Σ,F Σ)?L 2s (Σ,E Σ⊕F Σ).
Since this is a closed subspace of L 2s (Σ,E Σ⊕F Σ),the boundary condition for ?A
given by P (T )can be realized by a pseudodi?erential orthogonal projection,as noted in Remark 2.5.3.
To be more speci?c we recall that the realization A
P (T )of A with respect to the boundary condition P (T )has domain (4.4)D ( A P (T )):= f +f ? ∈L 21(M,E ⊕F ) f ?=T f + .
对翻译中异化法与归化法的正确认识 班级：外语学院、075班 学号：074050143 姓名：张学美 摘要：运用异化与归化翻译方法,不仅是为了让读者了解作品的内容,也能让读者通过阅读译作,了解另一种全新的文化,因为进行文化交流才是翻译的根本任务。从文化的角度考虑,采用异化法与归化法,不仅能使译文更加完美,更能使不懂外语的人们通过阅读译文,了解另一种文化,促进各民族人们之间的交流与理解。翻译不仅是语言符号的转换,更是跨文化的交流。有时,从语言的角度所作出的译文可能远不及从文化的角度所作出的译文完美。本文从翻译策略的角度，分别从不同时期来说明人们对异化法与归化法的认识和运用。 关键词:文学翻译；翻译策略；异化；归化；辩证统一 一直以来,无论是在我国还是在西方,直译(literal translation)与意译(liberal translation)是两种在实践中运用最多,也是被讨论研究最多的方法。1995年,美籍意大利学者劳伦斯-韦努蒂(Lawrence Venuti)提出了归化(domestication)与异化(foreignization)之说,将有关直译与意译的争辩转向了对于归化与异化的思考。归化与异化之争是直译与意译之争的延伸,是两对不能等同的概念。直译和意译主要集中于语言层面,而异化和归化则突破语言的范畴,将视野扩展到语言、文化、思维、美学等更多更广阔的领域。 一、归化翻译法 Lawrwnce Venuti对归化的定义是,遵守译入语语言文化和当前的主流价值观,对原文采用保守的同化手段,使其迎合本土的典律,出版潮流和政治潮流。采用归化方法就是尽可能不去打扰读者,而让作者向读者靠拢(the translator leaves the reader in peace, as much as possible, and moves the author towards him)。归化翻译法的目的在于向读者传递原作的基本精神和语义内容,不在于语言形式或个别细节的一一再现。它的优点在于其流利通顺的语言易为读者所接受,译文不会对读者造成理解上的障碍,其缺点则是译作往往仅停留在内容、情节或主要精神意旨方面,而无法进入沉淀在语言内核的文化本质深处。 有时归化翻译法的采用也是出于一种不得已,翻译活动不是在真空中进行的,它受源语文化和译语文化两种不同文化语境的制约,还要考虑到两种文化之间的
“异化”与“归化”之间的关系并评述 1、什么是归化与异化 归化”与“异化”是翻译中常面临的两种选择。钱锺书相应地称这两种情形叫“汉化”与“欧化”。A．归化 所谓“归化”（domestication 或target-language-orientedness），是指在翻译过程中尽可能用本民族的方式去表现外来的作品；归化翻译法旨在尽量减少译文中的异国情调，为目的语读者提供一种自然流畅的译文。Venuti 认为，归化法源于这一著名翻译论说，“尽量不干扰读者，请作者向读者靠近” 归化翻译法通常包含以下几个步骤：（1）谨慎地选择适合于归化翻译的文本；（2）有意识地采取一种自然流畅的目的语文体；（3）把译文调整成目的语篇体裁；（4）插入解释性资料；（5）删去原文中的实观材料；（6）调协译文和原文中的观念与特征。 B．“异化”（foreignization或source-language-orientedness）则相反，认为既然是翻译，就得译出外国的味儿。异化是根据既定的语法规则按字面意思将和源语文化紧密相连的短语或句子译成目标语。例如，将“九牛二虎之力”译为“the strength of nine bulls and two tigers”。异化能够很好地保留和传递原文的文化内涵，使译文具有异国情调，有利于各国文化的交流。但对于不熟悉源语及其文化的读者来说，存在一定的理解困难。随着各国文化交流愈来愈紧密，原先对于目标语读者很陌生的词句也会变得越来越普遍，即异化的程度会逐步降低。 Rome was not built in a day. 归化:冰冻三尺,非一日之寒. 异化:罗马不是一天建成的. 冰冻三尺,非一日之寒 异化:Rome was not built in a day. 归化:the thick ice is not formed in a day. 2、归化异化与直译意译 归化和异化，一个要求“接近读者”，一个要求“接近作者”，具有较强的界定性；相比之下，直译和意译则比较偏重“形式”上的自由与不自由。有的文中把归化等同于意译，异化等同于直译，这样做其实不够科学。归化和异化其实是在忠实地传达原作“说了什么”的基础之上，对是否尽可能展示原作是“怎么说”，是否最大限度地再现原作在语言文化上的特有风味上采取的不同态度。两对术语相比，归化和异化更多地是有关文化的问题，即是否要保持原作洋味的问题。 3、不同层面上的归化与异化 1、句式 翻译中“归化”表现在把原文的句式（syntactical structure）按照中文的习惯句式译出。
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翻译的归化与异化 作者：熊启煦 作者单位：西南民族大学,四川,成都,610041 刊名： 西南民族大学学报（人文社科版） 英文刊名：JOURNAL OF SOUTHWEST UNIVERSITY FOR NATIONALITIES（HUMANITIES AND SOCIAL SCIENCE） 年，卷(期)：2005,26(8) 被引用次数：14次 参考文献(3条) 1.鲁迅且介亭杂文二集·题未定草 2.刘英凯归化--翻译的歧路 3.钱钟书林纾的翻译 引证文献(15条) 1.郭锋一小议英语翻译当中的信达雅[期刊论文]-青春岁月 2011(4) 2.许丽红论汉英语言中的文化差异与翻译策略[期刊论文]-考试周刊 2010(7) 3.王笑东浅谈汉英语言中的差异与翻译方法[期刊论文]-中国校外教育（理论） 2010(6) 4.王宁中西语言中的文化差异与翻译[期刊论文]-中国科技纵横 2010(12) 5.鲍勤.陈利平英语隐喻类型及翻译策略[期刊论文]-云南农业大学学报(社会科学版) 2010(2) 6.罗琴.宋海林浅谈汉英语言中的文化差异及翻译策略[期刊论文]-内江师范学院学报 2010(z2) 7.白蓝跨文化视野下文学作品的英译策略[期刊论文]-湖南社会科学 2009(5) 8.王梦颖探析汉英语言中的文化差异与翻译策略[期刊论文]-中国校外教育（理论） 2009(8) 9.常晖英汉成语跨文化翻译策略[期刊论文]-河北理工大学学报（社会科学版） 2009(1) 10.常晖对翻译文化建构的几点思考[期刊论文]-牡丹江师范学院学报（哲学社会科学版） 2009(4) 11.常晖认知——功能视角下隐喻的汉译策略[期刊论文]-外语与外语教学 2008(11) 12.赵勇刚汉英语言中的文化差异与翻译策略[期刊论文]-时代文学 2008(6) 13.常晖.胡渝镛从文化角度看文学作品的翻译[期刊论文]-重庆工学院学报(社会科学版) 2008(7) 14.曾凤英从文化认知的视角谈英语隐喻的翻译[期刊论文]-各界 2007(6) 15.罗琴.宋海林浅谈汉英语言中的文化差异及翻译策略[期刊论文]-内江师范学院学报 2010(z2) 本文链接：https://www.doczj.com/doc/0e983039.html,/Periodical_xnmzxyxb-zxshkxb200508090.aspx
翻译作业10 Nov 15 一、请按归化法(Domestication)翻译下列习语。 Kill two birds with one stone a wolf in sheep’s clothing strike while the iron is hot. go through fire and water add fuel to the flames / pour oil on the flames spring up like mushrooms every dog has his day keep one’s head above water live a dog’s life as poor as a church mouse a lucky dog an ass in a lion’s skin a wolf in sheep’s clothing Love me, love my dog. a lion in the way lick one’s boots as timid as a hare at a stone’s throw as stupid as a goose wet like a drown rat as dumb as an oyster lead a dog’s life talk horse One boy is a boy, two boys half a boy, and three boys nobody. Man proposes, God disposes. Cry up wine and sell vinegar (cry up, to praise; extol: to cry up one's profession) Once bitten, twice shy. An hour in the morning is worth two in the evening． New booms sweep clean. take French leave seek a hare in a hen’s nest have an old head on young shoulder Justice has long arms You can’t teach an old dog Rome was not built in a day. He that lives with cripples learns to limp. Everybody’s business is nobody’s business. The more you get, the more you want. 二、请按异化法(foreignization)翻译下列习语。 Kill two birds with one stone a wolf in sheep’s clothing
归化和异化这对翻译术语是由美国著名翻译理论学家劳伦斯韦努蒂（Lawrence Venuti）于1995年在《译者的隐身》中提出来的。 归化：是要把源语本土化，以目标语或译文读者为归宿，采取目标语读者所习惯的表达方式来传达原文的内容。归化翻译要求译者向目的语的读者靠拢，译者必须像本国作者那样说话，原作者要想和读者直接对话，译作必须变成地道的本国语言。归化翻译有助于读者更好地理解译文，增强译文的可读性和欣赏性。 异化：是“译者尽可能不去打扰作者，让读者向作者靠拢”。在翻译上就是迁就外来文化的语言特点，吸纳外语表达方式，要求译者向作者靠拢，采取相应于作者所使用的源语表达方式，来传达原文的内容，即以目的语文化为归宿。使用异化策略的目的在于考虑民族文化的差异性、保存和反映异域民族特征和语言风格特色，为译文读者保留异国情调。 作为两种翻译策略，归化和异化是对立统一，相辅相成的，绝对的归化和绝对的异化都是不存在的。在广告翻译实践中译者应根据具体的广告语言特点、广告的目的、源语和目的语语言特点、民族文化等恰当运用两种策略，已达到具体的、动态的统一。 归化、异化、意译、直译 从历史上看，异化和归化可以视为直译和意译的概念延伸，但又不完全等同于直译和意译。直译和意译所关注的核心问题是如何在语言层面处理形式和意义，而异化和归化则突破了语言因素的局限，将视野扩展到语言、文化和美学等因素。按韦努蒂（Venuti）的说法，归化法是“把原作者带入译入语文化”，而异化法则是“接受外语文本的语言及文化差异，把读者带入外国情景”。（Venuti，1995:20）由此可见，直译和意译主要是局限于语言层面的价值取向，异化和归化则是立足于文化大语境下的价值取向，两者之间的差异是显而易见的，不能混为一谈。 归化和异化并用互补、辩证统一 有些学者认为归化和异化，无论采取哪一种都必须坚持到底，不能将二者混淆使用。然而我们在实际的翻译中，是无法做到这么纯粹的。翻译要求我们忠实地再现原文作者的思想和风格，而这些都是带有浓厚的异国情调的，因此采用异化法是必然；同时译文又要考虑到读者的理解及原文的流畅，因此采用归化法也是必然。选取一个策略而完全排除另一种策略的做法是不可取的，也是不现实的。它们各有优势，也各有缺陷，因此顾此失彼不能达到最终翻译的目的。 我们在翻译中，始终面临着异化与归化的选择，通过选择使译文在接近读者和接近作者之间找一个“融会点”。这个“融会点”不是一成不变的“居中点”，它有时距离作者近些，有时距离读者近些，但无论接近哪
姓名：徐中钧上课时间：T2 成绩： 翻译中的归化与异化 归化与异化策略是我们在翻译中所采取的两种取向。归化是指遵从译出语文化的翻译策略取向，其目的是使译文的内容和形式在读者对现实了解的知识范围之内，有助于读者更好地理解译文，增强译文的可读性。异化是指遵从译入语文化的翻译策略取向，其目的是使译文保存和反映原文的文化背景、语言传统，使读者能更好地了解该民族语言和文化的特点。直译和意译主要是针对的是形式问题，而归化和异化主要针对意义和形式得失旋涡中的文化身份、文学性乃至话语权利的得失问题，二者不能混为一谈。 在全球经济一体化趋势日益明显的今天，人们足不出门就能与世界其他民族的人进行交流。无论是国家领导人会晤、国际经济会议，还是个人聚会、一对一谈话，处于不同文化的两个民族都免不了要进行交流。如果交流的时候不遵从对方的文化背景，会产生不必要的误会，造成不必要的麻烦。在2005年1月20 日华盛顿的就职典礼游行活动中，布什及其家人做了一个同时伸出食指和小指的手势。在挪威，这个手势常常为死亡金属乐队和乐迷使用，意味着向恶魔行礼。在电视转播上看到了这一画面的挪威人不禁目瞪口呆，这就是文化带来的差异。人们日常生活中的习语、修辞的由于文化差异带来的差别更是比比皆是，如以前的“白象”电池的翻译，所以文化传播具有十分重要的意义。由此可见，借助翻译来传播文化对民族间文化的交流是很有必要的。 一般来说，采取归化策略使文章变得简单易懂，异化使文章变得烦琐难懂。归化虽然丢掉了很多原文的文化，但使读者读起来更流畅，有利于文化传播的广度。采取异化策略虽然保存了更多的异族文化，能传播更多的异族文化，有利于文化传播的深度，但其可读性大大降低。归化和异化都有助于文化的传播，翻译时应注意合理地应用归化与异化的手段。对于主要是表意的译文，应更多地使用归化手段，反之则更多地使用异化手段。 要使翻译中文化传播的效果达到最好，译者应该考虑主要读者的具体情况、翻译的目的、译入文的内容形式等具体情况而动态地采取归化与异化的手段。 文章字符数：920 参考文献： 论异化与归化的动态统一作者：张沉香（即讲义） 跨文化视野中的异化/归化翻译作者：罗选民https://www.doczj.com/doc/0e983039.html,/llsj/s26.htm
【摘要】《法国中尉的女人》自1969年面世以来，凭借着高度的艺术价值，吸引了国内外学者的眼球。在中国市场上也流传着此书的多个译本，本文就旨在透过《法国中尉的女人》中的两处题词的译作分析在具体翻译文本中如何对待归化与异化的问题。 【关键词】《法国中尉的女人》；题词；归化；异化 一、归化与异化 在翻译领域最早提出归化与异化两个词的学者是美国翻译学者韦努蒂。归化一词在《翻译研究词典》中的定义为：归化指译者采用透明、流畅的风格以尽可能减弱译语读者对外语语篇的生疏感的翻译策略。异化一词的定义为：指刻意打破目的语的行文规范而保留原文的某些异域特色的翻译策略。韦努蒂主张在翻译的过程中更多的使用异化，倡导看起来不通顺的译文，注重突出原文的异域风格、语言特色与文化背景。 二、如何看待归化与异化 鲁迅说对于归化与异化，我们不能绝对的宣称哪种是绝对的好的，另一种是绝对的不好的。翻译的过程中归化与异化是相辅相成互相补充的，具体使用哪种方法不仅需要依据翻译目的以及译文的读者群来判断。这也正是本文对待归化异化的一个态度。下面就通过分析刘蔺译本与陈译本在处理《法国中尉的女人》第五章的引言丁尼生《悼念集》时的例子来具体分析这一点。对于原文 at first as death， love had not been， or in his coarsest satyr-shape had bruised the herb and crush’d the grape， 刘蔺译本将此处翻译为 啊，天哪，提这样的问题 又有保益？如果死亡 首先意味着生命了结， 那爱情，如果不是 在涓涓细流中戛然中止， 就是一种平庸的友情， 或是最粗野的色迷 在树林中肆意饕餮， 全不顾折断茎叶， 揉碎葡萄―― 而此时陈译本却将此诗译作 哦，我啊，提出一个无益的问题 又有何用？如果人们认为死亡 就是生命的终结，那么，爱却不是这样， 否则，爱只是在短暂的空闲时 那懒散而没激情的友谊， 或者披着他萨梯粗犷的外套 已踩伤了芳草，并摧残了葡萄， 在树林里悠闲自在，开怀喝吃。 两篇译作各有千秋，从这首诗整体着眼刘蔺译本更趋向于归化手法，陈译本更倾向于异化处理。刘蔺译本将satyr一词译色迷采用了归化的技巧，而陈译本却将satyr一词译为萨梯，采用了异化的手法。satyr原指人羊合体的丛林之神，其实刘蔺译本与陈译本在处理satyr 时采用不同的方法就是因为翻译目的及读者文化群不同引起的。1986年出的刘蔺译本出版时，
中国旅游文本汉英翻译的归化和异化
中国旅游文本汉英翻译的归化和异化 内容摘要：自改革开放政策执行以来，我国旅游业一直呈现迅速繁荣发展趋势，旅游业在我国国民经济发展中扮演着十分重要的角色。与此同时，旅游类翻译已经被视为一条将我国旅游业引往国际市场的必经之路。为了能够让国外游客更清楚地了解中国的旅游胜地，我们应该对旅游翻译加以重视。尽管国内的旅游资料翻译如同雨后春笋般涌现，但在旅游翻译中仍存在许多问题，尤其是翻译策略的选择问题。归化与异化作为处理语言形式和文化因素的两种不同的翻译策略，无疑对旅游翻译的发展做出了巨大贡献。本文将重点讨论在汉英旅游翻译中，译者该如何正确的处理归化和异化问题，实现归化和异化的有机结合，从而达到理想的翻译水平。从归化与异化的定义、关系、以及归化和异化在旅游翻译中的应用与意义进行论述。 关键词：旅游文本归化翻译异化翻译 关于翻译中归化和异化的定义，国内外的专家学者已经做了很多的描述。为了使得本文读者更好地了解什么是翻译中的归化和异化，作者将介绍这两种翻译方法的关键不同，包括两者的定义以及两者之间的关系。
从他们的定义，很容易找到归化和异化之间的差异。通过对比，这两种策略有各自的目的、优点和语言样式。归化是指翻译策略中，应使作为外国文字的目标语读者尽可能多的感“流畅的风格，最大限度地淡化对原文的陌生感的翻译策略”（Shuttleworth & Cowie，1997：59）。换句话说，归化翻译的核心思想是译者需要更接近读者。归化的目的在于向读者传递最基本精神和内容含义。这种翻译策略明显的优势就是读者可以很容易地接受它流畅的语言风格。例如，济公是个中国神话中的传奇人物。不过，外国人可能没有在这种明确的翻译方式下获得信息。因此，适合的翻译方法是找出西方国家中类似济公这样的人物。因此最终的翻译为：济公是一个中国神话中的传奇人物（类似于西方文化中的罗宾汉）。这样的翻译更容易被目的语读者所接受。“异化是指在在翻译策略中，使译文打破目地语的约束，迁就外来文化的语言特点以及文化差异。”（Venuti，2001：240）这一战略要求“译者尽可能不去打扰作者，让读者向作者靠拢。”（Venuti，1995：19）归化可能保留源语的主要信息。异化的目的是使读者产生一种非凡的阅读体验。异化可以容纳外语的表达特点，将自身向外国文化靠拢，并将特殊的民族文化考虑在内。例如，如果将“岳飞庙和墓”翻译成英语，可以翻译为：“Yue Fei’s
2010届本科生毕业设计（论文） 毕业论文 Domestication or Foreignism-oriented Skills in Translation 学院：外国语学院 专业：姓名：指导老师：英语专业 xx 学号： 职称： xx 译审 中国·珠海 二○一○年五月
xx海学院毕业论文 诚信承诺书 本人郑重承诺：我所呈交的毕业论文Domestication or Foreignism-oriented Skills in Translation 是在指导教师的指导下，独立开展研究取得的成果，文中引用他人的观点和材料，均在文后按顺序列出其参考文献，论文使用的数据真实可靠。 承诺人签名： 日期：年月日
Domestication or Foreignism-oriented Skills in Translation ABSTRACT Translation, a bridge between different languages and cultures, plays an indispensable role in cross-culture communication. However, as a translator, we have to choose which strategy to deal with the cultural differences between the source language and the target language in the process of translation where there exists two major translation strategies--- domestication and foreignization. In this thesis, I will discuss these strategies and their application from translation, linguistics, and cross-cultural communication perspectives. In the thesis, I will first talk about the current research of the domestication and foreignization in the translation circle as well as point out the necessity for further research. Then, I will illustrate the relationship between linguistics and translation as well as between culture and translating and next systematically discuss these two translation strategies including their definitions, the controversy in history and their current studies. Besides, I will continue to deal with such neglected factors as may influence the translator’s choice of translation strategies: the type of the source text (ST), the translation purposes, the level of the intended readers, the social and historical background, the translator’s attitude and so on. Finally, from the linguistic and cultural perspectives, the thesis makes a comparative study of the application of domestication and foreignization by analyzing typical examples from the two English versions of Hong Lou Meng translated by Yang Xianyi and David Hawkes respectively. The thesis will conclude that these two translation strategies have their respective features and applicable value. I sincerely hope that this research into translation strategies will enlighten translators and make a little contribution to the prosperity of