On spectral projection for the complex Neumann problem Текст научной статьи по специальности «Математика»

УДК 517.55

On Spectral Projection for the Complex Neumann Problem

Ammar Alsaedy* Nikolai Tarkhanov^

Institute of Mathematics, University of Potsdam, Am Neuen Palais, 10, Potsdam, 14469

Germany

Received 08.04.2012, received in revised form 20.06.2012, accepted 20.07.2012 We .show that the L2 -.spectral kernel function of the д -Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.

Keywords: д -Neumann problem, strongly pseudoconvex domains, spectral kernel function.

Introduction

The d-Neumann problem appears naturally in studying the Dirichlet form for the Dolbeault complex on a compact complex manifold Z with boundary. More precisely, one minimizes the Dirichlet norm over the space of differential forms of bidegree (0, q) on Z whose complex normal parts on the boundary of Z vanish. The Euler-Lagrange equations of this variational problem just amount to the d -Neumann problem.

While the differential equation in Z in the d -Neumann problem is a generalized Laplace equation, the boundary conditions fail to satisfy the Shapiro-Lopatinskii condition. Hence, the elliptic regularity in Sobolev spaces on Z is violated. The main a priori estimate for (0,1) -forms on compact strongly pseudoconvex manifolds was proved by Morrey, see the references in [1]. When compared with a priori estimates for elliptic boundary value problems, the estimate of Morrey bears loss of 1 in the regularity. For differential forms of arbitrary bidegree (0, q) with q > 1 on strongly pseudoconvex manifolds the main a priori estimate was later proved by Kohn [2] who extended in this way the theory of harmonic integrals by W. Hodge (1941) and K. Kodaira (1953) to compact strongly pseudoconvex manifolds.

The d-Neumann problem initiated readily the study of so-called subelliptic operators which occured intensively in the 1970s and 1980s. In complex analysis this study was mostly focused upon the regularity of solutions of the d -Neumann problem in pseudoconvex domains of finite type. For a current survey in this direction we refer the reader to [3].

The most difficult part of [2] is the proof of regularity of solutions up to the boundary dZ. This proof was simplified by Kohn and Nirenberg in [4]. To this end, they had elaborated calculus of pseudodifferential operators which are nowadays referred to as classical ones.

The proof of regularity in the d -Neumann problem raised the problem of constructing explicit integral formulas for the solution. A satisfactory theory is nowadays available in [5]. We also mention an earlier paper [6] which studied estimates for the kernel function of the d-Neumann

*alsaedy@math.uni-potsdam.de ttarkhanov@math.uni-potsdam.de © Siberian Federal University. All rights reserved

operator. First steps towards calculus of pseudodifferential operators relevant to several complex variables were summarized in [7].

For compact strongly pseudoconvex manifolds Z the d -Neumann operator satisfies a pseudolocal estimate with gain 1 in the Sobolev scale. Combining this estimate with a familiar argument of topological tensor products shows that the spectral kernel function of the d-Neumann problem is smooth up to the boundary of Z x Z.

The present paper is motivated by the question of M. Shubin whether the spectral kernel function of the complex Laplacian under its natural boundary conditions is still Cœ up to the boundary of Z x Z, if Z is not compact. We answer the question in the affirmative. To this end we prove that a pseudolocal estimate holds even for non-compact strongly pseudoconvex manifolds. A close result was established in [8] using different techniques.

The spectral theory of the d-Neumann problem has been previously studied in [9,10] for compact manifolds with boundary. In [11], heat kernel asymptotics are developed for the heat kernel of a general elliptic operator with non-coercive boundary conditions.

1. The d-Neumann Problem

Let Z be a Hermitian complex manifold of dimension n with boundary dZ. We always think of Z as a closed sub domain of a larger Hermitian complex manifold Z' of the same dimension.

Suppose Z is strongly pseudoconvex, i.e., at each point of dZ the Levi form restricted to the tangent hyperplane has n — 1 positive eigenvalues. This slightly differs from the usual notation, for we don't control dZ at the points at infinity if there are any.

Let F be a Hermitian holomorphic vector bundle over Z'. For q = 0,1,..., n, we set Fq = F<g>cA T*(Z'). This bundle is the one we are interested in, since its sections are the differential forms of type (0, q) on Z' with coefficients in F. The operator d gives rise to a differential operator Bf on the F-valued differential forms by Of =1 < d.

A Hermitian metric on Z' induces a volume element dv on Z'. When combined with a Hermitian metric on F, this allows one to define a conjugate linear isomorphism of bundles * : Fq ^ Fq' by *v = (•, v)zdv. Here, Fq' = F' < f\n'n-qT*(Z') stands for the dual bundle of Fq.

Furthermore, in the space C*C^mp(Z', Fq) we can introduce an inner product by the formula

(u, v)l2(z',f) = / dv = / (*v,w}2

JZ' JZ'

for u,v e C^mp(Z', Fq). We say that a form u is square integrable on Z' if the function (u, u)z is integrable with respect to dv. As usual, the space of square integrable (0, q) -forms with coefficients in F on Z ' is denoted by L2(Z', Fq). The inner product (u,v)L2(Z;,F,) actually turns L2(Z ', Fq) into a Hilbert space with norm

í\\L2(Z',Fq ) := \J u)L2(Z',Fi ^

as is easy to check.

We now restrict our section spaces and operators thereon to the manifold Z, thus obtaining

Eq := CTO(Z, Fq), Lq := L2 (Z, Fq),

etc. It is obvious that Lq just amounts to the completion of {u G Eq : ||u||Lq < <»} in the norm ll-llo.

Let Dq be the set of all sections u G Lq, for which there is a sequence {uv} with the following properties:

1) uv G Lq nEq;

2) {uv} converges to u in Lq; and

3) {dpuv} is a Cauchy sequence in Lq+1.

The mapping T : Dq ^ Lq+1 defined by Tu = lim dpuv, where {uv} is a sequence with properties 1)-3), is called the maximal operator generated by dp.

Note that T is well defined. Indeed, if {uV} is another sequence satisfying 1)-3), and f = lim¿FuV, then for all g G CTO(Z, Fq+1') with a compact support in the interior of Z we get

(Tu - f, g) = lim (dpuv - dpuV, g) = lim (uv - uV, d'g) = 0,

whence Tu = f.

We will think of T as an unbounded operator from Lq to Lq+1, whose domain is Dq. Since Dq contains Lq n Eq 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 if u G Dq then Tu = dpu in the sense of distributions in the interior of Z.

Lemma 1.1. As defined above, T satisfies TDq C Dq+1 and T2 =0.

Proof. Assume that u G Dq and {uv} is a sequence with properties 1)-3). We set fv = <9puv. Then Tu = lim fv. And since dFfv = 0, we obtain that Tu G Dq+1 and T(Tu) = 0. □

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

L : 0 —► L0 L1 ... Ln —► 0. (1)

The L2 -cohomology of the Dolbeault complex on Z with coefficients in F is just the coho-mology of complex (1.1). More precisely, the cohomology at step q denoted by Hq(L ) is defined to be the quotient of the null-space of T: Dq ^ Lq+1 over the range of T: Dq-1 ^ Lq.

We now define T *, the adjoint of T, as usual for unbounded operators. Namely, let Dq * be the set of all forms g G Lq with the property that there is v G Lq-1 satisfying (Tu, g)Lq = (u, v)Lq-i for all u G Dq-1. We define T *: Dq * ^ Lq-1 by T *g = v.

The operator T* is well defined because the domain Dq-1 is dense in Lq-1. It is easy to see that if g G Dq * n Eq then T*g = d* g, where dp = *-1<9p * is the formal adjoint of dp.

Moreover, the Stokes theorem tells us that the elements of Dq *, which are smooth up to the boundary of Z, satisfy certain conditions on dZ. We write these in the form n(g) =0 on dZ, where n(g) is the complex normal component of g, cf. Section 3.2.2 in [12]. The equality n(g) = 0 means that the coefficients of g at each point of dZ satisfy a homogeneous system of linear equations, the latter varying smoothly over dZ.

Lemma 1.2. T*Dq * C Dq-1 and T*2 = 0.

q-2

Proof. Indeed, if g G Dq * and u G Dq then by the very definition and Lemma 1.1 we get

(Tu, T *g)£,-i = (T (Tu),g)£, =0. Therefore, T*g G Dq-1 and T*(T*g) = 0, as desired. □

Let us introduce an operator L on Lq with a domain Dq, which has the property that if u G Dq n £q then Lu = Au, where A = df + dfBFp is the Laplacian of the Dolbeault complex on Z' with coefficients in F. Namely, write Dq for the set of all u G D|, n D|,* with the property that Tu G Dq+1 and T*u G Dq-1. Then the operator L: Dq ^ Lq is defined by

Lu = T *Tu + TT *u,

cf. § 4.2 in [12].

The -Neumann problem on the manifold Z in the L2 setting consists in the following: Given a section f G Lq, when is there u G Dq such that Lu = f, and how does u depend on f ?

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

Hq = {u G Dq n Dq * : Tu = T*u = 0},

for q = 0,1,..., n. Since the operators T and T* are closed, Hq is a closed subspace of Lq. Denote by H: Lq ^ Hq the orthogonal projection of Lq onto Hq.

Lemma 1.3. u G Hq if and only if u G Dq and Lu = 0.

Proof. If u G Hq then obviously u G Dq and Lu = 0. If Lu = 0 then (Lu, u)^ = 0, and since

(Lu,u)£q = ||Tu||L„+i + ||T *uyLq-i

we have u G Hq. □

Lemma 1.4. The operator L is selfadjoint, and (L + 1)-1 exists, is bounded, and is everywhere in Lq defined.

Proof. Since T is a closed operator and the domain of T is dense, the same is also true for T*, and (T*)* = T.

It follows that the operators (TT* + 1)-1 and (T*T + 1)-1 exist, are bounded, selfadjoint and defined everywhere in Lq, cf. [13].

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

(L + 1)-1 = (TT * + 1)-1 + (T * T + 1)-1 - 1, which completes the proof. □

Corollary 1.1 (weak orthogonal decomposition). The range of L is orthogonal to Hq, and

Lq = Hq © LDL, (2)

where ¿Dq denotes the closure of ¿Dq in Lq.

Proof. This follows immediately from the selfadjointness of L and Lemma 1.3. □

In particular, if LDq is closed then we arrive at the "strong orthogonal decomposition"

Lq = Hq © t*TDq © TT*Dq. (3)

2. The d-Neumann Operator

The results of this and the next section go back at least as far as [2]. We bring them only for completeness.

Definition 2.1. Let LD® be closed and f e L, then f = Hf + Lu where u e . The dp-Neumann operator N: L« ^ D^ is defined by Nf = u — Hu.

Note that N is well defined. Indeed, if also f = Hf + Lu' where u' e D^ then L(u — u') = 0 whence

(u — Hu) — (u' — Hu') = (u — u') — H (u — u') = 0.

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

Lemma 2.1. Suppose LD^ is closed. Then the dp -Neumann operator N has the following properties:

1) N is bounded, selfadjoint, HN = NH = 0, and we have the orthogonal decomposition

f = Hf + T *TNf + TT *Nf (4)

for all f e L.

2) If f e Df, and Tf = 0 then TNf = 0. If moreover LDq+1 is closed then TNf = NTf.

3) If f e DT* and Tf = 0 then T*Nf = 0. If moreover LDq-1 is closed then T*Nf = NT *f.

Proof. See [12, 4.2.5] and elsewhere. □

The Laplacian A is well known to be an elliptic differential operator on Z'. Hence it follows that the harmonic differential forms u e H« are infinitely differentiable in the interior of Z, and the dp-Neumann operator N, if exists, preserves the interior regularity.

Beginning with its classical forms, the Dirichlet norm has been an important technical tool in studying the dp-Neumann problem. Given any u, v e D^ n D|,*, the Dirichlet inner product of these differential forms is defined by

D(u, v) = (Tu, Tv)l?+i + (T*u, T*v)l?-i + (u, v)l?,

and the Dirichlet norm is D(u) = \JD(u, u).

The space D^ n D|,* with the Dirichlet norm is a complete (Hilbert) space. It is denoted by

Dq.

Since D(u) > ||u||£q for all u e Dq there exists only one selfadjoint operator S with a domain D| CD«, such that if u e D| and v e Dq then

D(u, v) = (Su, v)l? . (5)

The following lemma gives a useful description of the operator L because our estimates will be in the norm D(u).

Lemma 2.2. The equalities hold D« = D| and L = S — 1, where the operator S is defined by (5).

Proof. If u e D« and v e D«, then D(u, v) = ((L + 1)u, v)Lq is fulfilled. Hence by the uniqueness of S, we have S = L + 1. □

3. Completely Continuous Norms

Let || • ||i and || • ||2 be two norms on a vector space L. We will say that the norm || • ||i is completely continuous with respect to the norm || • ||2 if every sequence in L which is bounded in the norm || • ||1 has a convergent subsequence in the norm || • ||2.

Lemma 3.1. If the norm D on Dq is completely continuous with respect to || • ||£q then Hq is finite dimensional.

Proof. Observe that if u,v G Hq then D(u,v) = (u,v)Lq. Suppose that the dimension of Hq is infinite. Then there exists an infinite sequence {uV} of orthonormal elements in Hq. Since D(uV) = ||uV ||Lq = 1 the sequence {uV} contains a convergent subsequence. But this is at variance with the fact that if v = ^ then ||uV — uM||Lq = a/2. □

Lemma 3.2. If the norm D on Dq is completely continuous with respect to || • then there exists a constant c > 0 such that for all u G Dq orthogonal to Hq, we have

||Tu||L, + l + ||T^ C |u|Lq .

Proof. Consider the Hilbert space Lq+1 x Lq-1 which is equipped with the norm

||{/,v}|| = (||/||Lq+i + ||v|Lq-1)1/2 .

Let M : Dq ^ Lq+1 x Lq-1 be the mapping defined by Mu = {Tu, T*u}. Note that M is a closed operator.

We will prove that the range of M is closed. Suppose that MDq is not closed. Then there exists a sequence {uV} in Dq, such that lim MuV = {/, v} and {/, v} G MDq.

Set uV = uV — HuV, then uV are orthogonal to Hq and limMuV = {/, v}. If ||uVare bounded then D(uV) = (|MuV||2 + ||uV||Lq)1/2 are bounded, too. Then by hypothesis {uV} has a convergent subsequence with a limit u, and since M is closed then Mu = {/, v} which contradicts the assumption that {/, v} G MDq. Thus by choosing a subsequence, if necessary, we may actually assume that lim ||uV||Lq =

Now set Uv = uV/|uV||£q. Then lim ||MUv|| = 0 and D(Uv) are bounded. Therefore {Uv} has a convergent subsequence {UVk} such that

lim Uvk = U, lim MUvfc = {0,0}.

Hence MU = 0 so that U G Hq. Since UV is orthogonal to Hq we have U = 0, but ||UV||Lq = 1. This contradiction proves that the range MDq is closed in Lq+1 x Lq-1.

Let R be the restriction of M to the orthogonal complement of Hq in Dq. Then R is one-to-one and has a closed range. By the closed graph theorem, the inverse R-1 is bounded. Hence there is c > 0 such that ||Ru||2 > c ||u||Lq. This proves the lemma. □

Theorem 3.1. If the norm D on Dq is completely continuous with respect to the norm || • , then LDqq is closed.

Proof. By Lemma 3.2, there exists c > 0 with the property that for all u G Dq which are orthogonal to Hq we have

(Lu,u)£q > c ||u||Lq ,

so that ||Lu||Lq > c ||u||Lq.

Set f = limLuv. We may assume that uv are orthogonal to H«, and then ||uv||Lq are uniformly bounded. Therefore, {uv} has a subsequence whose arithmetic means converge, cf. [13] Denoting this limit by u, we get f = Lu, which completes the proof. □

The question of when the norm D on D« is completely continuous with respect to the norm || • ||Lq, is very difficult in the general case and it requires special consideration. We present some consequences here.

Corollary 3.1. Suppose the norm D on D« is completely continuous with respect to the norm || • Ho . Then the dp -Neumann problem is solvable at step q in the sense that there exist operators H and N in L« with properties 1)-3) of Lemma 2.1.

Proof. This follows immediately from Lemma 2.1 and Theorem 3.1. □

For compact manifolds with boundary Z the subspace H0 is usually of infinite dimension. So by Lemma 3.1 the Dirichlet norm D may not be completely continuous with respect to the norm || • ||Lo on D0. However, the following result holds.

Theorem 3.2. If the norm D on D1 is completely continuous with respect to the norm || • ||li then LD 0 is closed.

Proof. See for instance [12, 4.2.6]. □

The next result immediately follows from Lemma 2.1 and Theorem 3.1. Recall that H0 = ker T0.

Corollary 3.2. Suppose the norm D on D1 is completely continuous with respect to the norm || • ||li . Then f = Hf + T*NTf for any section f e D°, where H: L0 ^ H0 is the orthogonal projection.

When acting on sections of F0 = F, the differential operator dp has injective symbol. Since

H0 = {u e L0 n C£C(Zo, F) : dpu = 0},

where Zo stands for the interior of Z, the operator H0 is a generalisation of the classical Bergman projector. Corollary 3.2 gives H0 = 1 — T* NT.

4. Pseudolocal Estimates

The regularity of the dp-Neumann operator near the boundary of Z is a much more delicate problem. It initiated the study of non-elliptic boundary value problems, thus motivating a development of pseudodifferential theory, cf. [4]. Kohn proved in [2] that if Z is a compact strongly pseudoconvex manifold then the norm D on D« is completely continuous with respect to the norm || • || Lq for all q = 1, ...,n. Moreover, the dp -Neumann operator preserves the regularity up to the boundary in the scale of Sobolev spaces Hs(Z, F«), with s = 0,1,..., in the sense that f e Hs(Z, F«) implies Nf e Hs(Z, F«). Kohn's original approach was considerably simplified in [4] in a very general framework via elliptic regularisation.

One says that a subelliptic estimate of order e > 0 holds for the dp-Neumann problem at step q in a neighbourhood U of a boundary point z0 e dZ if there is a constant c such that

||u||ff-(Z,pq) < cD(u) (6)

for every smooth form u which is supported in Z n U and satisfies the boundary condition n(u) =0 on dZ n U.

The systematic study of subelliptic estimates in [4] provides the following "pseudolocal estimate."

Theorem 4.1. Let Z be a compact pseudoconvex manifold with boundary. Suppose a subelliptic estimate (6) holds in a neighbourhood U of a boundary point z0. Pick arbitrary functions y, ^ G C£ (U), such that ^ = 1 in a neighbourhood of the support of y. Then, for every non-negative s there is a constant C with the property that

llyNf < C(||^/||h.(z,f«) + ||/ ||L)

for all / GLq n Hs(U, Fq).

Proof. See Theorem 4 and Remark 6.2 in [4]. This result is actually mentioned in [3], cf. Theorem 8. □

For a compact strongly pseudoconvex manifold Z, a subelliptic estimate (6) with e = 1/2 holds in a neighbourhood of every boundary point, provided 1 < q < n. It follows that for such manifolds the df-Neumann operator is continuous from Hs(Z, Fq) to Hs+1(Z, Fq).

If Z is not compact then the Of-Neumann problem on Z need not be solvable in the sense that the range LD^ is closed in Lq. In order to guarantee the normal solvability one has to arrange the problem with the points at infinity. As usual, this would require pseudodifferential analysis in weighted Sobolev spaces. Still, we may try to maintain the pseudolocal estimate of Theorem 4.1, thus showing the local regularity for the df-Neumann problem on a non-compact strongly pseudoconvex manifold Z.

Corollary 4.1. Assume that U is a neighbourhood of a boundary point z0, V a relatively compact open subset of U, and s a non-negative integer. If u G satisfies Lu G Hs(U, ) then u G Hs+1 (V, ) and

yuyHs+i(v,Fq) < C(yLu|Hs(u,Fq) + ||u||L2(u,Fq where C depends on U, V and s but not on u.

Proof. In case the closure of V does not meet dZ the assertion follows from the interior regularity of the Of-Neumann. Hence we can assume that V is small enough, for if not, we shrink it. Since each boundary point of Z possesses a neighbourhood whose closure is a compact strongly pseudoconvex manifold, we can assume without loss of generality that U is a compact strongly pseudoconvex manifold with boundary. It is convenient to choose U sufficiently small, so that the harmonic spaces on U be trivial.

For every q = 0,1,..., n, choose a parametrix of the Laplacian Aq on Z ', by a parametrix is meant an inverse modulo smoothing operators, see [12, 2.1.4] and elsewhere. This is a classical pseudodifferential operator of order —2 and type Fq ^ Fq on Z '. The Schwartz kernel KGq of

is a Csection of the bundle Fq kl Fq' away from the diagonal of Z ' x Z '.

Fix any z in the interior of U and denote by Cq(z, •) the unique solution of the Of-Neumann problem

ACq(z, •) = 0 in U,

n(Cq(z, •)) = n(*-1KG, (z, •)) on dU, (7)

n(dFCq(z, •)) = n(dF *-1 (z, •)) on dU

in U. The kernel

Kq(z, •) := *-1KG„ (z, •) - Cq(z, •)

gives a parametrix of the dp -Neumann problem at step q in U in the sense that the Green formula

u(z) = i (n(u),t(dpKq(z,-)))c + (n(dpu),t(Kq(z, 0))c ds + / (Au,Kq(z, -))c dv (8) jdU ./u

holds for all u G H2(U, Fq) up to a term Su, where S is a smoothing operator on U. By t(f) is meant the complex tangential component of f on dU, cf. Section 3.2.2 in [12].

Formula (8) is actually valid for all u G L2(U,Fq) with Au G L2(U,Fq). In this case the values n(u) and n(dpu) on dU are interpreted in a weak sense. To make it more precise it suffices to assume that the neighbourhood U is small enough. Using a local fundamental solution of A on Z' we find a differential form u0 G H2 (U, Fq) which satisfies Au0 = Au in U. Obviously, the traces of n(u0) and n(dpu0) on dU are well defined. Furthermore, the difference v = u — u0 lies in L2(U, Fq) and satisfies Av = 0 in U. Hence both n(v) and n(dpv) possess weak limit values on dU, cf. [14] and elsewhere. We set n(u) = n(u0) + n(u — u0) on dU, and similarly for n((9pu0).

Having disposed of this preliminary step, we can now return to the proof of the estimate. Let u G Dq be an arbitrary form with Lu G Hs(U, Fq). Write NU for the dp- -Neumann operator on the manifold U. By Theorem 4.1, u ' = NULu belongs to Hs+1(U, Fq) and satisfies

||u'||Hs+i(u,pq) < C' ||LuyHs(u,pq), (9)

where C is a constant independent of u. Since

Au = Lu in U, n(u') = 0 on dU, n(<9pu') = 0 on dU,

the difference u '' = u — u ' lies in L2(U, Fq) and fulfills Au ' ' = 0 weakly in the interior of U. Moreover, both n(u'') = n(u) and n(<9pu '') = n(<9pu) vanish on dZ n U. Applying (8) yields

u''(z) = i (n(u''),t(5^Kq(z, 0))c + (n(dpu''),t(Kq(z, ^)))c ds

JdU

up to a term Su ''. It follows that u '' G CTO(1/, Fq).

Since V CC U, there is a function x G CTO(?7) which is equal to 1 in a neighbourhood of dU \ dZ and vanishes near V. By the Stokes theorem, the above formula transforms to

u ' '(z) = /* (u '', A(xK(z, 0))c dv + Su ''

U

for all z G V. Hence

||u''|Hs+i(vip,) < C''(|u|L2(u,pq) + ||u'yl2(U,pq)) (10)

with C'' a constant independent of u. Combining (9) and (10) completes the proof. □

Perhaps, there is a direct proof of Corollary 4.1 using Theorem 4.1 but we have not been able to do this.

5. Spectral Projection

By Lemma 1.4, the operator L in is selfadjoint, and (L + 1)-1 is defined on all of . If the operator (L + 1)-1 is compact then the spectrum of L consists of at most countable many eigenvalues Aj > 0 which have no accumulation point but However, (L + 1)-1 fails to be compact for non-compact strongly pseudoconvex manifolds Z.

By the spectral theorem, for L there exists a unique orthogonal resolution Et, t > 0, of the identity on , such that

y(L)= / y(t) dEt

Jo-

for all admissible functions y on r. It is easy to see from this that the spectrum of L coincides with the union of the sets of points of increase of all functions (Etu,u)Lq, where u G .

The operator := E^+0 — E^ is an orthogonal projection of onto the corresponding eigenspace of L. The spectral function Et of L is thus a 'sum' of P^ over all A < t, i.e.,

Et = / PA dm(A) (11)

o-

where m(A) is a non-decreasing function on r.

For any interval I = [a, b], the operator E/ := Eb — Ea is an orthogonal projection in . It commutes with L, i.e., the equality E/L = LE/ holds on the domain of L. We see that L keeps invariant the range of E/.

By the spectral kernel function of the operator Lq is meant the Schwartz kernel KEq of the operator Ef. This is an element of D'(Z' x Z ', h ) with support in Z x Z, such that

(Efu,v)z = ,v <g> u)zxZ

for all u G C~mp(Z,) and v G C~mp(Z, ). Taking liberties one writes

Efu (z)=/ (KEq (z, 0,u)c (12)

Jz

for u G C£0mp(Z,). We next show that the integral makes sense for all distributions u G EZ(Z', ), i.e., for all generalized sections of with compact support in Z.

Theorem 5.1. The spectral kernel function of Lq is infinitely differentiable up to dZ, i.e., ke,q G C£C (Z x Z, h ).

Proof. Since

top.

q~(Z x Z, h ') = CO(Z, )ê n q~(Z, ) = £6(EZ (z', ), C~(Z, )),

the last equality being a consequence of the Schwartz kernel theorem, cf. for instance [12, § 1.5.1], it suffices to show that Et extends to a continuous map of EZ(Z', ) to Cj£C(Z, ). If we prove that Et extends to a continuous map of H—mp(Z, ) to HjOc(Z, ) for each non-negative integer s, the assertion readily follows.

As mentioned, Et is an orthogonal projection in . It follows that EtEt = Et and E* = Et. Using the connection between the adjoint and transposed operators, we arrive at the formula

Et = Et *-1 Et * . (13)

If Et maps L2(Z, Fq) continuously to Hfoc (Z, Fq), then the transpose E maps H-,mp (Z, Fq') continuously to L2(Z, Fq'). Hence the equality (13) allows one to extend Et to a continuous map of H-smp(Z,Fq) to Hjsoc(Z,Fq). We are thus reduced to proving that Et maps L2(Z, Fq) continuously to Hjsoc(Z, Fq) for each non-negative integer s.

To this end, pick an arbitrary form u G Lq. Using formula (11) for we easily find

Ls Etu = i AsPAudm(A)

Jo-

and

|Ls = / ||AsPAu||2, dm(A) < t2s PHlL,,

Jo-

which is due to the Pythagor theorem. Applying Corollary 4.1 we conclude that Etu G

HSoc(Z, Fq).

To estimate a seminorm of Etu in Hfoc(Z, Fq), we fix a relatively compact open set V C Z. Choose relatively compact open sets V1,..., Vs in Z with the property that

V cc Ui cc ... cc Us.

We now appeal to Corollary 4.1 to successively estimate the norm of PAu in Hs(V, Fq), namely

||PAu||H=(V,Fq) < Cs (||LPAu||H=-i(Ui,Fq) + ||PAu||L2(Ui p)) < < Cs (A ||PAu||Hs-i(Ui,p,) + y^HU^Ui.F,)^

and similarly

Hs-j (Uj ) ^ Cs-j (A ||Pau|| Hs j i(Uj + i, F q) + ||pAu|L2(Uj+iJF,)) for each j = 1,..., s — 1. Substituting these estimates into each other, we easily obtain

s

||Pau||hs(v,fq) < const(s) (]T Aj) ^uH^^p) <

j=o

s

< const(s)(]T Aj) |PAu|£q

j=0

whence

||Etu|Hs(v,Fq) < i ||Pau||h=(v,f q) dm(A) < const(s) / (V Aj) ||PAu||£q dm(A) < C ||u||£q.

Jo- Jo- y = 7

Here, the constant C depends on s, V and t but not on u. This completes the proof. □

Acknowledgements. The research of the first author was supported by the German Academic Exchange Service (DAAD).

References

[1] C.B.Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin et al., 1966.

[2] J.J.Kohn, Harmonic integrals on strongly pseudoconvex manifolds. I, II, Ann. Math., 78(1963), no. 1, 112-148; 79(1964), no. 3, 450-472.

[3] H.R.Boas, E.J.Straube, Global regularity of the д -Neumann problem: A survey of the L2-Sobolev theory, Several Complex Variables, 37(1999), 79-111.

[4] J.J.Kohn, L.Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math., 18(1965), 443-492.

[5] I.Lieb, J.Michel, The Cauchy-Riemann Complex, Vieweg, Braunschweig/Wiesbaden, 2002.

[6] R.Beals, N.K.Stanton, Estimates on kernels for the д-equation and the д-Neumann problem, Math. Ann., 289(1991), 73-83.

[7] A.Nagel, E.M.Stein, Lectures on Pseudo-Differential Operators: Regularity Theorems and Applications to Non-Elliptic Problems, Princeton University Press, Princeton, New Jersey, 1979.

[8] M.Englis, Pseudolocal estimates for <9 on general pseudoconvex domains, Indiana Univ. Math. J., 50(2001), 1593-1607.

[9] G.Metivier, Spectral asymptotics for the д-Neumann problem, Duke Math. J., 48(1981), 779-806.

10] R.Beals, N.K.Stanton, The heat equation for the д-Neumann problem. I, II, Comm. Part. Diff. Eq, 12(1987), 351-413; Canad. J. Math., 40(1988), 502-512.

11] D.M.McAvity, H.Osborn, Asymptotic expansion of the heat kernel for generalized boundary conditions, Classical Quantum Grav., 8(1991), 1445-1454.

12] N.Tarkhanov, Complexes of Differential Operators, Kluwer Academic Publishers, Dordrecht, NL, 1995.

13] F.Riesz, B.Sz.-Nagy, Lecons d'Analyse Fonctionnelle, Acad. des Sci. de Hongrie, Budapest, 1952.

14] Ya.A.Roitberg, Elliptic boundary value problems in the spaces of distributions, Kluwer Academic Publishers, Dordrecht, NL, 1996.

15] N.Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann., 195(1972), 149-158.

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спектральной проекции для комплексной задачи Неймана

Амар Олсиди Николай Тарханов

Мы показываем, что L2-спектральное ядро для 'решения д -задачи Неймана на некомпактном

строго псевдовыпуклом многообразии является гладким вплоть до границы.

Ключевые слова: д -задача Неймана, строго псевдовыпуклые области, спектральное ядро.