Duality by reproducing kernels

PII. S0161171203206037 http://ijmms.hindawi.com © Hindawi Publishing Corp.

DUALITY BY REPRODUCING KERNELS

A. SHLAPUNOV and N. TARKHANOV

Received 10 June 2002

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write᏿A(Ᏸ)for the space of solutions of the systemAu=0 in a domain ᏰX. Using reproducing kernels related to various Hilbert struc-tures on subspaces of᏿A(Ᏸ), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary ofᏰ, we spec-ify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known exam-ple being the ¯∂-Neumann problem. The duality itself takes place only for those domainsᏰwhich possess certain convexity properties with respect toA. 2000 Mathematics Subject Classification: 32C37, 58J10, 35N15, 46A20, 46E22.

1. Introduction. The present work continues our paper [12] and grows out of a desire to highlight the link of duality theorems for solutions of elliptic equations and basic problems of partial differential equations such as the ex-istence and regularity of solutions.

Let

V0 T→V1, V0←S V1 (1.1)

be two continuous mappings of Fréchet spaces such thatST=I on a closed subspaceU0_ofV0_{. In other words, the identity mapping ofU}0_{factors through} T, hence the restriction ofT toU0_{is one-to-one and the image ofU}0_underT is a closed subspace ofV1_.

Obviously,S mapsT U0_toU0_{. IfᏲis a continuous linear functional onU}0_, then

Ᏺ, u = SᏲ, T u (1.2)

for anyu∈U0_{, whereSis the transpose ofS:T U}0_→U0_{. Moreover,SᏲ=0}

impliesᏲ=0. We thus obtain a one-to-one mapping(U0_{)→(T U}0_{)given by} ᏲSᏲ. The problem of identifying the dual ofU0_{reduces to the description} of the range ofS.

differential equations. The crucial fact is that by the uniqueness theorem,C∞ -functions with compact support are not dense inU0_{, and so the functionals} of(U0_{)cannot be specified within distributions. To handleS, we therefore}

need much more refined analysis. For a deeper discussion, we refer the reader to [6] and [23, Chapter 3].

It is usually the case for hypoelliptic equations thatU0_{is in fact a nuclear} space. By the Schwartz kernel theorem, the mappingS:T U0_→U0_{has a kernel} KS∈U0⊗πˆ (T U0)(cf. [22, Section 1.4.1]). We callKS areproducing kernel for

u(x)=KS(x,·), T u

(1.3)

for allxin the domain ofu∈U0_.

The advantage of using reproducing kernels lies in the fact that it enables us to writeSᏲ= Ᏺ, KS(·, y). The right-hand side here is called theindicatrix of the functionalᏲ.

This concept was first studied in the particular cases of holomorphic and harmonic functions on certain explicitly given domains (cf. [2, 8,25]). How-ever, representing analytic functionals as analytic functions on domains inCn requires specific tools of the general theory of partial differential equations (cf. [3,4,19,26] and [20, Theorem 3.4]).

Our main result consists of the following. The correspondence Ᏺ SᏲ maps the dual space ofU0_{to the space of the solutions ofAv=0 in the same} domain, which grow near the boundary in a sense dual to the growth of the solutions inU0_{. This mapping is always one-to-one but not necessarily onto. Its} surjectivity is equivalent to the regularity of certain projection onto the space of solutions.

We evaluate the projection through the resolution operator of a generalised Neumann problem related toA. The desired regularity of the projection just amounts to that of the solution of the Neumann problem. We thus bring to-gether two different areas of analysis in which the problem of regularity turns out to be of key importance.

2. A general scheme

2.1. Spaces of the solutions of elliptic systems. LetXbe aC∞-manifold of dimensionnwith a smooth boundary∂X. The case∂X= ∅is also included. We tacitly assume thatXis embedded into a smooth closed manifold ˜Xof the same dimension.

For any smoothC-vector bundles Eand F over X, we write Diffm(X;E, F ) for the space of all linear partial differential operators of order≤mbetween sections ofEandF.

We pick a volume formdx onX, thus identifying the dual and conjugate bundles. ForA∈Diffm(X;E, F ), denote byA∈Diffm(X;F∗, E∗)the transposed operator and byA∗∈Diffm(X;F , E)the formal adjoint operator. We obviously have

A∗= ∗−1

E A∗F (2.1)

(cf. [21, Section 4.1.4] and elsewhere).

Writeσm_{(A)for the principal homogeneous symbol of ordermof the} op-eratorA,σm_{(A)living on the cotangent bundleT}∗_{XofX. From now on, we}

assume thatσm_{(A)is injective away from the zero section ofT}∗_{X. Hence, it}

follows that the Laplacian∆=A∗Ais an elliptic differential operator of order 2monX.

Given any open setUinX◦, the interior ofX, let᏿A(U )stand for the space of the solutions of the equationAu=0 inU with the topology of uniform convergence on compact subsets ofU. It is known that᏿A(U )is a Fréchet-Schwartz space.

Denote by᏿A(U )the dual space of᏿A(U ), that is, the space of all continu-ous linear functionals on᏿A(U ). As usual, we give᏿A(U )the strong topology, that is, the topology of uniform convergence of functionals on bounded sub-sets of᏿A(U ).

Throughout this paper, we assume that the Laplacian∆possesses the fol-lowing unique continuation property:

(U )s given any domain Ᏸ⊂

◦

X, ifu∈᏿∆(Ᏸ)vanishes on a nonempty open

subset ofᏰ, thenu≡0 inᏰ.

This property implies in particular the existence of a two-sided fundamental solution for∆in the interior ofX.

Natural domains for the solutions ofAu=0 are certainly open subsets of the interior ofX. However, some problems require to consider solutions on sets σ in X which are not open. Here, we are interested not simply in the restrictions of the solutions of the given set, but also in the local solutions of the systemAu=0 onσ. By these, we mean solutions of the system on various neighbourhoods ofσ depending on the solution.

Ifσis a closed subset ofX, then᏿A(σ )stands for the space of (equivalence classes of) local solutions ofAu=0 onσ. Two such solutions are equivalent if there is a neighbourhood ofσ where they are equal. In᏿A(σ ), a sequence {uν}is said to converge if there exists a neighbourhoodᏺofσ such that all the solutions are defined at least inᏺ and converge uniformly on compact subsets ofᏺ.

that the mappings᏿A(Uν)→᏿A(σ )are one-to-one. Then,᏿A(σ )is necessarily a Hausdorff space.

For an open setU⊂X, we denote byL2_{(U , E)the Hilbert space of all square} integrable sections ofEoverUwith scalar product

(u, v)L2_{(U ,E)}=

U

(u, v)xdx. (2.2)

More generally, writeHs_{(U , E),s∈Z}

+, for the Sobolev space of the sections of

EoverU, whose weak derivatives up to ordersbelong toL2_{(U , E). We define} Hs_{(U , E)withs= −1,−2, . . .to be the dual space forH}−s_{(U , E)with respect to} theL2_{(U , E)-pairing.}

We also denote by ᏿(s)A (U ), with any integer s, the closed subspace of Hs_{(U , E) consisting of all weak solutions of Au=0 inU. It is well known} that᏿(s)

A (U )is a separable Hilbert space with reproducing kernel (cf. [22] and elsewhere).

The union of the spaces᏿(s)A (U )over alls∈Zis perhaps of particular in-terest. For regularU, it consists of all the solutions ofAu=0 inU, which are of finite order of growth near the boundary ofU. This means thatufulfills an estimate

_u(x)≤ C

dist(x, ∂U )N (2.3)

for allx∈U, withNandCconstants depending onu. Write᏿(f )

A (U )for the space of the solutions ofAu=0 inU which have a finite order of growth near∂U. We give᏿(f )A (U )the inductive limit topology of the sequence᏿(A−s)(U ),s∈N.

Since the Dirichlet problem for the Laplacian∆=A∗AinUis uniquely solv-able, the topology of᏿(f )

∆ (U )can be equivalently described in the following

way. Pick a Dirichlet systemut(u)of orderm−1 on the boundary ofU provided the latter is smooth. By [18], for eachu∈᏿(f )

∆ (U ), the Dirichlet data

t(u)are well defined in

m−1

j=0

Ᏸ_{∂U , F}

j, (2.4)

Fjbeing some vector bundles in a neighbourhood of∂U.

Lemma_2.1. _{A sequence}{_uν}_{converges touin the space}᏿(f )

∆ (U )if and only

if t(uν)→t(u)in m−1

j=0 Ᏸ(∂U , Fj). Proof. _{SeeTheorem 2.32.}

2.2. Duality. LetΣ1be a vector subspace of᏿A(U )endowed with topology τ1which is not weaker than the Fréchet-Schwartz topology of᏿A(U ). Denote byΣ

Suppose thatVis a separable Hilbert space of functions in a domainUwith a scalar producth(·,·). Moreover, let there be a topological vector spaceΣ2, continuous linear mappings

Σ2 i2

V i1 Σ1, (2.5)

and a sesquilinear pairing

˜

h(·,·):Σ1×Σ2 →C, (2.6)

such that

(1) ˜h(·,·)is separately continuous;

(2) ˜h(i1u, v)=h(u, i2v)for allu∈Vandv∈Σ2.

Under these assumptions, the pairing ˜h(·,·)induces a continuous mapping

᏶:Σ2→Σ1by

(᏶v)(·):=Ᏺv(·):=h(˜ ·, v) (2.7)

for anyv∈Σ2.

Lemma_2.2. _{If i2:}Σ₂→_{Vis one-to-one, then the mapping}᏶_{given by (2.7) is} one-to-one.

Proof. IfᏲ_v(·)is identically zero then

Ᏺv

i1i2v

=hi2v, i2v

=0. (2.8)

Ash(·,·)is a scalar product onV, we conclude thati2v=0. Hencev=0 ifi2 is one-to-one.

By a priori estimates for the solutions of elliptic systems, it is easy to see that the inclusion

i=i1:V →᏿A(U )=:Σ1 (2.9)

is continuous if and only if all the evaluation functionalsxu(x),x∈U, are continuous onV. This latter just amounts to saying that the Hilbert spaceVhas a reproducing kernelK(x, y)∈V⊗V, that is, anyu∈V can be represented by the formula

u(x)=hu, K(x,·) (2.10)

for allx∈U(cf. [1]).

Theorem_2.3. _{LetVbe a Hilbert space with reproducing kernelK(}·_,·_).

Sup-pose thati2is one-to-one. Then, the mapping᏶given by (2.7) is onto if and only if

(1) i1i2(Σ2)is dense inΣ1; (2) for everyᏲ∈Σ

1, the sectionx∗−V1Ᏺ, i1K(x,·)belongs toi2(Σ2). Note that

hu(y),∗Vw(x)⊗v(y)=h(u, v)w(x) (2.11)

for eachx∈Uandu, v, w∈V, as is easy to check.

Proof

Necessity. Let Ᏺ be a continuous linear functional on Σ₁ vanishing on i1i2(Σ2). By the Hahn-Banach theorem, we prove thati1i2(Σ2)is dense inΣ1 once we show thatᏲ≡0.

By assumption, there is an elementv∈Σ2such thatᏲv=Ᏺ. It follows that

Ᏺv

i1i2v

=hi2v, i2v

=0, (2.12)

and sov=0. HenceᏲ≡0 as desired. Further, an easy calculation shows that

Ᏺ, i1K(x,·)

=_h˜_{i1K(x,·), v} =hK(x,·), i2v

= ∗Vhi2v, K(x,·)

= ∗Vi2v(x) ∈ ∗Vi2Σ2,

(2.13)

the fourth equality being due to the fact thatK(·,·)is a reproducing kernel ofV. This proves the necessity.

Sufficiency. Let conditions (1) and (2) of the theorem hold. The task is now to show that the mapping᏶:Σ2→Σ1is onto.

Lemma_2.4. _Letu∈Σ_{1. Then, the formula}

u(x)=_h˜_{u, i}−1

2 i1K(x,·)

(2.14)

is valid for allx∈U.

Proof. _{Indeed, by a priori estimates for elliptic systems all evaluation} func-tionalsδx(u)=u(x), x ∈U, are continuous on Σ1. The condition (2) then implies that

∗−1 V

δx, i1K(y,·)=i1∗−V1K(y, x)=i1K(x, y)∈E∗x⊗i2Σ2 (2.15)

for every fixed x ∈ U. It follows that the pairing ˜h(u, i−1

Pick a sequence{uν}inΣ2such that{i1i2uν}approximatesuinΣ1. Since K(·,·)is a reproducing kernel, we see that

i2uν(x)=hi2uν, K(x,·) (2.16)

whence

i1i2uν(x)=hi2uν, i2i−21i1,xK(x,·)=h˜i1i2uν, i−21i1,xK(x,·) (2.17)

for allx∈Uandν=1,2, . . . .Since the pairing ˜h(·,·)is separately continuous, the passage to the limit in (2.17), when ν→ ∞, yields (2.14). The lemma is proved.

We are now in a position to complete the proof ofTheorem 2.3. Suppose thatᏲ∈Σ

1. Then byLemma 2.4, we see that

Ᏺ(u)=Ᏺ_h˜_{u, i}−1

2 i1K(x,·)

=_{h(u, v),}˜ _(2.18)

where

v=i−21

Ᏺ, i1K(·, y)=i−21∗−V1

Ᏺ, i1K(y,·)∈Σ2. (2.19)

The last reasoning is an immediate consequence of condition (2), thus show-ing the theorem.

Corollary2.5. If Σ₁is a closed subspace of᏿_A(U ), then condition (2) of Theorem 2.3is equivalent to the following one:

(2) for each fixedy∈U, the sectioni1,y∗−V1K(·, y)belongs toi2(Σ2)⊗Ey.

Proof. _{That (2) implies (2) we have already established in the proof of} Lemma 2.4. It remains to show the implication (2)⇒(2).

Pick a continuous linear functionalᏲonΣ1. SinceΣ1is a closed subspace ofCloc(U , E), the space of continuous sections ofEoverU, this functional ex-tends, by the Hahn-Banach theorem, to anE∗-valued measuremwith a com-pact support inU. For anyx∈U,

∗−1 V

Ᏺ, i1K(x,·)=

suppm

i1,y∗−V1K(x, y), dm(y)

y∈i2Σ2 (2.20)

sincei1is continuous and suppmis a compact subset ofU. This completes the proof.

Corollary2.6. Leti₂be one-to-one. Suppose that the closed graph theo-rem is valid for mappings betweenΣ2andΣ1. Then, the mapping᏶:Σ2→Σ1 defined by (2.7) is a topological isomorphism between these spaces if and only if conditions (1) and (2) ofTheorem 2.3hold.

LetS1,S2, andVbe closed subspaces of the spacesΣ1,Σ2, andV, respectively. We thus get a commutative diagram

Σ2 i2

V i1 Σ1

∪ ∪ ∪

S2 i2

V i1 S1.

(2.21)

Once again, the pairing ˜h(·,·)induces the mappingJ:S2→S1which is to a certain extent the restriction of᏶toS2. We tacitly assume that the continuous mappings under study are characterised in terms of convergent sequences and that the closed graph theorem is valid for mappings betweenΣ2andΣ1.

Writeπ:V→Vfor the corresponding orthogonal projection.

Corollary_2.7. _{Leti2be one-to-one. Suppose that}᏶_{is a topological} isomor-phism of Σ2ontoΣ1. Then, the mappingJ is a topological isomorphism ofS2 ontoS

1if and only if (1) i1i2S2is dense inS1;

(2) the projectionπ mapsi2(Σ2)continuously intoi2(S2).

Proof. Sincei₂is one-to-one, so is the restriction ofi₂toS2, too. Hence, the mappingJis one-to-one byLemma 2.2. It remains to prove that conditions (1) and (2) ofTheorem 2.3, if applied toS1,S2, andV, are equivalent to conditions (1) and (2) of the present corollary. Of course, the conditions labelled with (1) coincide. Thus, we restrict our attention to the conditions labelled with (2).

Necessity. _{Pick a sequence}{_vν}_{converging in}Σ_{2 to a limitv. Then, the} corresponding sequence of functionalsᏲvν converges toᏲvinΣ1. Clearly, the restrictions ofᏲvν toS1converge in turn to the restriction ofᏲv toS1in the dual spaceS

1. If the mappingJ is a topological isomorphism ofS2 ontoS1, then there exists a sequencevν∈S2converging to a limitvin this space such that

Ᏺvν(u)=h˜

u,vν, Ᏺv(u)=h(˜ u,v) (2.22)

for allu∈S1. In particular, for allu∈V, we get

hu, π i2vν=hπu, i2vν =hu, i2vν =_h˜_{i1u, vν} =Ᏺvν

i1u =_h˜_i1u,vν =hu, i2vν

,

(2.23)

Sufficiency. _{Conversely, let conditions (1) and (2) ofCorollary 2.7hold.} Pick an orthonormal basis{bν} inV. It is well known that the reproducing kernel ofVis given in the form

K(x, y)= ν

∗Vbν(x)⊗bν(y). (2.24)

As every orthonormal basis inVcan be extended to an orthonormal basis inV, we see that

K(x, y)=πyK(x, y) (2.25)

for all x and y. By Theorem 2.3, the sectionx ∗−1

V Ᏺ, i1K(x,·)belongs toi2(Σ2)for allᏲ∈Σ1. By the Hahn-Banach theorem, every functionalF∈S1 actually extends continuously to a functionalᏲ∈Σ

1. Hence condition (2) yields

∗−1 V

F, i1K(x,·)

= ∗−1 V

Ᏺ, i1K(x,·)

=πx∗−V1

Ᏺ, i1K(x,·)

∈i2

S2

, (2.26)

the latter inclusion being due to the commutative diagram (2.21). We thus con-clude that condition (2) ofTheorem 2.3is fulfilled forJand that this mapping is onto. The topological arguments now follow from the closed graph theorem.

Note that if dimΣ1<∞, then conditions (1) and (2) ofTheorem 2.3imply thati1i2(Σ2)=Σ1and dimΣ1=dimΣ2. Conversely, suppose that these latter conditionsi1i2(Σ2)=Σ1and dimΣ1=dimΣ2are fulfilled. Hence, it follows that i2(Σ2)is a closed subspace ofV. ReplacingVbyi2(Σ2)andi1by its restriction toi2(Σ2), we still have the same mapping᏶. The reproducing kernelK(·,·)is given by

K(x, y)= dimΣ1

ν=1

i2bν∗(x)⊗i2bν(y), (2.27)

where{bν}is a basis inΣ2with the property that{i2bν}is an orthonormal basis ini2(Σ2). Given anyᏲ∈Σ1, we get

∗−1 V

Ᏺ, i1K(x,·)

= dimΣ1

ν=1

Ᏺi1i2bν

i2bν

(x)∈i2

Σ2

. (2.28)

Similar considerations apply to the commutative diagram (2.21). In this set-ting, the projectionπ always mapsVcontinuously intoi2(S2).

2.3. Neumann problem. In our applications,Vis usually a Hilbert space of the solutions of the equationAu=0 in a domainU⊂X.

Let the operatorAbe included into an elliptic compatibility complex of dif-ferential operatorsAi_∈Diffmi_(X;Ei_{, E}i+1_{),i=0,1, . . . , N, overX, withA}0_=A. Suppose thatVi_{Ᏸ(X, E}◦ i_{),i=0,1, . . . , N, are Hilbert spaces of the sections} ofEi_{overX, such that}

(1) Vi_∩C∞_{(X, E}i_{)is dense inV}i_{for alli=0,1, . . . , N;} (2) Ai_mapsVi_∩C∞_{(X, E}i_)toVi+1_∩C∞_{(X, E}i+1_). LetᏰi

T be the set of all sectionsu∈Vifor which there is a sequence{uν} with the following properties:

(1) uν∈Vi∩C∞(X, Ei); (2) {uν}converges touinVi_;

(3) {Auν}is a Cauchy sequence inVi+1_.

The mappingT:ᏰiT →Vi+1 defined byT u=limAuν, where{uν}is a se-quence with properties (1), (2), and, (3), is called the maximal operator gener-ated byA.

Note thatT is well defined. Indeed, if{uν}is another sequence satisfying (1), (2), and (3), andf=limAu_ν, then, for allg∈C∞(X, Ei+1∗_{)with a compact}

support in the interior ofX, we get

T u−f , g =limAuν−Auν, g

=limuν−uν, Ag

=0, (2.29)

whenceT u=f.

We think ofT as an unbounded operator fromVi_toVi+1_{whose domain is}

Ᏸi

T. SinceᏰiT containsVi∩C∞(X, Ei), the operator T is densely defined and closed.

From the lemma of Du Bois-Reymond and the uniqueness of a weak limit, it follows that ifu∈ᏰiT thenT u=Au in the sense of distributions in the interior ofX.

Lemma2.8. As defined above,T satisfiesTᏰi_T⊂Ᏸi_T+1andT2=0.

Proof. Letu∈Ᏸi_T and {uν}be a sequence with properties (1), (2), and (3). We setfν=Auν. Then,T u=limfν, and, sinceAfν=0, we obtain that T u∈Ᏸi+1

T andT (T u)=0.

Thus, we have the following complex of Hilbert spaces and their closed linear mappings:

V·: 0 →V0 T→V1 T→ ··· T→VN →0. (2.30)

The cohomology of the complex{Ei_{, A}i_{}evaluated by the spaces{V}i_{}is just} the cohomology of complex (2.30), that is,

Hi_V·₌ker

T:Ᏸi

T →Vi+1

TᏰiT−1

We now define T∗, the adjoint of T, as usual for unbounded operators. Namely, letᏰiT∗be the set of allg∈Viwith the property that there isv∈Vi−1 satisfying(T u, g)_Vi=(u, v)_Vi−1for allu∈Ᏸ_Ti−1. We defineT∗:Ᏸi_T∗→Vi−1by T∗g=v.

The operatorT∗is well defined because the domainᏰiT−1is dense inVi−1. It is clear thatT∗gis in general different fromA∗gin the sense of distributions in the interior ofX, forA∗is formally adjoint forAin the sense ofL2_-spaces onX.

Lemma_2.9. _{The operatorT}∗_satisfiesT∗Ᏸi_T∗⊂Ᏸi_T−_∗1_andT∗2=_0.

Proof. _{Indeed, ifg}∈Ᏸi_T∗andu∈Ᏸi_T−2_{, then by definition andLemma 2.8} we get

T u, T∗gVi−1=

T (T u), gVi=0. (2.32)

Therefore,T∗g∈Ᏸi−1

T∗ andT∗(T∗g)=0, which completes the proof.

Thus, we obtain the following (chain) complex of Hilbert spaces and their closed linear mappings:

V·∗: 0← V0_←T∗ _V1_←T∗ _···←T∗ _VN_{← 0. (2.33)}

The complex (2.33) is called the adjoint complex for (2.30), and its homology is denoted by

HiV·∗=

kerT∗:ᏰiT∗ →Vi−1 T∗_Ᏸi+1

T∗

. (2.34)

Introduce an operatorL on Vi _{with a domain Ᏸ}i

L, which better suits the Hilbert structure ofVi_{than the formal Laplacian∆=A}∗_A+AA∗_{of the}

com-plex{Ei_{, A}i_{}. Namely, writeᏰ}i

Lfor the set of allu∈ᏰiT∩ᏰiT∗with the property thatT u∈Ᏸi+1

T∗ andT∗u∈ᏰTi−1. Then, the operatorL:ᏰiL→Viis defined by

Lu=T∗T u+T T∗u (2.35)

(cf. [22, Section 4.2]).

The Neumann problem for the complex{Ei_{, A}i_{}in the spacesV}i_{consists in} the following:

(NP) given a sectionf∈Vi_{, when is thereu∈Ᏸ}i

Lsuch thatLu=f? And how doesudepend onf?

The weak orthogonal decomposition is actually the first step in solving the Neumann problem. Set

Ᏼi_=u∈Ᏸi

fori=0,1, . . . .Since the operatorsTandT∗are closed,Ᏼi_{is a closed subspace} ofVi_{. Denote byH:V}i_→Ᏼi_{the orthogonal projection ofV}i_ontoᏴi_.

Lemma_2.10. _Letu∈_Vi_{. Thenu}∈Ᏼi_{if and only if u}∈Ᏸi_LandLu=_0.

Proof. _{If u} ∈Ᏼi_{, then obviously u} ∈Ᏸi_{L and Lu} = _{0. If Lu} =_{0, then} (Lu, u)_Vi=0, and since

(Lu, u)Vi= T u2_Vi+1+T∗u 2

Vi−1, (2.37)

we haveu∈Ᏼi_.

Lemma_2.11. _{The operatorLis selfadjoint, and(L}+1₎−1_{exists, is bounded,} and is defined inVi_everywhere.

Proof. SinceTis a closed operator and the domain ofTis dense, the same is also true forT∗, and(T∗)∗=T.

It follows that the operators(T T∗+1)−1_and(T∗_T+1)−1_{exist, are bounded,} selfadjoint, and defined everywhere inVi_{(cf. [24, page 200]).}

We now easily verify that(L+1)−1_{exists, is bounded, is defined everywhere,} and is given by the formula

(L+1)−1_{=T T}∗₊₁−1

+T∗T+1−1−1 (2.38)

(cf. [22, Section 4.2.4] and elsewhere).

Corollary2.12(weak orthogonal decomposition). The range ofLis or-thogonal toᏴi_{, and}

Vi_=Ᏼi_⊕LᏰi

L, (2.39)

whereLᏰiLdenotes the closure ofLᏰiLinVi.

Proof. _{This follows immediately from the selfadjointness ofLandLemma} 2.10.

In particular, ifLᏰiLis closed, then we get the strong orthogonal decompo-sition

Vi_=Ᏼi_⊕T∗_TᏰi

L⊕T T∗ᏰiL. (2.40)

Definition_2.13. _LetLᏰi_{Lbe closed andf}∈_Vi_{, thenf}=_Hf+_Luwhere u∈Ᏸi

L. The Neumann operatorN:Vi→ᏰiLis defined byNf=u−Hu.

Note thatNis well defined. Indeed, if alsof=Hf+Luwhereu∈Ᏸi L, then L(u−u)=0, whence

We summarize the properties of the Neumann operator. They generalise those of the Green operator from Hodge theory, for the Neumann problem itself stems from the desire to extend the Hodge theory to the case of manifolds with boundary.

Lemma2.14. Suppose thatLᏰi_Lis closed. Then, the Neumann operatorNhas the following properties:

(1) Nis bounded, selfadjoint,HN =NH =0, and we have the orthogonal decomposition

f=Hf+T∗T Nf+T T∗Nf (2.42)

for allf∈Vi_; (2) iff∈Ᏸi

T andT f=0, thenT Nf=0. If, moreover,LᏰiL+1is closed, then T Nf=NT f;

(3) iff∈Ᏸi

T∗ andT∗f=0, thenT∗Nf =0. If, moreover,LᏰiL−1is closed, thenT∗Nf=NT∗f.

Proof. (1) The equalitiesHN=NH=0 and formula (2.42) follow immedi-ately from the definition ofN.

Further, by the closed graph theorem, there exists a constantc >0 such that ifu∈Ᏸi

Lis orthogonal toᏴithen we haveLu ≥cu. Applying this toNf, we obtain

Nf ≤1

cLNf = 1

cf−Hf ≤ 1

cf. (2.43)

HenceNis bounded.

Finally, the selfadjointness ofNfollows immediately fromLemma 2.11 be-cause

(Nf , g)_Vi=(Nf , Hg+LNg)_Vi=(Nf , LNg)_Vi =(LNf , Ng)Vi=(f , Ng)_Vi.

(2.44)

(2) Letf∈Ᏸi

L. Then from (2.42) andLemma 2.8, we getT∗T Nf ∈ᏰiT and T f=0 impliesT T∗T Nf=0. Hence, it easily follows thatT Nf=0.

If also LᏰiL+1 is closed, then for any f ∈ᏰiT we have T f =T T∗T Nf on the one hand, andT f=T T∗NT f on the other hand. Hence, it follows that L(T Nf−NT f )=0, and, sinceT Nf−NT f is orthogonal toᏴi+1_{, we deduce} thatT Nf−NT f=0, as required.

(3) The proof is analogous to that of part (2).

IfLi_{is a hypoelliptic pseudodifferential operator in the interior ofX, then} the harmonic space Ᏼi _{consists ofC}∞_{sections in the interior ofX and the}

Beginning with its classical forms, the Dirichlet norm has been an important technical tool in studying the Neumann problem.

Given anyu, v∈Ᏸi

T∩ᏰiT∗, the Dirichlet inner product ofuandvis defined by

D(u, v)=(T u, T v)Vi+1+

T∗u, T∗v_Vi−1+(u, v)Vi, (2.45)

and the Dirichlet norm isD(u)=D(u, u). The spaceᏰi

T∩ᏰiT∗ with the Dirichlet norm is a complete (Hilbert) space. It is denoted byᏰi_.

SinceD(u)≥ uVi for allu∈Ᏸi, there exists only one selfadjoint operator Swith a domainᏰi

S⊂Ᏸi, such that ifu∈ᏰiS andv∈Ᏸithen

D(u, v)=(Su, v)_Vi. (2.46)

The following lemma gives a useful description of the operatorLbecause our estimates will be in the normD(u).

Lemma2.15. The equalities holdᏰi_L=Ᏸi_S andL=S−1, where the operator Sis defined by (2.46).

Proof. _Ifu∈Ᏸi_{L and v}∈Ᏸi_{, then D(u, v)}=_((L+_{1)u, v)Vi is fulfilled.} Hence by the uniqueness ofS, we haveS=L+1.

Let·1and·2be two norms on a vector spaceV. We say that the norm ·1is completely continuous with respect to the norm·2if every sequence which is bounded in the norm·1has a convergent subsequence in the norm ·2.

Lemma2.16. If the normDonᏰiis completely continuous with respect to ·Vi, thenᏴiis finite dimensional.

Proof. _{Observe that if u, v} ∈Ᏼi _{then D(u, v)}=_{(u, v)Vi. Suppose that} the dimension ofᏴi _{is infinite. Then there exists an infinite sequence{uν}} of orthonormal elements inᏴi_{. SinceD(u}

ν)= uνVi=1, the sequence{uν} contains a convergent subsequence. But this is at variance with the fact that if ν≠µthenuν−uµVi=√2.

Lemma_2.17. _{If the normDon}Ᏸi_{is completely continuous with respect to} ·Vi, then there exists a constantc >0such that, for allu∈Diorthogonal to

Ᏼi_,

T u2

Vi+1+T∗u 2

Vi−1≥cu2_Vi. (2.47)

Proof. Consider the Hilbert spaceVi+1×Vi−1which is equipped with the norm

_{f , v}=f2

Vi+1+v2_Vi−1 1/2

LetM:Ᏸi_→Vi+1_×Vi−1_{be the mapping defined byMu= {T u, T}∗_{u}. We note}

thatMis a closed operator.

We prove that the range ofMis closed. Suppose thatMᏰi_{is not closed. Then} there exists a sequence{uν}inᏰi _{such that limMu}

ν= {f , v}and {f , v} ∈

MᏰi_.

Setu_ν=uν−Huν, thenuν are orthogonal toᏴi and limMuν= {f , v}. If uνVi are bounded, thenD(uν)=(Muν2+ uν2Vi)1/2 are bounded, too. Then, by hypothesis,{uν}has a convergent subsequence with a limitu, and sinceM is closed thenMu= {f , v}, which contradicts the assumption that {f , v} ∈MᏰi_{. Thus by choosing a subsequence, if necessary, we may assume} that limuνVi= ∞.

Now set Uν =uν/uνV i. Then limMUν =0 and D(Uν) are bounded. Therefore,{Uν}has a convergent subsequence{Uνk}such that

limUνk=U , limMUνk= {0,0}. (2.49)

HenceMU=0 so thatU∈Ᏼi_{. SinceU}

ν is orthogonal toᏴi, we haveU=0, but UνVi =1. This contradiction proves that the rangeMᏰi is closed in Vi+1_×Vi−1_.

Let R be the restriction ofMto the orthogonal complement ofᏴi _inᏰi_. ThenRis one-to-one and has a closed range. By the closed graph theorem, the inverseR−1_{is bounded. Hence there isc >0 such thatRu}2_≥cu2

Vi. This proves the lemma.

Theorem_2.18. _{If the normDon}Ᏸi_{is completely continuous with respect} to the norm·Vi, thenLᏰi_{is closed.}

Proof. ByLemma 2.17, there existsc >0 with the property that, for all u∈Ᏸi

Lwhich are orthogonal toᏴi, we have

(Lu, u)Vi≥cu2_Vi, (2.50)

so thatLuVi≥cuVi.

Setf =limLuν. We may assume that uν are orthogonal to Ᏼi, and then uνVi are uniformly bounded. Therefore, {uν} has a subsequence whose arithmetic means converge. (This actually puts some restrictions on the spacesVi_{under study.) Denoting this limit byu, we getf=Lu, which} com-pletes the proof.

The question of when the normDonᏰi_{is completely continuous with} re-spect to the norm · Vi is very difficult in the general case, and it requires special consideration. We present some consequences here.

Proof. _{This follows immediately from Lemma 2.14 and Theorem 2.18.}

For compact manifolds with boundaryX, the subspaceᏴ0_{is usually infinite} dimensional so, byLemma 2.16, the Dirichlet normDmay not be completely continuous with respect to the norm · V0 onᏰ0. But the following result holds.

Theorem2.20. If the normDonᏰ1is completely continuous with respect to the norm·V1, thenLᏰ0Lis closed.

Proof. It suffices to prove that there exists a constant c >0 such that LfV0≥cfV0for allf∈Ᏸ0_Lwhich are orthogonal toᏴ0.

First, ifu∈Ᏸ0

L, thenT u∈Ᏸ1andT u⊥Ᏼ1. Thus byLemma 2.17, we obtain T∗T u2_V0= Lu_V20≥cT u2V1. (2.51)

Further, sincef ⊥Ᏼ0_{, then, by the weak orthogonal decomposition (2.39),} f∈LᏰ0

L. Hence, for eachε >0, there existsu∈Ᏸ0Lsuch thatf−LuV0< ε. Thus,

f2

V0≤(Lu, f )V0+εfV0 ≤ T uV1T fV1+εfV0

≤1

cLuV0LfV0+εfV0

≤1_cfV0LfV0+ε ₁

cLfV0+fV0

.

(2.52)

Sinceεcan be made arbitrarily small by choosingLuclose enough tof, we obtainLfV0≥cfV0, which concludes the proof.

The next result follows from Lemma 2.14 and Theorem 2.18. Recall that

Ᏼ0_=kerT0_.

Corollary2.21. Suppose that the normDonᏰ1is completely continuous with respect to the norm·V1. Then,f=Hf+T∗NT ffor any sectionf∈Ᏸ0

T, whereH:V0_→Ᏼ0_{is the orthogonal projection.}

By assumption, the differential operatorA0_{=Ahas injective symbol. It} fol-lows thatA0_{is hypoelliptic in the interior ofXwhence}

Ᏼ0_=u∈V0_∩C∞

loc ◦

X, E0_{:Au=0, (2.53)}

that is, the operator H0 _{is a generalisation of the Bergman projector from} complex analysis.Corollary 2.21gives

As mentioned, a priori estimates for solutions of elliptic equations imply that, for each interior pointxofX, the evaluation functionalδx(u)=u(x)is bounded onᏴ0_{. Therefore,Ᏼ}0_{is a Hilbert space with reproducing kernel (cf.} [1]).

Let{eν}ν=1,2,...be some complete orthonormal system inᏴ0. Ifu∈Ᏼ0, then this section decomposes into the Fourier seriesu=cνeν which converges in the norm of the spaceV0_{and hence uniformly along with all derivatives on} compact subsets of the interior ofX. In the interior ofX×X, we consider the series

K(x, y)=KH(x, y)=

∞

ν=0

∗Eeν(x)⊗eν(y). (2.55)

Theorem2.22. Series (2.55) converges uniformly along with all derivatives

on compact subsets of the interior ofX×X, so that

KH∈Cloc∞ ◦

X×X, E◦ ∗E. (2.56)

Ifx∈X◦ is fixed, then this series actually converges in the norm of the space Ex∗⊗V0.

Proof. _{To shorten notation, we will restrict the discussion to the case} whereXis a closed domain inRn_.

Let

eν(x)=     

eν,1(x) .. . eν,k(x)

   

 (2.57)

be representations of the sectionseν, wherekis the rank ofE. Pick compact setsK1andK2in the interior ofX. Ifx∈K1is a fixed point, then, in view of the orthonormality of the system{eν}, we obtain forj=1, . . . , k

  N

ν=0

eν,j(x) 2

  2

≤

N

ν=0

eν,j(x)eν(x) 2

≤c1

N

ν=0

eν,j(x)eν(y) 2

V0

=c1 N

ν=0

_eν,j(x)2 ,

the constantc1>0 depending onAandKonly. Hence

N

ν=0

eν,j(x)2≤c1 (2.59)

for allx∈K1.

Therefore, denoting byc2the constant obtained by analogy for the setK2, we get for(x, y)∈K1×K2

N

ν=0

_∗Eeν(x)⊗eν(y)= N

ν=0

eν(x)eν(y)≤k√c1c2. (2.60)

This proves the absolute and uniform convergence of series (2.55) on com-pact subsets of the interior ofX×X.

Finally, (2.59) implies that, for fixedx∈X◦, equality (2.55) gives the expan-sion ofK(x, y)in the complete orthonormal system{eν}. To finish the proof, it is sufficient to observe thatxandyenter intoK(x, y)in a symmetric way.

Theorem2.23(Bergman formula). Ifu∈Ᏼ0, then

u(x)=u, K(x,·)V0 (2.61)

for allx∈X◦.

Proof. Letu=c_µeµ. Then, by the previous theorem, we get for fixedx in the interior ofX

u, K(x,·)_V0= µ,ν

cµeµ, eνV0eν(x)= ν

cνeν, eνV0eν(x)=u(x), (2.62)

and the proof is complete.

Thus, in order to discover the properties of π =H, we might study the Neumann operatorN1_{. However, “good” properties ofN}1_{is not what we can} generally expect. It is rather an instrument to produce examples for the general scheme.

2.4. Hodge theory on manifolds with boundary. Given a vector spaceV with norm · , we writeC(V , · )for the completion ofV under the norm ·.

Pick a Dirichlet systemBj,j=0,1, . . . , m−1, of orderm−1 on the boundary of X. More precisely, Bj is a differential operator of type E→Fj and order mj≤m−1 in a neighbourhoodUof∂X. Moreover, the ordersmjare pairwise different and the symbolsσ (Bj), if restricted to the conormal bundle of∂X, have ranks equal to the dimensions ofFj.

We actually assume that the dimensions ofFjare the same and equal to that ofE.

Let Cj, j =0,1, . . . , m−1, be the adjoint system for {Bj}with respect to Green’s formula (cf. [21]). Thus,Cj is a differential operator of typeF∗→Fj∗ and orderm−mj−1 in a smaller neighbourhoodUof∂X. We now set

t(u)= m−1

j=0

Bju, n(f )= m−1

j=0 ∗−1

F_jCj∗F (2.63)

foru∈Cm−1

loc (U , E)andf∈Clocm−1(U , F ).

Lemma 2.24(Green’s formula). For each u, v ∈H2m(X, E), the following formula holds:

∂X

t(u), n(Av)_x−n(Au), t(v)_xds=

X

(∆u, v)x−(u,∆v)x

dx.

(2.64)

Proof. See [21, Corollary 9.2.12].

GivenF, we consider the boundary value problem

∆u=F inX, t(u)=0 on∂X, (2.65)

which is an obvious generalisation of the classical Dirichlet problem (cf. [21, Section 9.2.4]).

Suppose thats >0. For sectionsu∈C∞(X, E)we define two types of nega-tive norms

u−s= sup v∈C∞(X,E)

(u, v)

vs , |u|−s= sup v∈C∞(X,E)

t(v)=0

(u, v)

vs , (2.66)

where(·,·)is the scalar product inL2_{(X, E). We denote the completions of} C∞(X, E) with respect to these norms byH−s_{(X, E) and C(C}∞_{(X, E),| · |}

−s), respectively. They are obviously Banach spaces and satisfy

H−s_{(X, E) CC}∞_{(X, E),|·|} −s

, (2.67)

We can define (u, v)for u∈H−s_{(X, E) and v ∈C}∞_{(X, E) as follows. By}

definition, there is a sequence{uν}inC∞(X, E)such thatuν−u−s→0 as ν→ ∞. Then

uν−uµ, v≤uν−uµ₋svs →0 (2.68)

asµ, ν → ∞. Set(u, v)=lim(uν, v). Clearly, this limit does not depend on the particular sequence{uν}, for ifuν−s→0, then|(uν, v)| ≤ uν−svs tends to zero, too. From the definition, it follows that for allu∈H−s_{(X, E)and} v∈C∞(X, E), we get

(u, v)≤ u−svs. (2.69)

In a similar way, we can define the pairing(u, v)foru∈C(C∞(X, E),|·|−s) andv∈C∞(X, E) witht(v)=0. Corresponding to (2.69), we obviously have |(u, v)| ≤ |u|−svs.

LetFbe inC(C∞(X, E),|·|−s−2m), wheres≥0. We say thatu∈H−s(X, E)is astrong solutionof (2.65) if there is a sequence of sectionsuν∈C∞(X, E)with t(uν)=0, such that

uν−u₋s →0, ∆uν−F₋s−2m →0 (2.70)

asν→ ∞.

Denote byᏴ(X)the set of allu∈C∞(X, E)that satisfy∆u=0 in the interior ofXandt(u)=0 on∂X. Since (2.65) is an elliptic boundary value problem,

Ᏼ(X)is finite dimensional. Moreover, for anyu∈Ᏼ(X), we actually obtain

0=(∆u, u)=(Au, Au) (2.71)

whenceAu=0 in X. Therefore, the space Ᏼ(X)consists of all u∈᏿A(

◦

X) which are C∞up to the boundary of X and which vanish up to the infinite order on∂X.

Lemma_2.25. _Lets≥_{0. IfF} ∈_C(C∞_{(X, E),}| · |_{−s−2m)andF} ⊥Ᏼ_{(X), then} there is a strong solutionu∈H−s_{(X, E)of (2.65) satisfyingu⊥Ᏼ(X)and}

u−s≤c|F|−s−2m, (2.72)

where the constantcdoes not depend onF andu.

Proof. _{See [14].}

Definition (2.70) of a strong solution of (2.65) obviously corresponds to an appropriate closureL:ᏰL→C(C∞(X, E),| · |−s−2m)of the Laplacian∆=A∗A (cf. [5, Chapter 2]). Namely, we denote byᏰLthe set of all sectionsu∈H−s(X, E), for which there is a sequence{uν}with the following properties:

(2) {uν}converges touinH−s_{(X, E);}

(3) {∆uν}is a Cauchy sequence inC(C∞(X, E),|·|−s−2m).

The closed densely defined operatorL:ᏰL→C(C∞(X, E),| · |−s−2m)given byLu=lim∆uν, where{uν}is any sequence with properties (1), (2), and (3), is called thestrong extensionof∆under the boundary conditionst(u)=0. It is clear thatu∈H−s_{(X, E)is a strong solution to problem (2.65) if and only if} Lu=F.

It is worth pointing out that the case∂X= ∅is formally permitted in the following theorem.

Theorem2.26. Suppose thats≥0. There are bounded linear operators

H:CC∞(X, E),|·|−s−2m →Ᏼ(X), G:CC∞(X, E),|·|−s−2m →ᏰL (2.73) such that

(1) Hhas the kernelKH(x, y)=νhν(x)⊗∗Ehν(y), where{hν}is an or-thonormal basis ofᏴ(X);

(2) AH=0andGH=HG=0; (3)

GLu=u−Hu for allu∈ᏰL,

LGF=F−HF for allF∈CC∞(X, E),|·|−s−2m

. (2.74)

Proof. _{This follows by the same method as in [15, Theorem 3.3], with} Lemma 3.2 therefrom replaced byLemma 2.25.

The operatorsHandGare actually independent ofssince they are unique extensions by continuity of these operators on the dense subspaceC∞(X, E) ofC(C∞(X, E),|·|−s−2m).

When restricted toL2_{(X, E), the operatorGis selfadjoint. Indeed, given any} F , v ∈L2_{(X, E), we may invoke the elliptic regularity of the Dirichlet} prob-lem (2.65) to conclude that bothGF andGv belong toH2m_{(X, E)and satisfy} the boundary conditiont(·)=0. It follows thatLGF=∆GF andLGv=∆Gv whence

(GF , v)=(GF , Hv+LGv)=GF , A∗AGv=A∗AGF , Gv=(F , Gv), (2.75)

which is due toTheorem 2.26. Hence the Schwartz kernel ofGis Hermitean, that is,KG(x, y)∗=KG(y, x)for all(x, y)away from the diagonal ofX×X.

Corollary_2.27. _{If, in addition,F}∈_H−s−2m_{(X, E), then there is a sequence} of sectionsuν∈C∞(X, E)witht(uν)=0, such that

uν−u₋s →0, ∆uν−F₋s−2m →0 (2.76)

From Lemma 2.24, we deduce that when u is smooth enough, it fulfills t(u)=0 if and only if (∆u, v)=(u,∆v)for all v satisfyingt(v)=0. This gives rise to the concept of aweak extensionof∆under the boundary condi-tionst(u)=0 (cf. [5, Chapter 2]). Given anF∈C(C∞(X, E),|·|−s−2m), a section uis said to be aweak solutionof (2.65) if it is inH−s_{(X, E) for somes≥0}

and

(u,∆v)=(F , v) (2.77)

for allv∈C∞(X, E)satisfyingt(v)=0.

Lemma_2.28. _{Suppose thatF} ∈_C(C∞_{(X, E),}| · |_{−s−2m)wheres}≥_{0. Ifu}∈ H−s_{(X, E)is a weak solution of (2.65), then actuallyu∈H}−s_{(X, E)and it is a} strong solution of (2.65). Moreover, there is a constantcnot depending onF or u, such that

u−s≤c

|F|−s−2m+u−s

. (2.78)

Proof. _{See [14].}

To study the Dirichlet problem with nonzero boundary datat(u)=u0, we need a result of [13]. Denote byH−s,B_{(X, E)the completion ofC}∞_{(X, E) with}

respect to the norm

u−s,B:= u−s+t(u)_⊕H−s−_mj−1/2_(∂X,F j)+

n(Au)

⊕H−s−2m+mj+1/2(∂X,F_j).

(2.79)

The advantage of using these spaces is that for each u ∈ H−s,B_{(X, E),} there is a sequence {uν} in C∞(X, E), such that uν → u in H−s(X, E), and {t(uν)}, {n(Auν)} are Cauchy sequences in ⊕H−s−mj−1/2(∂X, Fj) and ⊕H−s−2m+mj+1/2_{(∂X, Fj), respectively. Moreover,}{∆_uν}_{is a Cauchy sequence} inH−s−2m_{(X, E), which follows by manipulations of Green’s formula. Hence to} any elementu∈H−s,B_{(X, E), we can assign botht(u),n(Au)and∆udefined} in the above strong sense.

Lemma_2.29. _{For each pair}

u0∈ m−1

j=0

H−s−mj−1/2_{∂X, Fj, u1}∈ m−1

j=0

H−s−2m+mj+1/2_{∂X, Fj, (2.80)}

there is a sectionu∈H−s,B_{(X, E)with the property thatt(u)=u}

0andn(Au)= u1. Moreover, the mapping(u0, u1)Uis continuous in the relevant norms.

Proof. _{See [13, Lemma 6.1.2].}

Given any

F∈CC∞(X, E),|·|−s−2m

, u0∈ ⊕H−s−mj−1/2

∂X, Fj

we now consider the inhomogeneous Dirichlet problem

∆u=F inX, t(u)=u0 on∂X. (2.82)

A sectionuis said to be aweak solution of (2.82) if it is inH−s_{(X, E)for} somes≥0 and

(u,∆v)=(F , v)−

∂X

u0, n(Av)xds (2.83)

for allv∈C∞(X, E)satisfyingt(v)=0.

Theorem2.30. Suppose thats≥0. If F⊥Ᏼ(X), then there is a weak solu-tionu∈H−s_{(X, E)to (2.82) withu⊥Ᏼ(X). Moreover,u∈H}−s_{(X, E)satisfies} (2.82) in a strong sense, and there is a constantcindependent ofF,u0, andu, such that

u−s≤c

|F|−s−2m+u0_⊕H−s−_mj−1/2_(∂X,F j)

. (2.84)

Proof. UsingLemma 2.29, we reduce (2.83) to (2.77) with a suitable right sideF. To this end, we chooseU∈H−s,B_{(X, E)such thatt(U )=u}

0andn(AU )= u1,u1being arbitrary. By the definition ofH−s,B(X, E), there is a sequence{uν} inC∞(X, E)such that

uν →U inH−s(X, E),

tuν

→u0 in⊕H−s−mj−1/2

∂X, Fj

,

nAuν

→u1 in⊕H−s−2m+mj+1/2

∂X, Fj

,

(2.85)

and∆uν→FinH−s−2m(X, E). By Green’s formula, we get

uν,∆v

=∆uν, v

−

∂X

tuν

, n(Av)_xds (2.86)

for allv∈C∞(X, E)satisfyingt(v)=0. Lettingν→ ∞in this equality yields

(U ,∆v)=(F, v)−

∂X

u0, n(Av)xds. (2.87)

Subtracting (2.87) from (2.83), we obtain

(u−U ,∆v)=(F−F, v) (2.88)

for allv∈C∞(X, E)satisfyingt(v)=0, that is,u−Uis a weak solution of the Dirichlet problem (2.65) withF replaced byF−F. Moreover, it follows from (2.87) that

for allv∈Ᏼ(X). Combining Lemmas2.28and2.25thus results in the desired assertion.

We now derive a Poisson formula for solutions of the inhomogeneous Dirich-let problem.

To this end, we choose a Green operatorGA(·,·)forAonX(cf. [21, Section 9.2.1]). Given an oriented hypersurfaceS⊂X, we denote by[S]A_{the kernel on} X×Xdefined by

[S]A_{, g⊗u} X×X=

S

GA(g, u) (2.90)

for allg∈C∞(X, F∗)andu∈C∞(X, E)whose supports meet each other in a compact set.

In particular, the kernel [∂X]A _{is obviously supported on the diagonal of} ∂X×∂X.

For a sectionu∈C∞(X, E), we set

(Mu)(x)= −GA∗[∂X]A_{u= −}

∂X

GAKGA∗(x,·), u (2.91)

whenx∈X◦,KGA∗being the Schwartz kernel ofGA∗. The integral on the right-hand side is well defined, forKGA∗is aC∞-section ofEF∗outside the diagonal ofX×X.

Corollary2.31. As defined above,Minduces a continuous mappingP of ⊕H−s−mj−1/2_{(∂X, Fj)toH}−s_{(X, E) such thatP t(u)}=_{Mu. Moreover, for each} weak solutionuof (2.82) it follows that

u=Hu+G∆u+P t(u). (2.92)

Proof. _Letu∈_H−s_{(X, E)be a weak solution of (2.82). FromTheorem 2.30,} we deduce thatu∈H−s_{(X, E)satisfies (2.82) in a strong sense. More precisely,} there exists a sequence uν ∈C∞(X, E) which approximatesu inH−s(X, E), such thatt(uν)→t(u)and∆uν→∆uin the relevant norms. We now set

P u0:=lim ν→∞

uν−Huν−G

∆uν

=u−Hu−G(∆u), (2.93)

the limit existing inH−s_{(X, E)byTheorem 2.26. Moreover, it is independent} of the particular choice ofuwith a well-defined∆uandt(u)=u0, which is again due toTheorem 2.26.

Ifv∈C∞(X, E)has a compact support in the interior ofX, then byTheorem 2.26we get

P u0, v

=(u, v)−(u, Hv)−(∆u, Gv)

=u, v−Hv−∆(Gv)−

∂X

t(u), n(AGv)_xds

= −

∂X

t(u), n(AGv)_xds,

(2.94)

fort(Gv)=0. The right-hand side here just amounts to(−GA∗([∂X]A_{u), v),} provided thatuis smooth enough.

From (2.93), it follows thatP u0is the unique solution of the Dirichlet prob-lem

∆u=0 inX, t(u)=u0 on∂X, (2.95)

which is orthogonal to Ᏼ(X). We call P u0 the Poisson integral of u0. By

Theorem 2.30,

_{P u0}

−s≤cu0_⊕H−s−mj−1/2

(∂X,Fj) (2.96)

withca constant independent ofu0.

Theorem _2.32. _{The quotient space} ᏿(f ) ∆ (

◦

X)/Ᏼ(X)is topologically isomor-phic tom_j=−₀1Ᏸ_{(∂X, Fj).}

Proof. Given any u∈᏿(f )_∆ (

◦

X), the Cauchy datat(u) and n(Au), being first defined near∂X, have weak limit valuesu0 andu1on∂X belonging to m−1

j=0 Ᏸ(∂X, Fj)(cf. [21, Section 9.4]). Pick a regularisation ofuon∂X, that is, any sectionU∈H−s_{(X, E)which coincides withuin the interior ofX(cf. [21,} Section 9.3.6]).

Using the parametrixGof∆given byTheorem 2.26, we get by Green’s for-mula

u(x)−HU (x)= −

∂X

u0, nAKG(·, x)y−

u1, tKG(·, x)y

ds (2.97)

forx∈X◦. Sincet(KG(·, x))=0 for allx∈

◦

X, it follows thatu=HU+P u0, the sectionHU∈Ᏼ(X)being independent of the particular choice of the regular-isationU.

We have thus proved that any solutionu∈᏿(f )

∆ ( ◦

X)is representable through the weak limit valuest(u)on∂Xby the Poisson formula (2.92). Furthermore, m₋1

j=0 Ᏸ(∂X, Fj) is the inductive limit of the sequence m₋1

Since᏿(f )A (

◦

X)is obviously a closed subspace of᏿_∆(f )(X)◦ , the mapping

᏿(f ) A

◦

X

Ᏼ(X) →

m−1

j=0

Ᏸ_{∂X, F}

j (2.98)

given by ut(u) identifies the quotient space with a closed subspace of m₋1

j=0 Ᏸ(∂X, Fj).

2.5. Hardy spaces. Suppose thatUX◦is a domain withC∞boundary. Fix a Dirichlet systemB= {Bj}m−1

j=0 of orderm−1 on∂U, eachBjbeing a differential operator of ordermjand typeE→Fjin a neighbourhoodᏺof∂U. For a section uofEnear∂U, we set

t(u)= m−1

j=0

Bju∂U (2.99)

if defined.

Since∆=A∗Asatisfies the condition(U )sin the interior ofX, the sesquilin-ear form

h(u, v)=

∂U

t(u), t(v)xds (2.100)

defines a scalar product on᏿∆(U )∩C∞(U , E). Denote byH∆(B)(U )the

comple-tion of᏿∆(U )∩C∞(U , E)in the normuh(u, u). These spaces are called

theHardy spaces, by analogy to the classical Hardy spaces of harmonic func-tions. Alternatively,H_∆(B)(U )can be described as the space of allu∈᏿∆(U )of

finite order of growth, for which the weak boundary values oft(u)belong to m₋1

j=0 L2(∂U , Fj).

Lemma2.33. The spaceH_∆(B)(U )is a separable Hilbert space with a repro-ducing kernel.

Proof. _{By the very definition,H}(B)

∆ (U )can be identified as a closed

sub-space in⊕L2_{(∂U , F}

j). In particular,H_∆(B)(U )is a separable Hilbert space because ⊕L2_{(∂U , F}

j)is.

Theorem 2.32 implies that each element u0 ∈ H∆(B)(U ) can be actually

thought of as a solution from ᏿(f )

∆ (U ). To make this more precise, we

in-voke Theorem 2.30, withU in place of X. Since Ᏼ(U )is trivial in this case andL2_{(∂U , F}

j)H−mj(∂U , Fj)forj=0,1, . . . , m−1, there is a unique section u∈H1/2_{(U , E)satisfying∆u=0 andt(u)=u}

0in a strong sense. Moreover, we have

uH1/2(U ,E)≤cu0_⊕H−mj_{(∂U ,F}

j) (2.101)