aa r X i v : . [ m a t h . C V ] O c t THE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES JOE J PEREZ
Abstract.
Let G be a unimodular Lie group, X a compact manifold withboundary, and M the total space of a principal bundle G → M → X sothat M is also a strongly pseudoconvex complex manifold. In this work, weshow that if G acts by holomorphic transformations satisfying a local property,then the space of square-integrable holomorphic functions on M is infinite G -dimensional. Introduction
Basic definitions and statement of main theorem.
Let M be a complexmanifold with nonempty smooth boundary bM , ¯ M = M ∪ bM , so that M is theinterior of ¯ M , and dim C ( M ) = n . We will also assume for simplicity that ¯ M is aclosed subset in f M , a complex neighborhood of ¯ M so that the complex structureon f M extends that of M , and every point of ¯ M is an interior point of f M .Let us choose a smooth function ρ : f M → R so that M = { z | ρ ( z ) < } , bM = { z | ρ ( z ) = 0 } , and for all x ∈ bM , we have dρ ( x ) = 0. For any x ∈ bM define the holomorphictangent plane to the boundary at x by T C x ( bM ) = { w ∈ C n | n X k =1 ∂ρ∂z k (cid:12)(cid:12)(cid:12)(cid:12) x w k = 0 } . For x ∈ bM , define the Levi form L x by L x ( w, ¯ w ) = n X j,k =1 ∂ ρ∂z j ∂ ¯ z k (cid:12)(cid:12)(cid:12)(cid:12) x w j ¯ w k , ( w ∈ T C z ( bM )) . Then M is said to be strongly pseudoconvex if for every x ∈ bM , the form L x ispositive definite. Since ρ is real-valued, the Taylor expansion at x of ρ is(1) ρ ( z ) = ρ ( x ) + 2 R e f ( z, x ) + L x ( z − x, ¯ z − ¯ x ) + O ( | z − x | ) , ( z ∈ C n ) MR Classification numbers: 32E40; 32W05; 43A30.Keywords: ¯ ∂ -Neumann Problem, Subelliptic operators, Harmonic Analysis. with(2) f ( z, x ) = n X k =1 ∂ρ∂z k (cid:12)(cid:12)(cid:12)(cid:12) x ( z k − x k ) + 12 n X jk =1 ∂ ρ∂z j ∂z k (cid:12)(cid:12)(cid:12)(cid:12) x ( z j − x j )( z k − x k ) . This f is holomorphic in M ∩ U x , with U x a small neighborhood of x , and vanishesonly at x . We will see why later, but the point is the positivity of L x . It happensthat negative powers of f are also holomorphic in the neighborhood and blow upat x . A question we will address in this paper is under which conditions we cancorrect this locally defined f τ to obtain a global holomorphic function on M . Inthose conditions we will also say something about the size of O ( M ), the space ofall holomorphic functions on M .The motivation behind this question is described in detail in [Si]. Early in thestudy of functions of several complex variables, the idea of pseudoconvexity arosein [L].A point x ∈ bM is called a peak point for O ( M ) if there exists an f ∈ O ( M )unbounded on any neighborhood of x and bounded in the complement of thatneighborhood.The Oka-Grauert theorem [Gr] asserts that if ¯ M ⊂ C n is compact, has nonemptyboundary, and is strongly pseudoconvex, then every point of the boundary is apeak point for O ( M ). One way of proving this theorem and its variants will bedescribed in this introduction.A point x ∈ bM is called a local peak point for O ( M ) if there exist a function f ∈ O ( M ) and a neighborhood V of x in M such that f is unbounded on V , butbounded on V \ U x for any neighborhood U x of x in M . It was proven in [GHS]that if M is a strongly pseudoconvex complex manifold admitting a free cocompactholomorphic action of a discrete group, then every point in the boundary of M isa local peak point for L ∩ O ( M ), necessarily nontrivial.The goal of the present work is to extend this last result (that L ∩ O benontrivial) from [GHS] to general unimodular Lie group bundles. With a technicalassumption (that we call amenability) on the local properties of convolutions offunctions f τ we will demonstrate Theorem 1.1.
Assume that G is a unimodular Lie group and G → M → X aprincipal G -bundle. Assume further that the total space M is a strongly pseudocon-vex complex manifold on which G acts amenably by holomorphic transformationsand that X is compact. Then dim G L O ( M ) = ∞ . It is natural to assume the unimodularity of G in this context and not onlybecause it is an important tool in our formalism. In fact, [GHS] contains a G -bundle with non unimodular structure group having L ∩ O ( M ) = { } . However, HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 3 unimodularity is not the whole story. The same paper also describes a G -manifoldwith nonunimodular structure group and many holomorphic functions.Another word on the relationship between the results in [GHS] and ours: if inour setting the structure group G possesses a cocompact discrete subgroup Γ, then[GHS] is applicable and one obtains local peak points and nontriviality solving areduced problem:(3) G → M → X Γ → M → ( X × ( G/ Γ)) . Generically, however, it is not the case that a unimodular Lie group have such asubgroup, cf. [M].It does not seem to us that the methods in the present paper always allowdirect construction of unbounded holomorphic functions. Still, when holomorphicfunctions can be constructed here, they are not smooth in the boundary. Inparticular, they will not possess holomorphic extensions beyond the boundaryand so remain in the spirit of the early investigations of pseudoconvexity. Anatural source of examples of complex manifolds satisfying the hypotheses (exceptperhaps amenability) are the Grauert tubes of unimodular Lie groups G and ofreal-analytic manifolds of the form K × G with K compact.1.2. Compact case.
We begin by reviewing the case when M is compact, mod-ifying the argument used in [FK] to conform to our method. We discuss theconstruction of holomorphic functions with peak points because it turns out thatit is essentially our method of constructing any clearly nontrivial holomorphicfunctions in subsequent sections. Suppose M is a compact complex manifoldwhose boundary is strongly pseudoconvex and for a point x ∈ bM , we want aholomorphic function blowing up at x . Define the antiholomorphic exterior de-rivative ¯ ∂ : Λ , → Λ , in local coordinates ( z k ) k by ¯ ∂u = P ∂u∂ ¯ z k d ¯ z k . If it can beestablished that ¯ ∂u = φ has a smooth solution u whenever φ is a smooth antiholomorphic one-form thatsatisfies the compatibility condition ¯ ∂φ = 0, then we may construct the functiondesired. The first step is to use the pseudoconvexity property of the boundary toconstruct a function f , holomorphic in a neighborhood U x of x , that blows up justat x , as indicated before. Next, we can take a smooth function χ with supportin U x that is identically equal 1 close to x . Extending χf by zero on the rest of M , we obtain a function, which we also call χf , defined everywhere and smoothaway from x . Furthermore, ¯ ∂ ( χf ) = ( ¯ ∂χ ) f = 0 near x , so ¯ ∂χf can be extendedsmoothly to the boundary. If we can now find a smooth solution to ¯ ∂u = ¯ ∂χf ,then χf − u is holomorphic and must blow up at x since u is smooth up to theboundary. Let us describe the construction of solutions u ∈ L ( M ) to ¯ ∂u = φ with φ ∈ L ( M, Λ , ), ¯ ∂φ = 0. Note that solutions will only be determined modulothe kernel of ¯ ∂ consisting of square-integrable holomorphic functions. Also, it is JOE J PEREZ preferable to deal with self-adjoint operators, so since the Hilbert space adjoint¯ ∂ ∗ of ¯ ∂ satisfies im ¯ ∂ ∗ = (ker ¯ ∂ ) ⊥ , it is sufficient to seek u of the form u = ¯ ∂ ∗ v satisfying(4) ¯ ∂ ¯ ∂ ∗ v = φ. This is a self-adjoint operator. In order to eliminate the compatibility conditionon φ (and obtain an operator related to the Dolbeault cohomology of M also) letus add a term ¯ ∂ ∗ ¯ ∂v , thus obtaining(5) ( ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ ) v = φ, where φ need not be assumed to satisfy ¯ ∂φ = 0. Define the operator (cid:3) = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ .Notice that when ¯ ∂φ = 0 is true, equation (5) reduces to equation (4) becauseapplying ¯ ∂ to equation (5) gives ¯ ∂ ¯ ∂ ∗ ¯ ∂v = 0 which in turn implies0 = h ¯ ∂ ¯ ∂ ∗ ¯ ∂v, ¯ ∂v i = k ¯ ∂ ∗ ¯ ∂v k L ( M ) . Thus the new term in equation (5) vanishes when the compatibility conditionholds. So it is enough to prove the solvability of the equation (5). But in fact,as we will see below, it suffices to prove that the operator (cid:3) is Fredholm , i.e. thespaces ker (cid:3) and coker (cid:3) are finite-dimensional.The equation (cid:3) u = φ is a noncoercive boundary value problem. It has beenshown [K, FK] that on its domain in the antiholomorphic q -forms, when q > (cid:3) + 1 has the following regularity property. Let ζ , ζ be smoothcutoff functions for which ζ = 1 on supp( ζ ) and let H s ( M, Λ ,q ) be the integerSobolev space of sections in Λ ,q over M . Then (cid:3) v + v ∈ H s loc ( M, Λ ,q ) implies v ∈ H s +1loc ( M, Λ ,q ) and there exist constants C s so that(6) k ζ v k H s +1 ( M ) ≤ C s (cid:0) k ζ ( (cid:3) + 1) v k H s ( M ) + k ( (cid:3) + 1) v k L ( M ) (cid:1) uniformly in v . These inequalities imply that the operator ( (cid:3) + 1) − is boundedfrom L ( M, Λ ,q ) to H ( M, Λ ,q ) and so by Rellich’s theorem is a compact operatorin L ( M, Λ ,q ) because M is compact. Classical results of functional analysisallow one to conclude that (cid:3) has discrete spectrum with no finite limit point andeach eigenvalue has finite multiplicity. Hence (cid:3) has finite-dimensional kernel andcokernel and closed image ( i.e. it is a Fredholm operator).Now, one can solve equation (5) for all φ orthogonal to the finite-dimensionalkernel. As χf is unbounded, raising f to arbitrarily high powers generates linearlyindependent functions, still holomorphic in a neighborhood of x . Further, sincethe χf m have compact support, ¯ ∂ is injective on the vector space generated by { χf m | m = 1 . . . N } . It follows that for N sufficiently large, the image of thisspace under ¯ ∂ intersects the image of (cid:3) nontrivially: Q N = im (cid:3) ∩ span C { ¯ ∂χf m | m = 1 . . . N } 6 = { } . HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 5 This, together with the fact that Q N ⊂ im ¯ ∂ implies that ¯ ∂ ¯ ∂ ∗ u = φ can be solvedfor some φ ∈ Q N . Since all the forms ¯ ∂χf m are smooth, this φ will be smoothand so we proceed as indicated above.1.3. Regular coverings.
As we have mentioned, in [GHS] it was establishedthat all boundary points are local peak points when M is strongly pseudocon-vex and admits a free cocompact action of a discrete group Γ by holomorphictransformations. The proof above fails because when M is not compact, Rellich’stheorem no longer holds, so the dimension of the kernel and/or cokernel of (cid:3) maybe infinite-dimensional and the image of (cid:3) may not be closed. The von Neumanndimension of invariant subspaces of L (Γ) is used in order to measure the kerneland cokernel of (cid:3) in this setting as well as to measure the images of (cid:3) ’s spectralprojections. We describe this briefly. For a discrete group Γ, one forms L (Γ) = { ξ : Γ → C | X γ ∈ Γ | ξ ( γ ) | < ∞} . This is a Hilbert space with inner product h ξ, η i L (Γ) = P γ ∈ Γ ξ ( γ )¯ η ( γ ) and norm k ξ k L (Γ) = h ξ, ξ i . Now, Γ acts in L (Γ) by right translations R γ , γ ∈ Γ, definedby ( R γ ξ )( α ) = ξ ( αγ ) . Clearly, R γ is a unitary operator. A closed subspace L ⊂ L (Γ) is called invariant if it is invariant with respect to R γ for all γ ∈ Γ. It is true that if, in addition, ourinvariant subspace L is closed, then L is the image of a bounded left-convolutionoperator on the group: L = im L h where ( L h ξ )( α ) = X γ ∈ Γ h ( γ ) ξ ( γ − α )where h : Γ → C is called a convolution kernel. Furthermore, one can choose h sothat L h is a self-adjoint projection: L h = L ∗ h = L h . Here the adjoint L ∗ h is definedby h L ∗ h ξ, η i L (Γ) = h ξ, L h η i L (Γ) for all ξ, η ∈ L (Γ).Defining B ( L (Γ)) to be the continuous linear operators in L (Γ) and L Γ = { L h | h : Γ → C and L h ∈ B ( L (Γ)) } we see that L Γ consists of all operators in B ( L (Γ)) commuting with the right-translations. Von Neumann’s bicommutant theorem then gives that L Γ is a vonNeumann algebra. On L Γ there is a trace defined by(7) tr Γ ( L h ) = h ( e )and for a right-invariant subspace L = im L h with L h a self-adjoint projection, wedefine its Γ -dimension dim Γ ( L ) = tr Γ ( L h ) = h ( e ) . JOE J PEREZ
Notice that since the identity in B ( L (Γ)) is convolution with δ , the characteristicfunction of the identity, dim Γ ( L (Γ)) = tr Γ ( L δ ) = δ ( e ) = 1, though of coursedim C ( L (Γ)) = ∞ for infinite groups.Next, when Γ acts freely on a manifold M with compact quotient, X , onedecomposes the Hilbert space L ( M ) ∼ = L (Γ) ⊗ L ( X ) and defines a traceTr Γ = tr Γ ⊗ Tr B ( L ( X )) on the invariant operators. It is with the corresponding dimension that closed,invariant subspaces of L ( M ) are measured. In [GHS], it is shown that a variantof Kohn’s inequality (6) implies that the kernel of (cid:3) is finite-dimensional in thissense, though infinite-dimensional in the usual sense if nontrivial. Moreover (cid:3) isΓ-Fredholm in the sense that im (cid:3) contains a closed, Γ-invariant subspace of finiteΓ-codimension.The operator (cid:3) having the Fredholm property implies that the image of (cid:3) intersected with(8) L N = { X γ ∈ Γ N X m =1 c m,γ ( ¯ ∂χf m )( · γ − ) | X m,γ | c m,γ | < ∞}∼ = L (Γ) ⊗ span C { ¯ ∂χf, ¯ ∂χf , . . . , ¯ ∂χf N } ∼ = L (Γ) ⊗ C N contains closed, invariant subspaces Q of finite Γ-codimension in L N . Because ¯ ∂ is injective on the span of the χf m , m = 1 , , . . . , N , we have that dim Γ ( L N ) = N . As the kernel of (cid:3) has finite Γ-dimension, the image of (cid:3) contains closed,invariant subspaces of finite codimension, so the intersection im (cid:3) ∩ L N ⊂ L N willbe nontrivial if N is sufficiently large. Subsequently there exist closed, invariantnonempty subspaces Q ⊂ im (cid:3) ∩ L N . Picking a form φ = 0 in this Q , one sees thatit is smooth so (cid:3) u = φ is solvable and the rest of the argument is as previouslydescribed.1.4. G -bundles. In [GHS] it is shown that (cid:3) is Γ-Fredholm. In [Per], this theo-rem was adapted to the situation in which the discrete group Γ is replaced by aunimodular Lie group G (the reader is referred to [Per] for the relevant definitions).For a unimodular group with its biinvariant measure fixed, the left- and right-convolutions L ∆ , R ∆ by a distribution ∆ on G are defined as usual ( cf. § G on L G ⊂ B ( L ( G ))agreeing with tr G ( L ∗ h L h ) = Z G | h ( s ) | ds, whenever L h ∈ B ( L ( G )) and h ∈ L ( G ) . It is true that tr G ( A ∗ A ) < ∞ if andonly if there is an h ∈ L ( G ) for which A = L h ∈ B ( L ( G )).If we define ˜ h ( t ) = ¯ h ( t − ), and if h j , g j ∈ L ( G ), j = 1 , . . . , N , then theoperator L k = P N L ˜ h j L g j is in Dom(tr G ). Furthermore, k is continuous and HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 7 tr G ( L k ) = k ( e ), agreeing with the discrete case Equation (7). We will outline theconstruction of the invariant trace Tr G = tr G ⊗ Tr B ( L ( X )) below.As we have suggested, in [Per] we proved the following: Assume that G is aunimodular Lie group and G → M → X a principal G -bundle. Assume furtherthat the total space M is a strongly pseudoconvex complex manifold on which G acts by holomorphic transformations and that X is compact. Then, for q > , theoperator (cid:3) in Λ p,q ( M ) is G -Fredholm. The G -Fredholm property is similar to theΓ-Fredholm property described above, mutatis mutandis .In order to continue the program as in [GHS], the Γ-invariant spaces L N willneed to be replaced by G -invariant versions. These spaces will be constructedsimilarly to the L N in Equation (8), namely by taking (some) convolutions of ¯ ∂χf .As the bundle has a global right G action, we may write convolutions on M aswe would on just G . The spaces replacing the L N will be { R ∆ ¯ ∂χf | P δ ∆ = ∆ } ⊂ L ( M ) where P δ is some projection in B ( L ( G )) commuting with left-translations.With the P δ chosen appropriately, these spaces are closed, smooth, right-invariant,and of arbitrarily large G -dimension, analogously to the L N . Measuring thesenew spaces presents a new difficulty. In contrast to the discrete case in whichdim Γ L (Γ) = 1, there is now a complicated trace class for tr G . Our techniqueshere rely on methods developed in [Per] with the exception of the new materialin the present paper’s Sections 2 and 3.Similarly to the previous cases, for δ > (cid:3) ∩ { R ∆ ¯ ∂χf | P δ ∆ = ∆ } will contain nontrivial closed subspaces so we proceed as usual exceptfor one last contrast.In the compact case one constructs the function χf − u and the singularityof χf and smoothness of u guarantee χf − u = 0. In the discrete group case,nothing changes in this respect. In the present situation we will be faced with thepossibility that R ∆ χf be smooth to the boundary though χf has a singularitythere. In certain cases it is obvious that this cannot happen, but in others it is not(to us). We will handle a set of cases below in Section 4 and postpone a detaileddiscussion to a later paper. For now, let us say that our holomorphic action is amenable if there exists an x ∈ bM so that if f is the Levi polynomial at x , and F is either some negative power of f or the logarithm of f , then 1) χf ∈ L ( M ),2) k χF ( · , ξ ) k L ( G ) < ∞ for all ξ ∈ X , and 3) R ∆ χF / ∈ C ∞ ( ¯ M ) for all ∆ ∈ C ∞ ( G )(we have chosen a local section ξ : X → M in the support of χ ). In the eventthat the action be amenable, we have our result arguing similarly as is done inthe compact and covering space cases.1.5. Important examples.
In [GHS] a natural question is posed: is the co-compact unimodular group action relevant to the existence of holomorphic L -functions or is it just an toolmark of the method of proof? Now we might addanother question and ask if the existence of holomorphic functions on a G -bundle JOE J PEREZ has anything to do with amenability or if this is also just a useful tool in ourproofs.As we mentioned before, [GHS] presents an example with the following prop-erties. The complex dimension of M is 2, bM is strongly pseudoconvex, G is asolvable nonunimodular connected Lie group, dim R G = 3, G has a free action on¯ M which is holomorphic on M , ¯ M /G = [ − , L O ( M ) = { } .The point here is that if we only impose bounded geometry conditions anduniformly strong pseudoconvexity, then the space of holomorphic L -functionsmay be trivial.Now a further property, the amenability, is involved that may or may not trulybe relevant to the existence of holomorphic functions on such a manifold. Clearlymore examples need be constructed and analyzed. We will postpone this for thefuture.1.6. Other approaches and results.
Recent works on covering spaces extend-ing [GHS] are related to the Shafarevich conjecture (which asserts that the univer-sal covering of a projective complex manifold is holomorphically convex) and canbe found in [Br1, Br2, Br3]. The paper [TCM1] deals with the case in which M is only assumed weakly pseudoconvex. Using cohomological techniques and holo-morphic Morse inequalities, the authors obtain a lower bound for the Γ-dimensionof the space of L sections and upper bounds for the Γ-dimensions of the highercohomology groups. In [TCM2], it is shown that the von Neumann dimensionof the space of L holomorphic sections is bounded below under weak curvatureconditions on M .In the present work, Section 2 contains a method for determining that a closed, G -invariant subspace of L ( M ) be infinite G -dimensional. Section 3 describes amethod of constructing large, smooth, invariant subspaces of L ( M ). In Section4 we construct local expressions for functions and convolutions and briefly discussamenability. In Section 5 we prove that dim G L O ( M ) = ∞ . Section 6 discusses amethod by which the problem may be adjusted so as to give holomorphic functionswith stronger singularities.2. Paley-Wiener Theorems
This section is a small modification of a part of [AL].
Definition 2.1.
Let M be a G -manifold with an invariant measure. For f ∈ L ( M ) , define h f i ⊂ L ( M ) to be the L -closure of the complex vector spacegenerated by right-translates of f by G . In symbols, h f i = ( finite X k α k f ( · t k ) | α k ∈ C , t k ∈ G ) L ( M ) . HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 9 Theorem 2.2. [AL]
Let G be a locally compact unimodular group containing aclosed, noncompact, connected set. Let f be in L ( G ) such that meas (supp( f )) Let G → M → X be a principal G -bundle with G a unimodularLie group. If = h ∈ L ( M ) has sufficiently small support, then dim G h h i = ∞ .Proof. Let the support of h lie in a trivialization G × U , U ⊂ X of M and choosea section so that we may write h = h ( t, x ), t ∈ G , x ∈ X . Also let P be aself-adjoint invariant projection whose image contains h h i . By invariance P R t h = R t h for any t ∈ G . By Lemma 1.2 of [AL], there exists a sequence ( t k ) k ⊂ G for whichthe functions ( R t k h ) k are linearly independent and for which S = ∪ k supp( R t k h )has finite measure. Denote by χ S the characteristic function of S . The operator u χ S P u then has an infinite-dimensional eigenspace span { R t k h | k ∈ N } cor-responding to the eigenvalue one. We conclude that χ S P must not be a compactoperator.Let us compute the Hilbert-Schmidt norm of χ S P . Since P is invariant, its rep-resentation in terms of its distributional kernel κ takes the form( P u )( t, x ) = Z G × X dsdy κ ( st − ; x, y ) u ( s, y ) . If ( ψ k ) k is an orthonormal basis for L ( X ), the kernel of χ S P can be expanded ina Fourier series χ S ( t ) κ ( st − ; x, y ) = χ S ( t ) X kl H kl ( st − ) ψ k ( x ) ¯ ψ l ( y ) . Since ( ψ k ⊗ ¯ ψ l ) kl forms an orthonormal basis for L ( X × X ), H kl is the kl th Fourier coefficient of κ with respect to the decomposition L ( G × X × X ) ∼ = L kl ( L ( G ) ⊗ ψ k ⊗ ¯ ψ l ). We obtain k χ S P k HS = Z G × G dsdt | χ S ( t ) | X kl | H kl ( st − ) | = X kl k H kl k Z G dt | χ S ( t ) | = meas ( S ) X kl k H kl k and conclude that P kl k H kl k = + ∞ , for if not, we would have a Hilbert-Schmidt(and thus compact) operator χ S P with an infinite-dimensional eigenspace corre-sponding to eigenvalue one. We describe the invariant trace in L ( M ), ( cf. [T]).Again using the orthonormal basis ( ψ k ) k of L ( X ), we have(9) L ( M ) ∼ = L ( G ) ⊗ L ( X ) ∼ = M k ∈ N L ( G ) ⊗ ψ k . Denoting by P k the projection onto the k th summand in (9), we obtain a matrixrepresentation of any operator A ∈ B ( L ( M )) with elements A kl = P k AP l ∈B ( L ( G )). If A ∈ B ( L ( M )) G , we recover the H kl from above as matrix elements A ↔ [ A kl ] kl = [ L H kl ] kl . The G -trace of such an operator is given byTr G ( A ) = X k tr G ( L H kk ) . If P is a self-adjoint projection, we compute Tr G ( P ∗ P ) = P kl tr G ( L ∗ H kl L H kl ) = P kl k H kl k L ( G ) by normality of tr G and the definition of tr G . Thus dim G h h i =Tr G ( P ) = P kl k H kl k = ∞ . (cid:3) Smooth Invariant Closed Subspaces The group intrinsically. We gather some algebraic results. Define ˜ α ( t ) = α ( t − ) for any distributions α, β on G . The right-convolutions satisfy( R α β )( t ) def = Z G ds α ( s ) β ( ts ) = Z G ds β ( s ) α ( t − s ) = ( R β α )( t − ) , so R α β = g R β α , and if G is unimodular, then k R α β k L ( G ) = k R β α k L ( G ) . Usingthe definition ( L s α )( t ) = α ( s − t ), we obtain the identity( R α R β γ )( t ) = Z G ds α ( s ) (cid:20)Z G dr β ( r ) γ ( tsr ) (cid:21) = Z G dr (cid:20)Z G ds α ( s ) β ( s − r ) (cid:21) γ ( tr ) = ( R [ L α β ] γ )( t ) . In this subsection, assume H ∈ C ∞ c ( G ) and consider h H i ⊂ L ( G ). Any g ∈ h H i satisfies g = lim m g m with g m = R ∆ m H for some sequence (∆ m ) m ⊂ C ∞ c ( G ).Equivalently, ( g m ) m is Cauchy, thus(10) k g m − g n k = k ( R ∆ m − R ∆ n ) H k = k R H (∆ m − ∆ n ) k −→ . HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 11 Definition 3.1. Let R H = U | R H | be the polar decomposition of R H , with U apartial isometry, and let | R H | = R C λdE λ be the spectral decomposition of | R H | .For δ ∈ [0 , C ] ∪ { + } , let P δ = R Cδ dE λ and define h H i δ = { g ∈ h H i | P δ U ∗ ˜ g = U ∗ ˜ g } . Lemma 3.2. If δ > , then g ∈ h H i δ implies that g = R ∆ H for some ∆ ∈ L ( G ) .Consequently, h H i δ ⊂ H ∞ ( G ) .Proof. As in (10), let R ∆ m H → g ∈ h H i δ . Then R H ∆ m → ˜ g and U P δ U ∗ R H ∆ m → U P δ U ∗ ˜ g = ˜ g. The composition P δ U ∗ R H = P δ | R H | = P δ | R H | P δ , when restricted to the orthog-onal complement of ker P δ , is an injection with bounded inverse, as is U P δ U ∗ R H .Therefore there exists a Cauchy sequence (∆ ′ m ) m in L ( G ) ⊖ ker P δ with limit∆ g ∈ L ( G ) ⊖ ker P δ so that g = R ∆ g H. Noting that ∆ g ∈ L ( G ) for all g ∈ h H i δ and H ∈ H ∞ c ( G ), we have h H i δ ⊂ H ∞ ( G ). (cid:3) Remark 3.3. Since im | R H | = im( R ∗ H R H ) ⊂ C ∞ ( G ), we have im P δ ⊂ C ∞ ( G ) forall δ ∈ (0 , C ]. Lemma 3.2 and Corollary 6.4 of [Per] provide that dim G h H i δ < ∞ for δ > 0. The previous lemma gives that, if δ > 0, then h H i δ ⊂ { R ∆ H | ∆ ∈ im P δ } . In fact, the spaces are equal: Lemma 3.4. Let | R H | = R C λdE λ and P δ = R Cδ dE λ as before. Then, for any δ > , we have h H i δ = { R ∆ H | ∆ ∈ im P δ } .Proof. For δ > 0, all g ∈ h H i δ satisfy˜ g = U P δ U ∗ ˜ g = U P δ U ∗ R H ∆ g = U P δ | R H | ∆ g = U | R H | P δ ∆ g = R H P δ ∆ g , so each g ∈ h H i δ is of the form R ∆ g H for ∆ g ∈ im P δ . Conversely, if ˜ g = R H P δ ∆ g for ∆ g ∈ im P δ , the above chain of equalities can be read right to left, obtaining˜ g = R H P δ ∆ g = U P δ U ∗ ˜ g . (cid:3) Theorem 3.5. For δ ∈ (0 , C ] , the spaces h H i δ ⊂ h H i are closed, smooth, right-invariant, and dim G h H i δ → ∞ as δ → + .Proof. The invariance condition on h H i δ is equivalent to the statement g = R ∆ H for ∆ ∈ im P δ if and only if R t g = R t R ∆ H = R [ L t ∆] H ∈ h H i δ ( t ∈ G ) . Since P δ is a function of R H , it commutes with all left-translations so L t ∆ ∈ im P δ .For a moment consider the case in which δ = 0 + . The projection P + is ontothe closure of the image of | R H | = U ∗ R H and so P + U ∗ = U ∗ . The conditionrestricting h H i δ , P + U ∗ ˜ g = U ∗ ˜ g , is therefore vacuous, so h H i + = h H i . By Corollary 2.3, dim G ( h H i + ) = ∞ . Now, under the map g ˜ g we obtain anisomorphism(11) h H i δ = { R P δ ∆ H | ∆ ∈ L ( G ) } ∼ = { R H P δ ∆ | ∆ ∈ L ( G ) } = g h H i δ . Note that this isomorphism interchanges a left invariant subspace with a rightinvariant one. Let us see why they have the same G -dimension. If h H i δ is theimage of L h , a self-adjoint projection, then g h H i δ is the image of the self-adjointprojection R h . This is because L h g = g if and only if R h ˜ g = ˜ g . We concludethat tr G ( L h ) = tr G ( R h ) = k h k L ( G ) , which implies that h H i δ and g h H i δ have thesame G -dimension, though one is a left module and the other is a right module.In the polar decomposition R H = U | R H | , the partial isometry U commutes withleft-translations so, as left G -modules, g h H i δ = { R H P δ ∆ | ∆ ∈ L ( G ) } ∼ = {| R H | P δ ∆ | ∆ ∈ L ( G ) } ( δ > . It is obvious that the right hand side of the above expression is equal to im P δ , socombining these observations with Equation (11), we obtain dim G h H i δ = tr G ( P δ ),valid for δ > 0. Since ( P δ ) δ are a spectral family, P δ → P + strongly (implyingdim G h H i δ = tr G P δ → tr G P + ), and so normality of the trace gives the result assoon as we obtain tr G P + = ∞ . To wit, U im P + = im U P + = im R H and thishas infinite G -dimension since it contains H . (cid:3) Actions. For a function h ∈ C ∞ c ( ¯ M ) with small enough support, we maychoose a section and write h as a smooth function of ( t, x ) ∈ G × U where U ⊂ X .Since M has a global right G -action, we may abbreviate a convolution by ∆, R ∆ ⊗ L ( X ) , simply writing R ∆ . We obtain an expression for k R ∆ h k L ( M ) by firstdecomposing h as in Equation (9). With H k ( t ) = h h ( t, · ) , ψ k i L ( X ) , the function h = P k H k ⊗ ψ k and(12) R ∆ h = X k ( R ∆ H k ) ⊗ ψ k so k R ∆ h k L ( M ) = X k k R ∆ H k k L ( G ) . Remark 3.6. Let δ > R H k = U k | R H k | , | R H k | = R Cδ λdE kλ and the projections P kδ = R Cδ dE kλ . Then, for each l ∈ N for which R ∆ H l = 0 we have k R ∆ h k L ( M ) = X k k R ∆ H k k L ( G ) ≥ k R ∆ H l k L ( G ) ≥ δ k ∆ k L ( G ) (∆ ∈ im P lδ ) . This implies that im P lδ ∋ ∆ R ∆ h is boundedly invertible as long as R ∆ H l = 0.Let us then take D lδ = im P lδ for R ∆ H l = 0 and define h h i δ,l = { R ∆ h | ∆ ∈ D lδ } . Lemma 3.7. For δ > , the spaces h h i δ,l are closed, invariant, and smooth.Furthermore, dim G h h i δ < ∞ . HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 13 Proof. The previous remark and Lemma 3.4 give that the space h h i δ,l is closed.For δ > 0, Lemma 3.4 also provides that D lδ ⊂ C ∞ ∩ L ( G ). Consider the estimate k R ∆ h k L ( M ) = Z X dx Z G dt (cid:12)(cid:12)(cid:12)(cid:12)Z G ds ∆( s ) h ( ts, x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ∆ k L ( G ) Z X dx k h ( · , x ) k L ( G ) (13) . k ∆ k L ( G ) (cid:12)(cid:12)(cid:12)(cid:12)Z X dx k h ( · , x ) k L ( G ) (cid:12)(cid:12)(cid:12)(cid:12) = k ∆ k L ( G ) k h k L ( M ) , where A . B means that for some C > | A | ≤ C | B | uniformly. Recall the right-invariant Sobolev spaces as in [Per]. There, the derivatives defining the spacesessentially commute with right translations. Thus the above estimate implies k R ∆ h k H s ( M ) . k ∆ k L ( G ) k h k L s ( M ) where k · k L s ( M ) is the L norm of the derivativesup to order s . Since all derivatives of h are in L ( G ), we have h h i δ,l ⊂ H ∞ ( ¯ M ).Corollary 6.4 of [Per] states that if a space is closed, invariant, and in H ∞ ( ¯ M ),then it has finite G -dimension. (cid:3) Lemma 3.8. As δ → + , dim G ( h h i δ,l ) −→ + ∞ .Proof. By Equation (12), the space h h i δ,l has an orthogonal decomposition h h i δ,l = M k { R ∆ H k | ∆ ∈ D lδ } ⊗ ψ k . Now, since H l ( t ) = h h ( t, · ) , ψ l i L ( X ) ∈ C ∞ c ( G ), Theorem 3.5 holds and providesthat h H l i δ ⊗ ψ l = { R ∆ H l | ∆ ∈ D lδ } ⊗ ψ l is a closed, invariant subspace of h h i δ,l whose G -dimension is unbounded as δ → + . (cid:3) Levi’s Function and Its Convolutions As discussed in the introduction, we need to know when convolutions of thesingular functions gotten by taking the Levi polynomial to negative powers arenot smooth in the boundary. We will not fully answer this question here butprovide some tools and some simple examples, postponing a full analysis of thesituation. We start with an analysis of f itself. This first bit is in [GHS].Without loss of generality (replacing ρ by e λρ − λ > M so that the Levi form L x ( w, ¯ w ) is positivefor all nonzero w ∈ C n (and not only for w ∈ T cx ( bM )) and at all points x ∈ bM .Let us also assume that the defining function ρ is constant on the orbits of pointsof M and reconsider the Levi polynomial in Equation (2). The complex quadrichypersurface S x = { z | f ( z, x ) = 0 } has T C x ( bM ) as its tangent plane at x . Thestrong pseudoconvexity property implies that ρ ( z ) > f ( z, x ) = 0 and z = x is close to x . This means that near x the intersection of S x with bM contains only x . The function 1 /f ( · , x ) is therefore holomorphic in U ∩ M (where U is aneighborhood of x in f M ) and x is its peak point. Since ρ < M , (1) impliesthat Re f ( z, x ) < x ∈ bM and z ∈ M is sufficiently close to x . It follows thatwe can choose a branch of log f ( z, x ) so that g x ( z ) = log f ( z, x ) is a holomorphicfunction in z ∈ M ∩ U x where U x is a sufficiently small neighborhood of x in bM . Consequently all powers of f are also well-defined and holomorphic in aneighborhood of zero. Thus define a = ∂ρ∂z k (cid:12)(cid:12)(cid:12)(cid:12) x , M = 12 ∂ ρ∂z k ∂z l (cid:12)(cid:12)(cid:12)(cid:12) x , f τ ( z ) = (cid:18) a · z + M z · z (cid:19) τ ( τ > , and f ( z ) = log( a · z + M z · z ) where a · b = P a k b k . The functions f τ areholomorphic in a neighborhood of 0 and blow up only at 0. Lemma 4.1. Take coordinates in which x is zero in the above. Then for z suffi-ciently near zero in ¯ M there are constants C, D > so that C | z | ≤ | a · z + M z · z | ≤ D | z | . Proof. This is true because2 | a · z + M z · z | ≥ − Re ( a · z + M z · z ) ≥ ρ ( z ) − Re ( a · z + M z · z ) = L ( z, ¯ z ) + O ( | z | )and the Levi form has a smallest eigenvalue λ > 0, so L ( z, ¯ z ) > λ | z | . The otherestimate is obvious. (cid:3) Let U be a neighborhood of a point x = 0 of the boundary and choose a cut-offfunction χ ∈ C ∞ c ( U ), so that χ = 1 in a neighborhood of 0. Locally definedfunctions cut off by χ will be considered extended by zero tacitly. Lemma 4.2. Let χ ∈ C ∞ c ( ¯ M ) with small support near zero. Then χf τ ∈ L p ( M ) whenever τ ∈ [0 , n/p ) .Proof. For χf τ ∈ L p we only need f τ ∈ L ploc . By Lemma 4.1, there is a constant C ′ so that, with r = | z | (14) Z B ǫ | f τ | p dV ≤ C ′ Z ǫ r n − r pτ dr < ∞ whenever τ < n/p . The case of the logarithm is similar. (cid:3) Remark 4.3. Note that in the estimate (13), the quantity k h ( · , x ) k L ( G ) playsan important role. Later we will need a similar quantity k χf τ ( · , x ) k L ( G ) . Theestimate above gives that for all x ∈ X , k χf τ ( · , x ) k L ( G ) < ∞ as long as 2 τ < dim R G . Corollary 4.4. Let d = dim R G , τ ∈ [0 , d/ and ∆ ∈ L ( G ) . Then R ∆ χf τ ∈ L ( M ) . HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 15 Definition 4.5. Let ξ : X → M be a piecewise continuous section of G → M p → X so that ξ | p (supp χ ) is continuous. The action of G on M is called amenable ifthere exists an x ∈ bM and τ ≥ so that if f is the Levi polynomial at x , then1) χf τ ∈ L ( M ) , 2) k χf τ ( · , ξ ) k L ( G ) < ∞ for all ξ ∈ X , and 3) R ∆ χf τ / ∈ C ∞ ( ¯ M ) for all nonzero ∆ ∈ C ∞ ( G ) . Examples. Here we will present some simple examples regarding amenabil-ity and the nonunimodular examples from [GHS].4.1.1. The strip. Consider the tube of R : S = { z ∈ C | Im ( z ) ∈ ( − , } . The defining function is ρ ( z ) = Im ( z ) − 1, and with the basepoint x = a + ib ∈ bS for the Taylor expansion of ρ , the Levi polynomial is f ( z, ˜ ξ ) = − i b ( z − ˜ ξ ) − ( z − ˜ ξ ) . Let us take the basepoint x = i ↔ ( a, b ) = (0 , f ( z ) = f ( z, i ) = − i ( z − i ) − ( z − i ) = − ( z + 1) . The strip S has an obvious translation invariance, R ∋ t : z z + t and so convolutions of functions make sense on S . We can verify that Re f < i ∈ bS , so f can be raised to arbitrary real powers. Considerthen the convolution(∆ ∗ χf τ )( z ) = Z R dt ∆( t ) χ ( z + t ) f τ ( z + t )and its limits as S ∋ z → bS , or more specifically, z → i . These are approximatelylim z → i Z ǫ − ǫ dt [( z + t ) + 1] τ where ǫ ≈ diam(supp χ ) and we have ignored the details of ∆ ∈ C ∞ ( R ). Thesefunctions are well-behaved at the boundary, ( cf. [CS], Theorem 2.1.3) and so thisis not an amenable action.Of course, if we wanted to investigate L O in the strip, with the action of Γ = Z by translations along R , we could use [GHS], which applies. In other words, thestructure group R can be reduced to Z . Also, in this case L holomorphic functionscan be obtained by explicit construction, taking e.g. f ( z ) = 1 / ( a + z ) where a > 1, or exp( − z ) log( z − i ). Countably many strips. It may happen that G is nonunimodular but thereare many L -holomorphic functions on M . Consider the group G ∼ = Z ⋉ R ofmatrices g = (cid:20) n x (cid:21) ( n ∈ Z , x ∈ R )and let M be the tube of G consisting of all matrices h = (cid:20) m z (cid:21) ( m ∈ Z , z ∈ C , | Im z | < . Then M is a disjoint countable union of strips { z ∈ C | | Im z | < } . Consider M as a (non connected) complex manifold with boundary with ¯ M obtained bytaking closure of each strip, so that ¯ M consists of the matrices of the same formwith | Im z | ≤ 1. It is clear that M is strongly pseudoconvex.The action of G on M is obtained by left multiplication of the matrices: g · h = gh . It amounts to interchanging the strips and real translations along the stripsdepending on the strip. This action is obviously holomorphic and free. It is easyto see that ¯ M /G = [ − , C ), we obtain an invariant metric. The L holomorphicfunctions on each strip constructed before, can be extended to M by 0. Here,questions of amenability are similar to the single strip case.4.1.3. Trivial L O . Let Ω ⊂ C be the Siegel domain of the second kind, Ω = { ( z , z ) ∈ C | Im z > | z | } equipped with its Bergman metric. For ǫ > M ǫ ⊂ Ω given by M ǫ = { ( z , z ) | y > x + y /ǫ } on which the matrix group G consisting of matrices g = λ iλξ t + iξ λ ξ ( ξ, t ∈ R , λ > z , z , T . Easy computations show that foreach ǫ > M ǫ is the strongly pseudoconvex total space of a G -bundle withdim R G = 3 acting in M ǫ by holomorphic transformations and ¯ M /G ∼ = [ − , L O on this space is trivial ( cf. Section 3 of [GHS]). Remark 4.6. The examples in [GHS] both have the property that the orbitsof the singularity of the Levi function fill the boundary. It seems to us thatamenability is impossible in this situation. HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 17 The thickened cylinder. Here we try to get amenability by taking the Carte-sian product with a circle, not considered a group. Consider the thickening of( S × R ), ( S × R ) C ∼ = C / Z × C ⊂ C . With z k = x k + iy k , ( k = 1 , 2) the defining function ρ : ( S × R ) C → R we cantake to be ρ ( z , z ) = y + y − 1. In particular, ρ is invariant under translations R ∋ t : ( z , z ) ( z , z + t ) . Put T = { ( z , z ) ∈ ( S × G ) C | ρ ( z ) < } . Then T ∼ = S x × B y ,y ) × R x and thequotient T /G ∼ = S × B , the solid torus, R −→ T −→ S × B . So pick a ξ ∈ bM with coordinates (˜ x , ˜ y , ˜ x , ˜ y ), (˜ y + ˜ y = 1) and consider itsorbit: G · ξ = { (˜ x , ˜ y , t, ˜ y ) | t ∈ R } ⊂ b T . The Levi form is L = 2 δ jk , ( jk = 0 , p ∈ bM with coordinates ˜ ξ = (˜ x k + i ˜ y k ) is f ( z, ˜ ξ ) = − i X ˜ y k ( z k − ˜ ξ k ) − X ( z k − ˜ ξ k ) . Now choose p ↔ ˜ ξ = ( i, i.e. y = 1, so that f ( z ) = − i ( z − i ) − ( z − i ) − z . and the group only translates z . A point on the orbit of the singularity will be p ↔ z = ˜ ξ so we take the convolution and take a limit as we approach that point:( R ∆ χf τ )( z ) = Z G dt ∆( t ) χ ( zt ) (cid:2) − i ( z − i ) − ( z − i ) − ( z + t ) (cid:3) τ Put r = − i ( z − i ) − ( z − i ) and note that as z → ˜ ξ , r → 0. In terms of r , theconvolution is ( R ∆ χf τ )( z ) = Z G dt ∆( t ) χ ( zt ) (cid:2) r − ( z + t ) (cid:3) τ . Coming toward the base point (and thus the path of the singularity) from insidethe manifold, we can take p r with coordinates z = 0 and r → R ∆ χf τ )( p ) = lim r → Z G dt ∆( t ) χ ( p r t ) (cid:2) r − t (cid:3) τ ≈ lim r → Z ǫ − ǫ dt (cid:2) r − t (cid:3) τ ≈ lim r → Z ǫ dt (cid:2) r − t (cid:3) τ which will diverge for τ ≤ − / 2. For τ > − / τ = 0 (meaning the logarithm),then the integral converges. Thus T is τ amenable for τ ≤ − / Reduction of the structure group. We have seen that in some casesit is possible to make amenable actions from those that are not by reducing thestructure group. Sometimes it may be possible to construct holomorphic functionswith stronger singularities at the boundary by a similar reduction. It is thus in ourinterest while solving the Levi problem on a G -bundle M to choose the structuregroup H ⊂ G with dimension as small as possible. To this end we note that if H ⊂ G then (cid:3) is H -invariant too, but not necessarily H -Fredholm unless G/H is compact ( H is unimodular by Theorem 8.36 [Kn]). In this case (as in thereduction in the Expression (3)) we may profit by working the problem in theform H → M → X × ( G/H ) instead of G → M → X .5. Main Theorem Assume that the action of G on M is amenable and choose f = f τ , the Levifunction at x ∈ bM with R ∆ χf / ∈ C ∞ ( ¯ M ). Consider the Fourier expansion of h = ¯ ∂χf , h = P k H k ⊗ ψ k with ( ψ k ) k an orthonormal basis for L ( X ) as in § l ∈ N for which R ∆ H l = 0 and let D δ = D lδ . Further, let hh χf ii δ = { R ∆ χf | ∆ ∈ D δ } . Since D δ ⊂ L ( G ), Corollary 4.4 and Remark 4.3 imply that hh χf ii δ ⊂ L ( M ).Furthermore, χf ∈ L ( M ) is in the domain of the Hilbert space operator ¯ ∂ H ( cf. [Per] or [FK]). Amenability guarantees that hh χf ii δ ∩ C ∞ ( ¯ M ) = { } . Lemma 5.1. The restricted antiholomorphic exterior derivative ¯ ∂ : hh χf ii δ →h ¯ ∂χf i δ is a bijection.Proof. Since ¯ ∂ is elliptic with analytic coefficients, its kernel contains only analyticfunctions. The small X -support of the members of hh χf ii δ imply therefore thatkernel of ¯ ∂ | hh χf ii δ is trivial. Since R ∆ χf is smooth in the interior of M for ∆ ∈ D δ ,we have ¯ ∂R ∆ χf = R ∆ ¯ ∂χf . Since hh χf ii δ ⊂ L ( M ), ¯ ∂ and ¯ ∂ H coincide there. (cid:3) Theorem 5.2. Assume that the action of G in M is amenable. Then the space L ∩ O ( M ) is infinite- G -dimensional. Remark 5.3. The method is similar to using a Friedrichs mollifier on the equation (cid:3) u = ¯ ∂χf . The group convolution R ∆ (cid:3) u = R ∆ (cid:3) u = ¯ ∂R ∆ χf , ∆ ∈ C ∞ ( G ), byinvariance. Proof. Let f = f τ be a function with properties verifying amenability of theaction. Theorem 6.6 of [Per] provides that the operator (cid:3) ( cf. (5)) on its domainis G -Fredholm. Lemma 3.8 allows us to conclude that, for δ > L δ ⊂ im (cid:3) ∩ h ¯ ∂χf i δ HE LEVI PROBLEM ON STRONGLY PSEUDOCONVEX G -BUNDLES 19 of arbitrarily large G -dimension. For such δ , let 0 = g ∈ L δ and solve (cid:3) u = g. By Lemma 3.7, h ¯ ∂χf i δ ⊂ C ∞ ( ¯ M , Λ , ), so g is smooth. The regularity of (cid:3) thengives that u ∈ C ∞ ( ¯ M ). Since the images of ¯ ∂ and ¯ ∂ ∗ are orthogonal, we have (cid:3) u = ¯ ∂ ¯ ∂ ∗ u = g and g = ¯ ∂φ for some φ ∈ hh χf ii δ by Lemma 5.1. Form the holomorphic functionΦ = φ − ¯ ∂ ∗ u. Amenability gives hh χf ii δ ∩ C ∞ ( ¯ M ) = { } , from which φ / ∈ C ∞ ( ¯ M ). We concludethat Φ / ∈ C ∞ ( ¯ M ) and thus is nonzero. (cid:3) The holomorphic function Φ in the proof cannot be extended smoothly beyond bM . Corollary 5.4. In the setting above, let x ∈ bM be the base point of the Levipolynomial f . Then there exists a holomorphic function Φ x which cannot be holo-morphically extended beyond x .Proof. Obvious. (cid:3) Acknowledgments The author wishes to thank Gerald Folland, Jonathan Rosenberg, Matt Stenzel,and Alex Suciu for numerous conversations and Mikhail Shubin for the suggestingthe problem and many years of friendship and mathematical advice. References [AL] Arnal, D.; Ludwig, J.: Q.U.P. and Paley-Wiener Properties of Unimodular, EspeciallyNilpotent, Lie Groups, Proceedings of the AMS , , no 4, April 1997, 1071-1080[Br1] On holomorphic L functions on coverings of strongly pseudoconvex manifolds. Publ. Res.Inst. Math. 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