Hardy Spaces for a Class of Singular Domains
Anne-Katrin Gallagher, Purvi Gupta, Loredana Lanzani, Liz Vivas
aa r X i v : . [ m a t h . C V ] S e p HARDY SPACES FOR A CLASS OF SINGULAR DOMAINS
A.-K. GALLAGHER, P. GUPTA, L. LANZANI, AND L. VIVASA bstract . We set a framework for the study of Hardy spaces inherited by com-plements of analytic hypersurfaces in domains with a prior Hardy space structure.The inherited structure is a filtration, various aspects of which are studied in spe-cific settings. For punctured planar domains, we prove a generalization of a famousrigidity lemma of Kerzman and Stein. A stabilization phenomenon is observedfor egg domains. Finally, using proper holomorphic maps, we derive a filtrationof Hardy spaces for certain power-generalized Hartogs triangles, although thesedomains fall outside the scope of the original framework.
1. I ntroduction
In this paper, we construct Hardy spaces for a class of domains, which includesthe punctured unit disk D ∗ = D \ { } and the product domain D × D ∗ as partic-ularly simple, but enlightening, examples. Although our class of domains is notbiholomorphically invariant, it is possible to push the construction forward undercertain biholomorphisms. This allows us to construct Hardy spaces for the Hartogstriangle, H = { ( z , z ) ∈ C : | z | < | z | < } , and compute the relevant Szeg˝o ker-nels. In fact, this was the original motivation for this work. The Hartogs triangleis of classical importance in several complex variables, see [29], and serves as animportant example of a singular domain since its boundary fails to be even locallygraph-like at one point. While H and its generalizations have received a lot of at-tention from the point of view of the ∂ -problem, e.g., [8, 21, 6, 20, 9], and Bergmanspaces, e.g., [7, 13, 14, 15, 5, 17, 23], Hardy spaces for H were considered for thefirst time only recently by Monguzzi in [22]. Independently of Monguzzi, we hadconstructed a di ff erent Hardy space for the Hartogs triangle, and this discrepancy L. Lanzani and L. Vivas were supported in part by the National Science Foundation (DMS-1901978and DMS-1800777). P. Gupta was supported in part by a UGC CAS-II grant (Grant No. F.510 / / CAS-II / ff International Research station during a workshopof the
Women in Analysis (WoAn), an AWM Research Network. We are grateful to the Institute forits kind hospitality and to the Association of Women in Mathematics for its generous support. Wealso wish to thank Mei-Chi Shaw for providing the inspiration for this work, and Bj¨orn Gustafsson foro ff ering helpful feedback on an earlier version of this manuscript. led us to recognize the central phenomenon of this paper. Before we describe thisphenomenon, we clarify the main terminology used in this work.Since there is no unified notion of a Hardy space in the literature, we state hereour minimum criteria for using this term. A Hilbert space of functions H on theboundary of a domain is deemed a Hardy space only if there is a reproducing kernelHilbert space (in the sense of Aronszajn in [1]) X of holomorphic functions on thedomain such that( a ) functions in a dense subspace A ⊂ X admit boundary values in H , and( b ) this identification of A with a subspace of H is an isometry that extends toan isometric isomorphism between X and H .We note that in all the explicit examples in this paper, X is directly defined in termsof an exhaustion procedure on the domain, see Sections 2, 5, and 6. However, ourgeneral setting is not conducive to this process, and X is only abstractly defined,for more details, see (3.1) and subsequent paragraphs.To describe the class of domains under consideration, we start with a domain Ω ⋐ C n and a Borel measure ν supported on its boundary, b Ω , that admits a Hardyspace structure. This structure is then inherited by domains that are obtainedfrom Ω by removing analytic hypersurfaces that are component-wise minimallydefined, see Definition 3.10. We refer to any such domain as a ‘variety-deleteddomain’, and denote it by Ω ∗ . We call this process the ‘inheritance scheme’, andthe pair ( Ω , ν ) the ‘parent space’.In a notable departure from the classical theory, it turns out that under appro-priate assumptions on the parent space, any variety-deleted domain is associatedto a filtration of Hardy spaces, as opposed to a single such space. This is dueto the fact that functions holomorphic on Ω ∗ can be singular along the deletedvariety, but all orders of singularities cannot be captured in a single Hardy space,see the discussion at the beginning of Subsection 2.2. We demonstrate via explicitexamples that this filtration may or may not stabilize, depending on the choice of ν and the deleted variety.1.1. Function-theoretic context.
In [26], Poletsky and Stessin give a constructionof Hardy spaces for hyperconvex domains in C n . We note that, while Ω ∗ is pseudo-convex whenever Ω is, it is never hyperconvex. Our construction therefore coversa new class of domains.Note that this class of domains is however uninteresting from the point of viewof Bergman space theory, since the Bergman space for Ω ∗ equals the Bergmanspace for Ω , see [24, Proposition 1.14]. Additionally, our approach does not leadto meaningful Hardy spaces of harmonic functions because b Ω is not, in general, a ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 3 uniqueness set for harmonic functions on Ω ∗ . For instance, if Ω ∗ = D \ { } , thenRe z and Re z are both harmonic on D ∗ but coincide on b D .1.2. Boundary-based approach to Hardy spaces.
The lack of a general exhaustionprocedure to construct X shifts the burden of the construction to the dense subspace A . In the classical setting of the unit disk, A is the disk algebra, i.e., the space ofholomorphic functions on D that are continuous up to the boundary. If we extendthis definition verbatim to the punctured disk, since D ∗ = D , it would lead tothe same Hardy space, which does not capture a significant class of holomorphicfunctions on D ∗ . Our construction overcomes this issue. When ( Ω , ν ) is the parentspace, we consider A to be A ( Ω , ν ) : = O ( Ω ) ∩ C ( Ω ∪ supp ν ) . Moreover, for ( Ω ∗ , ν ), we work with subspaces of O ( Ω ∗ ) ∩ C ( Ω ∗ ∪ supp ν ) which haveprescribed singularity along the deleted variety. Under appropriate assumptions(see Definition 3.5 for details), the L ( ν )-completion of A| supp ν is a reproducingkernel Hilbert space on the domain in consideration. Hence, we call it a Hardyspace and refer to its reproducing kernels as a Szeg˝o kernels.We point out that there may be kernel functions c ( z , · ) that have the reproducingproperty for A , namely, for all z in the domain(1.1) F ( z ) = Z supp ν F ( w ) · c ( z , w ) d ν ( w ) ∀ F ∈ A , but are not the Szeg˝o kernel for the associated Hardy space. For instance, this isthe case for the Cauchy kernel of any smoothly bounded planar domain Ω , D .Our boundary-based approach is particularly suited to obtaining such boundaryintegral representation formulas.1.3. Description of results.
We first state conditions on the parent space ( Ω , ν )that lead to a Hardy space for Ω , see Definitions 3.1 and 3.4. Then we provide theinheritance scheme that gives a filtration of Hardy spaces for Ω ∗ , see Theorem 3.12.For each level of the filtration, we produce new kernels that have the reproducingproperty (1.1). Moreover, we give a su ffi cient condition for these kernels to agreewith the Szeg˝o kernels, see Proposition 3.14. We then proceed to analyse theframework via some examples.In Theorem 4.2, we consider simply connected planar domains with finitelymany points removed. For this class of domains, we formulate and extend afamous rigidity lemma of Kerzman and Stein [18], i.e., if Ω ⋐ C is simply connectedthen the Cauchy kernel on Ω coincides with the Szeg˝o kernel for Ω if and onlyif Ω is a disc. We next identify a family of domains for which the filtration of A.-K. GALLAGHER, P. GUPTA, L. LANZANI, AND L. VIVAS
Hardy spaces stabilizes. These are egg domains, sometimes known as complexellipsoids, in C from which a single variety has been deleted, and we observe thatthe stabilization occurs at di ff erent levels depending on the choice of boundarymeasure, see Theorem 5.1. Finally, we use proper holomorphic maps to transferthe filtered Hardy space structure on D × D ∗ to a class of non variety-deleteddomains, i.e., the Hartogs triangle and its rational power generalizations that werefirst introduced in [13, 14]. We also produce explicitly the Szeg˝o kernels for thesedomains in Theorems 6.1 and 6.2.1.4. Structure of this paper.
In Section 2, we consider the punctured disk as thisexemplifies the general construction of the filtration of Hardy spaces. In Section 3,we provide the general framework and prove the main inheritance results. Section4 is specialized to the setting of planar domains, for which more explicit formulascan be proved by means of conformal mapping, along with the aforementionedrigidity result. The egg domains are dealt with in Section 5, and D × D ∗ , theHartogs triangle and its rational power generalizations are treated in Section 6.2. M otivating example We consider the open unit disk D and the arc-length measure σ S on b D asthe parent space, and the punctured disk D ∗ : = D \ { } as the variety-deleteddomain. Using the basic descriptions for the L -Hardy space for the disk detailedin Subsection 2.1, we derive a filtration of Hardy spaces for ( D ∗ , σ S ) in Subsection2.2. Throughout this section, we omit σ S from the notation for the relevant functionspaces.2.1. Hardy Space for the unit disk.
The classical L -Hardy space H ( D ) is thespace of holomorphic functions on D that are finite in the norm given by k F k H ( D ) : = sup < r < π Z π | F ( re i θ ) | d θ ! . Note that for any F ∈ H ( D ) with F ( z ) = P ∞ j = a j z j , it follows that k F || H ( D ) = (cid:16) ∞ X j = | a j | (cid:17) < ∞ . This characterization facilitates the identification of H ( D ) as a reproducing kernelHilbert space, by way of considering the inner product h F , G i : = lim r → π Z π F ( re i θ ) G ( re i θ ) d θ for F , G ∈ H ( D ) , ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 5 and the evaluation operators F F ( z ) for z ∈ D and F ∈ H ( D ). Moreover, atruncation of power series argument gives that the disk algebra A ( D ) = O ( D ) ∩C ( D ) is a dense subspace of H ( D ). Next, the restriction to the boundary mapfrom A ( D ) ⊂ H ( D ) to A ( D ) | b D ⊂ L ( b D ) extends to an isometric isomorphism,up to a multiplicative constant, Φ : H ( D ) −→ A ( D ) | b D L ( b D ) F ( z ) = ∞ X j = a j z j Φ ( F )( e i θ ) = ∞ X j = a j e ij θ , where P ∞ j = a j e ij θ is the representation of Φ ( F ) as its Fourier series. We call theclosure of A ( D ) | b D in L ( b D ) the Hardy space H ( D ) for ( D , σ S ). Note that if we set X as ( H ( D ) , √ π || . || H ( D ) ) and A as A ( D ), then H = H ( D ) satisfies the minimumcriterion of a Hardy space stated in the introduction.The Szeg˝o kernel s for H ( D ) may now be derived from the Cauchy integralformula for F ∈ A ( D ), which says that F ( z ) = π i Z b D F ( w ) w − z dw = π Z b D F ( w )1 − zw d σ S ( w ) . Since s is uniquely determined by such a reproducing property and the fact that s ( z , . ) ∈ H ( D ) for z ∈ D , see Proposition 3.3, we have that s ( z , w ) = π − zw for z ∈ D , w ∈ b D . Hardy spaces on the punctured disk.
In an attempt to develop a Hardy spacetheory for the punctured disk, one might first consider O ( D ∗ ) ∩ C ( D ∗ ). However, D ∗ = D , so this approach would only lead to the rediscovery of the Hardy space onthe unit disk. One might also try to construct a Hardy space for D ∗ by consideringthe closure of (cid:16) O ( D ∗ ) ∩ C ( D ∗ ∪ b D ) (cid:17) | b D with respect to L ( b D ). This fails, too, aspointwise evaluation on this class of L ( b D )-functions is not bounded for any pointin D ∗ . To wit, consider the functions F k ( z ) : = k X j = jz j , k ∈ N . Clearly, F k ∈ O ( D ∗ ) ∩ C ( D ∗ ∪ b D ) , while k ( F k ) | b D k L ( b D ) ≤ √ π (cid:16) ∞ X j = j − (cid:17) < ∞ ∀ k ∈ N . Since F k ( z ) diverges as k → ∞ for any z ∈ D ∗ , it follows that the pointwise evaluationoperator is not a bounded operator on (cid:16)(cid:16) O ( D ∗ ) ∩ C ( D ∗ ∪ b D ) (cid:17) | b D , k . k L ( b D ) (cid:17) for anypoint in D ∗ . This failure stems from allowing holomorphic functions on D ∗ with A.-K. GALLAGHER, P. GUPTA, L. LANZANI, AND L. VIVAS essential singularities at the origin. Thus, we allow poles of prescribed order at theorigin, that is, for k ∈ N , consider the following subset of O ( D ∗ ) ∩ C ( D ∗ ∪ b D ) A k ( D ∗ ) = n F : D ∗ ∪ b D −→ C : F ( z ) = (cid:16) z − k G ( z ) (cid:17) | D ∗ for some G ∈ A ( D ) o . (2.1)For each k ∈ N define H k ( D ∗ ) to be the closure of A k ( D ∗ ) | b D with respect to L ( b D ).It immediately follows from z | b D , H k ( D ∗ ) = n f ∈ L ( b D ) : f = z − k g for some g ∈ H ( D ) o . In particular, any function f ∈ H k ( D ∗ ) is represented by its Fourier series P ∞ j = − k ˆ f j e ij θ where P ∞ j = − k | ˆ f j | < ∞ . Note that H ( D ∗ ) = H ( D ), H k ( D ∗ ) ( H k + ( D ∗ ) for any k ∈ N ,and S ∞ k = H k ( D ∗ ) is dense in L ( b D ).We can also derive the Szeg˝o kernel s k for H k ( D ∗ ) directly from the Szeg˝o kernel s for H ( D ). That is, for F ∈ A k ( D ∗ ) given, let G ∈ A ( D ) such that F ( z ) = z − k G ( z ) for z ∈ D ∗ . Then for z ∈ D ∗ , we get z k F ( z ) = G ( z ) = Z b D G ( w ) s ( z , w ) d σ S ( w ) = Z b D w k F ( w ) s ( z , w ) d σ S ( w ) . Thus, the kernel given by s k ( z , w ) = w k z k s ( z , w ) = π w k z k (1 − zw ) = π zw ) k (1 − zw )exhibits the reproducing property for H k ( D ∗ ), and s k ( z , . ) ∈ H k ( D ∗ ) for all z ∈ D ∗ .Hence s k is the Szeg˝o kernel for H k ( D ∗ ).Lastly, we remark that H k ( D ∗ ) satisfies the minimum criteria, laid out in Section 1,for a space H to be called a reproducing kernel Hilbert space. Here A correspondsto A k ( D ∗ ), while X is the space H k ( D ∗ ) consisting of F ∈ O ( D ∗ ) which satisfy k F k H k ( D ∗ ) : = sup < r < r k π Z π (cid:12)(cid:12)(cid:12) F ( re i θ ) (cid:12)(cid:12)(cid:12) d θ ! < ∞ . It follows that H k ( D ∗ ) = n F ∈ O ( D ∗ ) : F ( z ) = ( z − k G ( z )) | D ∗ for some G ∈ H ( D ) o . (2.2)Moreover, the Laurent series for any function in H k ( D ∗ ) is of the form P ∞ j = − k a j z j with P ∞ j = − k | a j | < ∞ . This implies that H k ( D ∗ ) is a Hilbert space. Furthermore,pointwise evalution is bounded on H k ( D ∗ ). This follows from pointwise evalutionbeing bounded on H ( D ), characterization (2.2), and the fact that z | D ∗ ,
0. Thus, H k ( D ∗ ) is a reproducing kernel Hilbert space. Finally, H k ( D ∗ ) and H k ( D ∗ ) can beseen to be isometrically isomorphic, up to a constant factor, by mapping the j -thLaurent series coe ffi ent of F ∈ H k ( D ∗ ) to the j -th Fourier coe ffi cient of F | b D for all j ≥ k . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 7
3. H ardy spaces on variety - deleted domains The construction of the Hardy spaces for D ∗ suggests a general inheritancescheme for the construction of Hardy spaces for domains that are obtained byremoving certain complex varieties from a given domain. As is the case of D ∗ inSection 2, one starts with a domain Ω and a boundary measure ν that togethercarry their own Hardy space structure. We henceforth refer to such a pair ( Ω , ν ) asa parent space .We detail requirements on the parent space ( Ω , ν ) in Subsection 3.1. In Sub-section 3.2, we describe the class of complex varieties that will be removed from Ω to produce the so-called variety-deleted domain Ω ∗ . The inheritance scheme isdescribed in Subsection 3.3.3.1. Requirements on the parent space.
We consider a domain Ω ⋐ C n equippedwith a finite Borel measure ν on its topological boundary b Ω . We denote thesupport of ν by T , and set Ω T : = Ω ∪ T . We discuss some conditions that allow us to identify reproducing kernel Hilbertspaces of holomorphic functions on Ω that admit boundary values on T for, atleast, a dense subspace. Definition 3.1.
Let ( Ω , ν ) be such that F ( Ω , ν ) is a family of complex-valued func-tions on Ω T . Then F ( Ω , ν ) is said to be weakly admissible if and only if(i) F | T ∈ L ( ν ) for any F ∈ F ( Ω , ν ), and(ii) for any compact set K ⊂ Ω , there exists a C K > n | F ( z ) | : z ∈ K o ≤ C K k F | T k L ( ν ) for all F ∈ F ( Ω , ν ) . If we further assume that F ( Ω , ν ) is closed under subtraction, then each elementof F ( Ω , ν ) is uniquely determined by its values along T .We focus on the family of holomorphic functions given by A ( Ω , ν ) : = O ( Ω ) ∩ C ( Ω T ) . Note that A ( Ω , ν ) is an algebra over C . It satisfies condition (i) in Definition 3.1because C ( T ) ⊂ L ( ν ) whenever ν is a finite Borel measure. Definition 3.2.
Let ( Ω , ν ) be such that A ( Ω , ν ) is weakly admissible. We define the pre-Hardy space associated to ( Ω , ν ) as H ( Ω , ν ) : = A ( Ω , ν ) | TL ( ν ) , A.-K. GALLAGHER, P. GUPTA, L. LANZANI, AND L. VIVAS where A ( Ω , ν ) | T : = n f : T → C , f = F | T for some F ∈ A ( Ω , ν ) o . Proposition 3.3, and the subsequent discussion, justifies the nomenclature in-troduced in Definition 3.1. Note that despite the nonstandard terminology, thefollowing proposition is standard in functional analysis.
Proposition 3.3.
Suppose that A ( Ω , ν ) is weakly admissible. Then for any z ∈ Ω , thereexists a unique bounded linear functional Ev z : H ( Ω , ν ) → C such that Ev z ( F | T ) = F ( z ) for any F ∈ A ( Ω , ν ) . Furthermore, there exists a unique functions : Ω × T → C such that (1) s ( z , . ) ∈ H ( Ω , ν ) for all z ∈ Ω , and (2) Ev z and s ( z , . ) are related through the integral representation given by Ev z ( f ) = h f ( . ) , s ( z , . ) i L ( ν ) = Z T f ( w ) s ( z , w ) d ν ( w ) for any f ∈ H ( Ω , ν ) . We refer to the function s as the Szeg˝o kernel for H ( Ω , ν ). Proof.
Note that A ( Ω , ν ) | T is a normed vector space when endowed with the normfor L ( ν ). The existence of Ev z ( f ) follows from the Bounded Linear ExtensionTheorem applied to the evaluation F | T F ( z ) for F ∈ A ( Ω , ν ) | T . An applicationof the Riesz Representation Theorem then yields the existence and uniqueness of s ( z , . ). (cid:3) In the literature, Hardy spaces are considered as examples of reproducing kernelHilbert spaces on Ω . Note that H ( Ω , ν ) contains functions that a priori are definedonly on T ⊆ b Ω . With an additional assumption on A ( Ω , ν ), H ( Ω , ν ) may be iden-tified with a function space on Ω , and hence may be considered as a reproducingkernel Hilbert space on Ω .To identify the appropriate function space on Ω for a given weakly admissible A ( Ω , ν ), we note first that Ev ( . ) ( f ) is holomorphic on Ω for all f ∈ H ( Ω , ν ). Thisis obvious if there exists an F ∈ A ( Ω , ν ) such that F | T = f . It is also true forgeneral f ∈ H ( Ω , ν ) because the uniform boundedness of the evaluation operatorson compacta, see (ii) in Definition 3.1, says that Ev ( . ) ( f ) is the normal limit ofholomorphic functions. Thus, the map I : H ( Ω , ν ) −→ O ( Ω )(3.1) f F , where F ( z ) : = Ev z ( f ) ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 9 is well-defined. Denote by X ( Ω , ν ) : = I (cid:16) H ( Ω , ν ) (cid:17) ⊂ O ( Ω ). The injectivity of I can be stated through a condition on certain Cauchy sequences in A ( Ω , ν ). Weformulate this condition for general function spaces as follows. Definition 3.4.
Let ( Ω , ν ) be such that F ( Ω , ν ) is weakly admissible. Then F ( Ω , ν )is said to be strongly admissible if for any sequence { F n } n ∈ N ⊂ F ( Ω , ν ) for which { ( F n ) | T } n ∈ N is Cauchy in L ( ν ) and F n → Ω as n → ∞ ,the sequence { ( F n ) | T } n ∈ N converges to 0 in L ( ν ) as n → ∞ .Now suppose ( Ω , ν ) is such that A ( Ω , ν ) is strongly admissible. Then we mayequip X ( Ω , ν ) with a reproducing kernel Hilbert space structure via I . This allowsus to identify H ( Ω , ν ) with a reproducing kernel Hilbert space on Ω , and hence wecan make the following definition. Definition 3.5.
Let ( Ω , ν ) be such that A ( Ω , ν ) is strongly admissible. The Hardyspace of ( Ω , ν ) is H ( Ω , ν ).We note that we do not have an independent description of X ( Ω , ν ) in this gen-eral setting of strongly admissible function spaces. However, in all the examplesconsidered in this paper, X ( Ω , ν ) is independently described using an exhaustion-based approach, see the spaces denoted by H ( . ) in Sections 2, 5 and 6.Examples of ( Ω , ν ) for which A ( Ω , ν ) is strongly admissible include(1) ( Ω , σ ), where Ω ⊂ C is a C ,α -smooth bounded domain, and σ is the arc-length measure on b Ω , see the discussion at the beginning of Section 4.(2) ( D n , σ S × ... × σ S ), where σ S is the arc-length measures of the unit circle inthe j -th coordinate, and T = ( b D ) n , and(3) ( Ω , σ ), where Ω ⊂ C n is a C -smooth bounded domain, σ is the surfacemeasure of b Ω , and T = b Ω , see [30].On the other hand, recall from Subsection 2.2 that A ( D ∗ , σ S ) is not even a weaklyadmissible subspace of L ( b D , σ S ). Conditions analogous to weak and strongadmissibility, albeit in a broader context, were identified in [1, Theorem p. 347].An example is also given therein to demonstrate the inequivalence of the twoconditions, see [1, p. 349].3.2. Requirements on the variety.
We first recall some standard notions fromanalytic geometry. Let K ⋐ C n be a bounded set. Definition 3.6.
Denote by O ( K ) the set of equivalence classes of (cid:8) ( f , ω ) : ω is an open neighborhood of K and f : ω → C is holomorphic (cid:9) modulo the equivalence relation ( f , ω ) ∼ ( f , ω ) if and only if there is an openneighborhood ω ⊂ ω ∩ ω of K such that f | ω = f | ω . The equivalence class of ( f , ω )will be denoted simply by f , which we call the germ of an analytic function on K .Note that O ( K ) forms a ring under multiplication and addition. Definition 3.7.
Let ω ⊂ C n be an open set. A subset V of ω is an analytic variety in ω if for any z ∈ ω there exists a neighborhood U ( z ) ⊂ ω such that U ( z ) ∩ V is thecommon zero set of some f , . . . , f k ∈ O ( U ( z )) for some k ∈ N . We say that V is a locally principal variety in ω if k may be chosen equal to 1 for any z ∈ ω . Definition 3.8.
Define V ( K ) to be the set of equivalence classes of (cid:8) ( V , ω ) : ω is an open neighborhood of K , V ( ω is a locally principal variety in ω (cid:9) modulo the equivalence relation ( V , ω ) ∼ ( V , ω ) if and only if there is an openneighborhood ω ⊂ ω ∩ ω of K such that V | ω = V | ω . The equivalence class of( V , ω ) will be denoted simply by V , which we call the germ of an analytic hypersurfacein K .We next focus on the situation when K = Ω for Ω ⋐ C n is a domain. Note thatthe zero set of any nontrivial f ∈ O ( Ω ) gives rise to an element V ∈ V ( Ω ), butnot every element in V ( Ω ) arises this way. If V ∈ V ( Ω ) is indeed the zero set ofa single f ∈ O ( Ω ), then V is called principal and such an f a defining function forV . A principal germ V is called minimally defined if it admits a defining function f ∈ O ( Ω ) such that, whenever U ⊂ Ω is an open set (in the relative topology) and g ∈ O ( U ) vanishes on U ∩ V , then f | U divides g in O ( U ). We call such an f a minimaldefining function of V in O ( Ω ). It follows from a standard argument that minimaldefining functions are unique up to non-vanishing holomorphic factors. We statethis as a lemma for easy reference. Lemma 3.9.
Let V be a minimally defined germ of an analytic hypersurface in Ω . Supposef , g ∈ O ( Ω ) are two minimal defining functions of V. Then there is an h ∈ O ( Ω ) such thatf = hg, and h does not vanish on Ω . Finally, V ∈ V ( Ω ) is said to be irreducible if it cannot be expressed as V ∪ V forelements V , V ∈ V ( Ω ) distinct from V . Note that for any V ∈ V ( Ω ), there is an m ∈ N such that V ∩ Ω = ∪ mj = ( V j ∩ Ω ), where each V j is an irreducible germ of ananalytic hypersurface in Ω , see [10, § ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 11
Definition 3.10.
Let Ω ⋐ C n be a domain. Let V ∈ V ( Ω ) be a finite union ofirreducible, minimally defined germs of analytic hypersurfaces on Ω . Then Ω ∗ = Ω \ V is called a variety-deleted domain .We now discuss some examples of variety-deleted domains. In the planar case,if Ω ⋐ C is a domain and V ⊂ V ( Ω ), then Ω ∩ V = { a , ..., a m } for some a , ..., a m ∈ Ω and m ∈ N . It is immediate to see that f j ( z ) = z − a j is a minimal defining functionof { a j } in O ( Ω ). Thus, Ω \ V = Ω \ { a , ..., a m } is a variety-deleted domain.A further class of examples, which includes bounded convex domains in C n ,is provided by the following result. Note that the result implies that for such Ω , Ω \ V is a variety-deleted domain for any V ∈ V ( Ω ). Proposition 3.11.
Let n > . Suppose Ω ⋐ C n is a domain such that Ω admits a Steinneighborhood basis and H ( Ω ; Z ) = . Then any irreducible V ∈ V ( Ω ) is minimallydefined.Proof. The proof is well-known. For the reader’s convenience, we highlight themain steps of the argument. Recall that a Cousin II distribution on the compactset Ω is a collection { ( U ι , f ι ) } ι ∈ I , where { U ι } ι ∈ I is a (relatively) open cover of Ω , and f ι ∈ O ( U ι ) with f ι | U ι ∩ U = h ι · f | U ι ∩ U for some nonvanishing h ι ∈ O ( U ι ∩ U ).The hypothesis on Ω implies that, given such a Cousin II distribution, there is an f ∈ O ( Ω ) such that f ι = h ι · f | U ι for some nonvanishing h ι ∈ O ( U ι ), for all ι ∈ I , i.e., Ω is a Cousin II set, see [11].Let V ∈ V ( Ω ) be irreducible. Then V admits a local minimal defining functionat each point of V ∩ Ω , see [10, § { U i , f i } i ∈{ ,..., m } , such that f i is a minimal definingfunction of V ∩ U i for i ∈ { , ..., m } . We claim that the Cousin II solution, f ∈ O ( Ω ),for this distribution is a minimal defining function of V in O ( Ω ). First observe that f | U i ∩ V = ( h − i · f i ) | U i ∩ V = i ∈ { , ..., m } . Thus, f vanishes on V . Next, let U ⊂ Ω be a (relatively) open subset and g ∈ O ( U ) be such that g vanishes on U ∩ V . Sinceeach f i is minimal, it follows that each f i divides g in O ( U ∩ U i ). Furthermore, f | U i divides f i in O ( U i ), in particular f | U ∩ U i divides f i in O ( U ∩ U i ) for each i . Therefore f | U ∩ U i divides g in O ( U ∩ U i ). That is, f | U divides g locally and hence in O ( U ) since f and g are globally defined in U . (cid:3) In general, if V ∈ V ( Ω ) is principal, then any defining function f ∈ O ( Ω ) of V isminimal if and only if { z ∈ ω : det D f ( z ) = } is nowhere dense in V ∩ ω for some open neighborhood ω of Ω , see [10, § Ω \ V , where V is an a ffi ne hyperplane, is always a variety-deleted domain.3.3. The inheritance scheme.
We first construct Hardy spaces for triples of theform ( Ω , ν, V ), such that(i) Ω ⋐ C n is a domain, ν is a finite Borel measure on b Ω ,(ii) V is an irreducible, minimally defined germ of an analytic hypersurface in Ω , and(iii) Ω ∩ V , ∅ and ν ( T ∩ V ) = Ω T = Ω ∪ T , Ω ∗ = Ω \ V , and A ( Ω , ν ) is as in Definition 3.1. We alsoset T ∗ : = T \ V . Let ψ ∈ O ( Ω ) be a minimal defining function of V . Then for anynon-negative integer k , we consider the following subset of O ( Ω ∗ ) ∩ C ( Ω ∗ ∪ T ∗ ) A k ( Ω ∗ , ν ) : = n F : Ω ∗ ∪ T ∗ → C : F = ( ψ − k G ) | Ω ∗ ∪ T ∗ for some G ∈ A ( Ω , ν )(3.2) and F | T ∗ ∈ L ( ν ) o . Note that it follows from Lemma 3.9, that A k ( Ω ∗ , ν ) does not depend on the choiceof minimal defining function of V . Hence, we make no reference to ψ in ournotation and work with a fixed choice of ψ for the purpose of our proofs.We identify A k ( Ω ∗ , ν ) with a function space on Ω ∗ ∪ T by extending its memberstrivially, by zero, to T ∩ V , which is a measure-zero set. Then the space of boundaryvalues of A k ( Ω ∗ , ν ), i.e., A k ( Ω ∗ , ν ) | T = { F | T : F ∈ A k ( Ω ∗ , ν ) } is a subspace of L ( ν ). Note that as subspaces of L ( ν ), A k ( Ω ∗ , ν ) | T = A k ( Ω ∗ , ν ) | T ∗ .This allows us to speak of the notion of weak and strong admissibility for A k ( Ω ∗ , ν ).The spaces A k ( Ω ∗ , ν ) always inherit the properties of weak and strong admissibilityfrom A ( Ω , ν ). Theorem 3.12.
For ( Ω , ν, V ) satisfying ( i ) , ( ii ) and ( iii ) above, the following holds. (1) If A ( Ω , ν ) is weakly admissible, then so is A k ( Ω ∗ , ν ) for any k ∈ N . (2) If A ( Ω , ν ) is strongly admissible, then so is A k ( Ω ∗ , ν ) for any k ∈ N .Proof. For the proof of part (1), fix a k ∈ N and suppose that A ( Ω , ν ) is weaklyadmissible. We need to show that for any compact set K ⊂ Ω ∗ , there exists a ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 13 constant c K > z : A k ( Ω ∗ , ν ) −→ C F Ev z ( F ) : = F ( z ) , z ∈ K , are uniformly bounded on K . For that, let F ∈ A k ( Ω ∗ , ν ). Then F = ( ψ − k G ) | Ω ∗ ∪ T ∗ for some G ∈ A ( Ω , ν ) and F | T ∈ L ( ν ). Since A ( Ω , ν ) is weakly admissible and K iscompact in Ω ∗ , hence in Ω , it follows that there exists a constant C K > | Ev z ( G ) | ≤ C K k G | T k L ( ν ) ∀ z ∈ K . Therefore, | Ev z ( F ) | = (cid:12)(cid:12)(cid:12) ψ − k ( z ) (cid:12)(cid:12)(cid:12) · | Ev z ( G ) | ≤ C K (cid:12)(cid:12)(cid:12) ψ − k ( z ) (cid:12)(cid:12)(cid:12) k G | T k L ( ν ) ∀ z ∈ K . Since K ⊂ Ω ∗ , ψ is continuous and nonvanishing on K , and ν ( V ∩ T ) =
0, thereexists a constant f C K > | Ev z ( F ) | ≤ f C K k G | T k L ( ν ) = e C k (cid:13)(cid:13)(cid:13) ( ψ k · F ) | T (cid:13)(cid:13)(cid:13) L ( ν ) ∀ z ∈ K . As ψ | T is bounded and F | T ∈ L ( ν ), there is a constant c K such that | Ev z ( F ) | ≤ c K k F | T k L ( ν ) . This concludes the proof of part (1).To prove part (2), let k ∈ N and suppose that A ( Ω , ν ) is strongly admissible. Let { ( F n ) } n ∈ N ⊂ A k ( Ω ∗ , ν ) be a sequence such { F n | T } n ∈ N is Cauchy in L ( ν ) and F n −→ Ω ∗ . Then for any n ∈ N , F n = ( ψ − k G n ) | Ω ∗ ∪ T ∗ for some G n ∈ A ( Ω , ν ). Therefore, k ( G n − G m ) | T k L ( ν ) = (cid:13)(cid:13)(cid:13) ( ψ k · F n − ψ k · F m ) | T (cid:13)(cid:13)(cid:13) L ( ν ) . Since ψ is bounded on T , it follows that { ( G n ) | T } n ∈ N is a Cauchy sequence in L ( ν ).Furthermore, A ( Ω , ν ) is weakly admissible, and so for any compact set K ⊂ Ω ,there exists a constant C K > | G n ( z ) − G m ( z ) | ≤ C K k ( G n − G m ) | T k L ( ν ) ∀ z ∈ K , i.e., { G n } n ∈ N converges uniformly on compacta in Ω . Thus, there exists a G ∈ O ( Ω )such that G n ( z ) −→ G ( z ) for all z ∈ Ω as n → ∞ . However, for z ∈ Ω ∗ , G n ( z ) = ψ k ( z ) F n ( z ) −→ n → ∞ . Therefore, G ( z ) = z ∈ Ω ∗ . This implies that G ≡ Ω , and G n −→ Ω . Since A ( Ω , ν ) is stronglyadmissible, it follows that ( G n ) | T −→ L ( ν ) as n → ∞ . This in turn implies that( F n ) | T −→ L ( ν ) as n → ∞ . Thus, A k ( Ω ∗ , ν ) is strongly admissible. (cid:3) We are now set to define the central objects of this discussion.
Definition 3.13.
Let ( Ω , ν ) be such that A ( Ω , ν ) is weakly admissible and k ∈ N . The k-th pre-Hardy space H k ( Ω ∗ , ν ) is the closure of A k ( Ω ∗ , ν ) | T in L ( ν ). If A ( Ω , ν ) isstrongly admissible, we call H k ( Ω ∗ , ν ) the k-th Hardy space of ( Ω , ν, V ).Note that A ( Ω ∗ , ν ) = A ( Ω , ν ) | Ω ∗ ∪ T ∗ , i.e., A ( Ω ∗ , ν ) does not lead to a new space. Furthermore,(3.3) A ( Ω ∗ , ν ) ⊆ A ( Ω ∗ , ν ) ⊆ . . . ⊆ A k ( Ω ∗ , ν ) ⊆ . . . , and, for any ℓ ∈ N , the spaces ψ ℓ A k ( Ω ∗ , ν ) : = n ψ ℓ · F : F ∈ A k ( Ω ∗ , ν ) o satisfy theinclusions(3.4) ψ ℓ A k ( Ω ∗ , ν ) ⊆ A k − ℓ ( Ω ∗ , ν ) whenever ℓ ≤ k . The collection { H k ( Ω ∗ , ν ) } k inherits these properties. That is, H ( Ω ∗ , ν ) = H ( Ω , ν ).Furthermore, H ( Ω ∗ , ν ) ⊆ H ( Ω ∗ , ν ) ⊆ · · · ⊆ H k ( Ω ∗ , ν ) . . . , (3.5)as well as ψ ℓ H k ( Ω ∗ , ν ) ⊆ H k − ℓ ( Ω ∗ , ν ) whenever ℓ ≤ k . (3.6)Applying Proposition 3.3 to A k ( Ω ∗ , ν ), we see that H k ( Ω ∗ , ν ) possesses a Szeg˝okernel s k for any k ∈ N . Moreover, the Szeg˝o kernel s for H ( Ω , ν ) generates afamily of kernels with the reproducing property for H k ( Ω ∗ , ν ). Proposition 3.14.
Let ( Ω , ν ) be such that A ( Ω , ν ) is weakly admissible. Let ϕ ∈ A ( Ω , ν ) be such that ϕ = h ψ where ψ is a minimal defining function of V and h ∈ A ( Ω , ν ) isnonvanishing on Ω T \ V. Thenc k ,ϕ ( z , w ) : = ϕ k ( w ) ϕ k ( z ) s ( z , w ) , z ∈ Ω ∗ , w ∈ T (3.7) has the reproducing property for H k ( Ω ∗ , ν ) . Moreover, if h is nowhere vanishing on Ω T and | ϕ | is constant on T, then c k ,ϕ is the Szeg˝o kernel for H k ( Ω ∗ , ν ) for all k ∈ N .Proof. Let k ∈ N and F ∈ A k ( Ω ∗ , ν ). Then there is a G ∈ A ( Ω , ν ) such that F = ( ψ − k G ) | Ω ∗ ∪ T ∗ and F | T ∈ L ( ν ). Since h k G ∈ A ( Ω , ν ), it follows that for any z ∈ Ω ∗ F ( z ) = ϕ − k ( z ) ϕ k ( z ) F ( z ) = ϕ − k ( z ) h k ( z ) G ( z ) = ϕ − k ( z ) Z T h k ( w ) G ( w ) s ( z , w ) d ν ( w ) . Thus, F ( z ) = Z T F ( w ) c k ,ϕ ( z , w ) d ν ( w ) for any z ∈ Ω ∗ . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 15
The reproducing property of c k ,ϕ for H k ( Ω ∗ , ν ) then follows from the density of A k ( Ω ∗ , ν ) | T in H k ( Ω ∗ , ν ) with respect to L ( ν ).It remains to show that if h is nonvanishing on Ω T and | ϕ | equals some constant c ≥ T , then c k ,ϕ ( z , . ) ∈ H k ( Ω ∗ , ν ) for any z ∈ Ω ∗ . Note first that c , h nor ψ vanish on T ∗ . Since s ( z , . ) ∈ H k ( Ω , ν ) for any z ∈ Ω , it follows thatthere exists a sequence { S n ( z , . ) } n ∈ N such that S n ( z , . ) ∈ A ( Ω , ν ) for all n ∈ N and (cid:13)(cid:13)(cid:13) s ( z , . ) − S n ( z , . ) | T (cid:13)(cid:13)(cid:13) L ( ν ) −→ n → ∞ , ∀ z ∈ Ω . This, and the fact that ϕ k ( . ) ϕ − k ( z ) is bounded on T for any fixed z ∈ Ω ∗ , implies that (cid:13)(cid:13)(cid:13) c k ,ϕ ( z , . ) − (cid:16) ϕ k ( . ) ϕ − k ( z ) S n ( z , . ) (cid:17) | T (cid:13)(cid:13)(cid:13) L ( ν ) −→ n → ∞ , ∀ z ∈ Ω ∗ . To see that ϕ k ( . ) ϕ − k ( z ) S n ( z , . ) is in A k ( Ω ∗ , ν ) for any z ∈ Ω ∗ , we first note ϕ k ( w ) = c k ϕ − k ( w ) ∀ w ∈ T . It then su ffi ces to show that ϕ − k ( . ) S n ( z , . ) is in A k ( Ω ∗ , ν ) for any z ∈ Ω ∗ . Since h ∈ A ( Ω , ν ) is nonvanishing on Ω T , it follows that h − k ( . ) S n ( z , . ) ∈ A ( Ω , ν ). Thus, bythe definition of A k ( Ω ∗ , ν ), it remains to show that (cid:16) ψ − k ( . ) h − k ( . ) S n ( z , . ) (cid:17) | T is in L ( ν ).This membership holds because ψ · h = ϕ is a nonvanishing continuous functionon T . This concludes the proof of c k ,ϕ being the Szeg˝o kernel for H k ( Ω ∗ , ν ). (cid:3) Remark 3.15.
Note that replacing the Szeg˝o kernel for H ( Ω , ν ) in (3.7) with any other kernel with the reproducing property for H ( Ω , ν ), yields yet another familyof kernels with the reproducing property for H k ( Ω ∗ , ν ).We briefly discuss the especially favorable situation when V ∩ T = ∅ . In thiscase the requirement that F | T ∗ ∈ L ( ν ) in the definition of A k ( Ω ∗ , ν ) is redundant.Moreover, the containment relations in (3.3) and (3.5) are strict, i.e., A ℓ ( Ω ∗ , ν ) ( A k ( Ω ∗ , ν ) and H ℓ ( Ω ∗ , ν ) ( H k ( Ω ∗ , ν ) , when ℓ ≤ k , ℓ ∈ N , and those in (3.4) and (3.6) are equalities, i.e., ψ ℓ A k ( Ω ∗ , ν ) = A k − ℓ ( Ω ∗ , ν ) and ψ ℓ H k ( Ω ∗ , ν ) = H k − ℓ ( Ω ∗ , ν ) , when ℓ ≤ k , ℓ ∈ N . Theorem 5.1 provides examples of ( Ω , ν, V ) that exhibit the dual phenomenon, i.e.,the containments (3.5) stabilize to equalities, while the containments in (3.6) arestrict.In the classical construction, the Hardy space H ( Ω , ν ) is a module over thealgebra A ( Ω , ν ). This phenomenon cannot percolate to H k ( Ω ∗ , ν ) as, in general, A k ( Ω ∗ , ν ) is not even an algebra. However, when V ∩ T = ∅ , the union S ∞ k = A k ( Ω ∗ , ν )is a filtered algebra over C since A k ( Ω ∗ , ν ) · A j ( Ω ∗ , ν ) ⊆ A k + j ( Ω ∗ , ν ) , j , k ∈ N . The space S ∞ k = H k ( Ω ∗ , ν ) is then a filtered module over this filtered algebra since A k ( Ω ∗ , ν ) · H j ( Ω ∗ , ν ) ⊆ H k + j ( Ω ∗ , ν ) , j , k ∈ N . We now consider the general case, i.e., V = V ∪ · · · ∪ V m , where each V j is an irreducible, minimally defined germ of an analytic hypersurface in Ω . Let ψ j ∈ O ( Ω ) be a minimal defining function of V j , j ∈ { , . . . , m } . Then ψ = ψ · . . . · ψ m ∈O ( Ω ) is a minimal defining function of V . One could proceed as in Definition 3.2using ψ . However, this approach leads to an incomplete picture of the relevantspaces as each irreducible germ can independently yield a one-parameter familyof spaces. For instance, consider the example Ω ∗ P at the beginning of Section 4, andcompare the spaces in (4.1) to the above definition where all the factors of ψ wouldappear with the same exponent.To remedy this issue we proceed inductively. We write Ω ∗ ℓ = Ω \ ( V ∪ . . . ∪ V ℓ ) , ℓ ∈ { , . . . , m } , and define A k ( Ω ∗ , ν ) as in Definition 3.2 for k ∈ N . For ℓ ≥
2, consider multi-indices k = { k , . . . , k ℓ } and k ′ = { k , . . . , k ℓ − } with k j ∈ N , and define A k ( Ω ∗ ℓ , ν ) : = n F : Ω ∗ ∪ T ∗ → C : F = ( ψ − k ℓ ℓ G ) | Ω ∗ ∪ T ∗ for some G ∈ A k ′ ( Ω ∗ ℓ − , ν )and F | T ∗ ∈ L ( ν ) o . The inductive nature of this definition allows for the iterated application of Theo-rem 3.12 and Proposition 3.14. In particular, if ( Ω , ν ) is such that A ( Ω , ν ) is stronglyadmissible, then(3.8) H k ( Ω ∗ , ν ) : = A k ( Ω ∗ m , ν ) | T L ( ν ) is a reproducing kernel Hilbert space on Ω for any k = N m , and we call it the k -thHardy space of ( Ω , ν, V ). 4. P lanar domains In this section, we apply the scheme described in Subsection 3.3 to variety-deleted planar domains. Note that the case of the punctured disk is covered inSubsection 2.2.Recall that any variety-deleted planar domain may be written as Ω ∗ P = Ω \ P , P = { p , ..., p m } ⊂ Ω , see Definition 3.10 and the subsequent discussion. We henceforth refer to Ω ∗ P asan m-punctured domain . Here, we consider Ω ⋐ C of class C ,α for α ∈ (0 , σ on b Ω so that V ∩ supp( σ ) = ∅ . Under these assumptions, A ( Ω , σ ) is strongly admissible. This is because A ( Ω , σ ) ⊂ E ( Ω ), the classical ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 17
Smirnov–Hardy space of Ω , which is strongly admissible due to the existenceof nontangential limits in L ( σ ), see [12, Theorem 10.3 & Section 10.5]. In fact, A ( Ω , σ ) | b Ω is dense in E ( Ω ) | b Ω , see [12, Theorem 10.6 & Section 10.5]. Thus, theHardy space H ( Ω , σ ) coincides with the classical Hardy space on Ω . Now wecan either apply the inductive scheme of Section 3.3 or, equivalently, consider theclosure in L ( σ ) of the strongly admissible space of boundary values of(4.1) A k ( Ω ∗ P , σ ) = { F ∈ O ( Ω ∗ P ) : ( z − p ) k · ... · ( z − p m ) k m F ( z ) ∈ A ( Ω , σ ) } for k = ( k , ..., k m ) ∈ N m . Either construction gives a family of Hardy spaces n H k ( Ω ∗ P , σ ) o k ∈ N m such that H k ( Ω ∗ P , σ ) ( H k ′ ( Ω ∗ P , σ ) whenever k j ≤ k j ′ , j ∈ { , ..., m } . Note that each H k ( Ω ∗ P , σ ) is the space of L -boundary values of holomorphic func-tions on Ω that have poles of order k , ..., k m at p , ..., p m , respectively. ApplyingProposition 3.14 and Remark 3.15 iteratively, we obtain the following result. Proposition 4.1.
Let ϕ = ( φ , ..., φ m ) ∈ A ( Ω , σ ) m , k = ( k , . . . , k m ) ∈ N m and ϕ ± k = φ ± k · ... · φ ± k m m . Suppose each φ j vanishes only at p j , j = , ..., m. Suppose c ( z , w ) is akernel with the reproducing property for H ( Ω , σ ) . Then ϕ ( z ) − k c ( z , w ) ϕ ( w ) k , z ∈ Ω ∗ P , w ∈ b Ω , has the reproducing property for H k ( Ω ∗ P , σ ) . Further, if each φ j has a simple zero at p j and | φ j | is constant on b Ω , thenc k , ϕ ( z , w ) : = ϕ ( w ) k ϕ ( z ) k s ( z , w ) , z ∈ Ω ∗ P , w ∈ b Ω , is the Szeg˝o kernel for H k ( Ω ∗ P , σ ) for all k ∈ N m . In addition to the Szeg˝o kernel, we discuss a generalization of the Cauchy kernelfor H k ( Ω ∗ P , σ ), k ∈ N m . Recall that the classical Cauchy kernel C ( z , w ) = π i w − z is a holomorphic function on C × C \ { z = w } such that j ∗ (cid:16) C ( z , w ) dw (cid:17) d σ ( w )has the reproducing property for H ( Ω , σ ), where j : b Ω → C is the inclusion map.Applying Proposition 4.1 to this kernel, we obtain the following analog of the Cauchy integral formula for m -times punctured domains F ( z ) = π i Z b Ω ( w − p ) k · · · ( w − p m ) k m ( z − p ) k · · · ( z − p m ) k m ( w − z ) | {z } = :2 π i C k ( z , w ) F ( w ) dw for F ∈ A k ( Ω ∗ P , σ ) and z ∈ Ω ∗ P . We call C k ( z , w ) the Cauchy k -kernel for m punctures .Note that it is a meromorphic function on C × C \ { z = w } whose poles dependsolely on the location of the punctures. When written with respect to σ , the integralkernel in the above formula is, in fact, C Ω ∗ P k ( z , w ) : = C k ( z , w ) ˙ γ ( w ) , where w = γ ( t ) is the arc-length parametrization of b Ω . It follows that C Ω ∗ P k ( z , w ) hasthe reproducing property for H k ( Ω ∗ P , σ ). In contrast to C Ω ∗ P k , the Szeg˝o kernel, s k , of H k ( Ω ∗ P , σ ) is, in general, not known explicitly. However, for simply connected Ω ,Theorem 4.2 below gives a formula for s k in terms of the Szeg˝o kernel for H ( Ω , σ ).It also shows that the two kernels, s k and C Ω ∗ P k , coincide if and only if Ω ∗ P is a diskpunctured at its center. This rigidity result extends the Kerzman–Stein Lemma([18, Lemma 7.1]) to the case of m -punctured domains. Theorem 4.2.
Let Ω ⋐ C be a C ,α -smooth simply connected domain, and P = { p , ..., p m } ⊂ Ω . Let µ : Ω → D be a biholomorphism with q j = µ ( p j ) , j = , ..., m. (1) The Szeg˝o kernel for H k ( Ω ∗ P , σ ) is given bys k ( z , w ) = ϕ − k ( z ) s ( z , w ) ϕ − k ( w ) , z ∈ Ω ∗ P , w ∈ b Ω , where, ϕ = (cid:16) M q ◦ µ, ..., M q m ◦ µ (cid:17) for M q ( ζ ) = ζ − q − q ζ , ( q , ζ ) ∈ D × D . (2) C Ω ∗ P k ( z , w ) = s k ( z , w ) for some k ∈ N m if and only if Ω ∗ P is a disk punctured at itscenter. To prove Theorem 4.2, we use the fact that the Szeg˝o kernel s of H ( Ω , σ ) is S | Ω × b Ω ,where S is the continuous extension of the Szeg˝o kernel for E ( Ω ) to Ω × Ω \ { ( z , z ) : z ∈ b Ω } . Note that S ( z , w ) = S ( w , z ) for z , w ∈ Ω × Ω \{ ( z , z ) : z ∈ b Ω } . The continuousextension of the Szeg˝o kernel for E ( Ω ) follows from three facts. Firstly, this is truefor the classical Szeg˝o kernel S D of the disk. Secondly, the derivative of anybiholomorphism β from Ω onto D admits a continuous nonvanishing square rooton Ω , see [26, Theorem 3.5]. And lastly, the Szeg˝o kernel for E ( Ω ) can be expressedin terms of S D and p β ′ , see the transformation law in [19, Lemma 5.3]. Proof of Theorem 4.2.
Since µ extends continuously to Ω , we have that ϕ ∈A ( Ω , σ ) m . Moreover, since | M q | ≡ b D , and M q only has a simple zero at q , thesame is true of each φ j on b Ω and at p , respectively. Thus, by Proposition 4.1, c k , ϕ ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 19 is the Szeg˝o kernel for H k ( Ω ∗ P , σ ). Since M q ◦ µ = ( M q ◦ µ ) − on b Ω , the first claimfollows.Next, observe that S k ( z , w ) = ϕ ( z ) − k S ( z , w ) ϕ ( w ) − k extends s k continuously to( Ω \ P ) \ { ( z , z ) : z ∈ b Ω } , and S k ( z , w ) = S k ( w , z ). Thus, if C Ω ∗ P k = s k , it must be thatfor z , w ∈ b Ω , z , w ,(4.2) C Ω ∗ P k ( S ( z , w )) − C Ω ∗ P k ( S ( z , w )) = π i e ( S ( z , w )) w − z ˙ γ ( w ) − | e ( S ( z , w )) | g ˙ γ ( z ) ! = , where e ( S ( z , w )) = ( w − p ) k · · · ( w − p m ) k m ( z − p ) k · · · ( z − p m ) k m , and g ˙ γ ( z ) = ˙ γ ( z ) w − zw − z is the vector ob-tained from reflecting ˙ γ ( z ) in the chord determined by w and z . Thus, as in theproof of the classical Kerzman–Stein Lemma, (4.2) implies that for any two distinctpoints z , w ∈ b Ω , the chord connecting w and z meets the boundary curve withthe same angle at both points. But this can only happen if b Ω is a circle [27], i.e., Ω = D r ( a ) = { z ∈ C : | z − a | < r } for some a ∈ C and r >
0. In this case | ˙ γ ( w ) | = | g ˙ γ ( z ) | for all z , w ∈ b D r ( a ), and so | e ( S ( z , w )) | ≡ z , w ∈ b D r ( a ). If k ∈ N m , this yieldsthat | ( w − p ) · · · ( w − p m ) | is constant on b D r ( a ), which is only possible if P = { a } . (cid:3) Theorem 4.2 is stated only for simply connected domains because of the limitedapplicability of Proposition 4.1. In particular, if Ω is multiply connected, then theconditions on φ j , assumed in Proposition 4.1, may not be attainable. For example,if Ω = { z ∈ C : 1 < | z | < } and V = { a } for some a ∈ Ω , then there is no φ ∈ A ( Ω , σ )that has a simple zero at a and is such that | φ | ≡ C on b Ω . This is because, owingto the argument and maximum principles, N ( ξ ) : = π i R b Ω φ ′ ( w ) φ ( w ) − ξ dw is a continuous,integer-valued function on D C (0) and hence a constant. If φ had a simple zero,then N ≡ D C (0), forcing φ to be a homeomorphism between Ω and D C (0),which is impossible.However, in the case when Ω is finitely connected, the Szeg˝o kernel for H k ( Ω ∗ P , σ )enjoys a transformation law under biholomorphisms. The proof goes along clas-sical arguments in [4, Ch. 12] and [19, Lemma 5.3], after taking into account theboundary regularity of conformal maps between C ,α -smooth domains, see [2,App. A]. Theorem 4.3.
Suppose Ω , D ⋐ C are C ,α -smooth domains, and µ : Ω → D is abiholomorphism. Then, for k ∈ N m ,s Ω ∗ P k ( z , w ) = p µ ′ ( z ) (cid:18) s D ∗ µ ( P ) k (cid:0) µ ( z ) , µ ( w ) (cid:1)(cid:19) p µ ′ ( w ) , z ∈ Ω ∗ P , w ∈ b Ω , where s Ω ∗ P k and s D ∗ µ ( P ) k denote the Szeg˝o kernels for H k ( Ω ∗ P , σ ) and H k ( D ∗ µ ( P ) , σ ) , respectively.
5. V ariety - deleted egg domains as examples of finite stabilization In this section, we consider triples of the form (cid:16) E p , ν, { z = } (cid:17) , p ∈ N , where(5.1) E p = n ( z , z ) ∈ C : | z | p + | z | p < o , and the measure ν on b E p is either( a ) σ , the Euclidean surface area measure, or( b ) ω p , the Monge–Amp`ere boundary measure associated to the exhaustionfunction ϕ p ( z , z ) = p log (cid:16) | z | p + | z | p (cid:17) . Note that ω p is also the Leray–Levi measure associated to the defining function ρ p ( z , z ) = π p (cid:16) | z | p + | z | p − (cid:17) . In the case of the ball, or p =
1, the two measures coincide and H ( E , σ ) = H ( E , ω ).In all other cases, H ( E p , σ ) ( H ( E p , ω p ). We show that this discrepancy, owing todi ff erent choices of measure, is amplified in the case of E ∗ p = E p \{ z = } . Moreover,this setting yields examples of nontrivially stabilizing filtrations of Hardy spaces.For some context, note that H ( E p , σ ) is the space of boundary values of theclassical Hardy space on E p as defined by Stein in [30], while H ( E p , ω p ) is the spaceof boundary values of the Poletsky–Stessin Hardy space associated to ϕ p on E p ,see [25]. The latter spaces have been studied by Hansson in [16], S¸ahin in [28], andBarrett–Lanzani in [3]. Later, we encounter the limiting case of (cid:16) E p , ω p , { z = } (cid:17) as p → ∞ . To wit, if E ∞ = lim p →∞ E p in the Hausdor ff metric, and ω ∞ is the Monge–Amp`ere measure corresponding to the function ϕ ∞ ( z , z ) = lim p →∞ ϕ p ( z , z ) = log max {| z | , | z |} , then E ∗∞ = E ∞ \ { z = } is D × D ∗ and ω ∞ = σ S × σ S , see [25, § { z = } does not intersect supp( ω ∞ ) = b D × b D , { H k ( E ∗∞ , ω ∞ ) } k ∈ N does not stabilize, and z ℓ H k ( E ∗∞ , ω ∞ ) = H k − ℓ ( E ∗∞ , ω ∞ ) for ℓ ≤ k . The behavior of this filtration is quitedi ff erent when p < ∞ . Theorem 5.1.
Let p ∈ N . Then { H k ( E ∗ P , σ ) } k ∈ N stabilizes at k = , i.e., H k ( E ∗ p , σ ) = H ( E ∗ p , σ ) , ∀ k ∈ N . On the other hand, { H k ( E ∗ P , ω p ) } k ∈ N stabilizes at k = p − , i.e., H ( E ∗ p , ω p ) ( H ( E ∗ p , ω p ) ( · · · ( H p − ( E ∗ p , ω p ) = H k ( E ∗ p , ω p ) , ∀ k ≥ p . Moreover, H ( E ∗ p , ω p ) ) z H ( E ∗ p , ω p ) ) · · · ) z k H k ( E ∗ p , ω p ) ) · · · . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 21
In order to prove Theorem 5.1, we describe the relevant Hardy spaces. Here, j : b E p → C denotes the inclusion map, and d c is the real operator i ( ∂ − ∂ ). Droppingthe subscripts of ϕ p and ρ p , we have that ω p = j ∗ ( d c ϕ ∧ dd c ϕ ) = j ∗ ( ∂ρ ∧ ∂∂ρ )(2 π i ) = − det ρ z ρ z ρ z ρ z z ρ z z ρ z ρ z z ρ z z π |∇ ρ | σ, on b E p , where ρ z j is the first order partial derivative of ρ with respect to z j , j ∈ { , } , and ρ z j z k is the second order partial derivative of ρ with respect to z j and z k , j , k ∈ { , } .For ease of computation, we parametrize ( b E p ) ∗ : = b E p \ { ( z , z ) ∈ C : z z = } as(5.2) ϑ : ( s , θ , θ ) (cid:16) s p e i θ , (1 − s ) p e i θ (cid:17) , ( s , θ , θ ) ∈ (0 , × [0 , π ) . Since b E p ∩ { z z = } is a set of measure zero for both σ and ω p , we have that(5.3) L ( b E p , ν ) (cid:27) L (cid:16) (0 , × [0 , π ) ; ϑ ∗ ν (cid:17) , ν = σ, ω p , via the map f f | ( b E p ) ∗ ◦ ϑ . It is easy to check that ϑ ∗ d σ = p q ( s ) − p + (1 − s ) − p s − p (1 − s ) − p ds d θ d θ ≈ ds d θ d θ s − p (1 − s ) − p , and ϑ ∗ ω p = ds d θ d θ . Here, a ( r ) ≈ b ( r ) means that there are constants c , C > cb ( r ) ≤ a ( r ) ≤ Cb ( r )for all r . For the sake of brevity, we drop all references to ϑ , use ( s , θ , θ ) ascoordinates on b E p , and abbreviate || f || L ( b E p ,ν ) to || f || ν . We now provide descriptionsof the spaces H ( E p , σ ) and H ( E p , ω p ) in terms of L -convergent series expansions. Proposition 5.2.
Let p ∈ N . Then H ( E p , σ ) = X j ,ℓ ≥ a j ,ℓ s j p (1 − s ) ℓ p e i ( j θ + ℓθ ) : X j ,ℓ ≥ | a j ,ℓ | β j + p , ℓ + p ! < ∞ , (5.4) H ( E p , ω p ) = X j ,ℓ ≥ a j ,ℓ s j p (1 − s ) ℓ p e i ( j θ + ℓθ ) : X j ,ℓ ≥ | a j ,ℓ | β jp + , ℓ p + ! < ∞ , (5.5) where β ( x , y ) = R s x − (1 − s ) y − ds is the Euler beta function. In particular, H ( E , σ ) = H ( E , ω ) , and H ( E p , σ ) ( H ( E p , ω p ) when p > .Proof. We first prove (5.5). In view of (5.3), any f ∈ L ( b E p , ω p ) may be written as(5.6) f ( s , θ , θ ) = X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ ( s ) e i ( j θ + ℓθ ) , where { ˆ f j ,ℓ ( s ) } ( j ,ℓ ) ∈ Z are the Fourier coe ffi cients of f ( s , ., . ), and P ( j ,ℓ ) ∈ Z || ˆ f j ,ℓ || L (0 , < ∞ .Now, for F ∈ A ( E p , ω p ), we may write F ( z , z ) = P j ,ℓ ≥ a j ,ℓ z j z ℓ , if ( z , z ) ∈ E p , P j ,ℓ ∈ Z b F j ,ℓ ( s ) e i ( j θ + ℓθ ) , if ( z , z ) = (cid:16) s p e i θ , (1 − s ) p e i θ (cid:17) ∈ b E p , where in E p , the power series converges uniformly on compact subsets, and on b E p , the series converges in L ( ω p ). Next, for s ∈ (0 , F is continuous on the closedpolydisk n ( z , z ) ∈ C : | z | ≤ s / p , | z | ≤ (1 − s ) / p o . Hence, b F j ,ℓ ( s ) = s j p (1 − s ) ℓ p (2 π i ) lim r → − " | w | p = r (1 − s ) | w | p = rs F ( w , w ) w j + w ℓ + dw dw = a j ,ℓ s j p (1 − s ) ℓ p , j , ℓ ≥ , , otherwise.Moreover, X j ,ℓ ∈ Z || b F j ,ℓ || L (0 , = X j ,ℓ ≥ Z | a j ,ℓ | s jp (1 − s ) ℓ p ds = X j ,ℓ ≥ | a j ,ℓ | β jp + , ℓ p + ! < ∞ . Thus, we obtain the characterization in (5.5) for a dense subspace. By taking L ( ω p )-limits of sequences in A ( E p , ω p ), the expansion for any f ∈ H ( E p , ω p ) can beestablished. The argument for (5.4) runs along similar lines.Now, since β (cid:16) jp + , ℓ p + (cid:17) ≤ β (cid:16) j + p , ℓ + p (cid:17) for all j , ℓ ≥
0, we have that H ( E p , σ ) ⊆ H ( E p , ω p ), with equality when p =
1. To show strict containment for any p >
1, weconsider the series f ( s , θ , θ ) = P j ,ℓ ≥ a j ,ℓ s j p (1 − s ) ℓ p e i ( j θ + ℓθ ) , with a j ,ℓ = β ( m + , n + − / mn , when jp = m ∈ N , ℓ p = n ∈ N , , otherwise . Then || f || ω p = π P j ,ℓ ≥ | a j ,ℓ | β (cid:16) jp + , ℓ p + (cid:17) = π P m , n ≥ ( mn ) − < ∞ , but since || f || σ ≈ X j ,ℓ ≥ | a j ,ℓ | β j + p , ℓ + p ! ≥ c X m , n ≥ m n m − p n − p ( m + n ) p − , f does not converge in L ( σ ). (cid:3) Proof of Theorem 5.1.
Fix a p ∈ N . First, we consider σ ≈ s p − (1 − s ) p − ds d θ d θ . Itis clear that z − k (cid:12)(cid:12)(cid:12) b E p = (1 − s ) − k / p e − ik θ ∈ L ( b E p , σ ) ⇐⇒ k < . Now suppose there is a g ∈ H ( E ∗ p , σ ) \ H ( E ∗ p , σ ). Then( i ) g ∈ L ( E p , σ ) \ H ( E ∗ p , σ );( ii ) ( z g ) | b E p = P j ,ℓ ≥ a j ,ℓ s j p (1 − s ) ℓ p e i ( j θ + ℓφ ) with P j ,ℓ ≥ | a j ,ℓ | β (cid:16) j + p , ℓ + p (cid:17) < ∞ . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 23
Writing g = P j ,ℓ ∈ Z ˆ g j ,ℓ ( s ) e i ( j θ + ℓθ ) , we obtain from ( ii ) thatˆ g j ,ℓ = a j ,ℓ + , if j ≥ , ℓ ≥ − , otherwise . Thus, || g || σ ≈ X j ≥ | a j , | Z s j + p − ds − s + X j ,ℓ ≥ | a j ,ℓ + | β j + p , ℓ + p ! , which is finite only if a j , = j ≥
0, and P j ,ℓ ≥ | a j ,ℓ + | β (cid:16) j + p , ℓ + p (cid:17) < ∞ . In thatcase, g ∈ H ( E ∗ p , σ ), which contradicts ( i ). Thus, H ( E ∗ p , σ ) = H ( E ∗ p , σ ). A similarargument shows that H k ( E ∗ p , σ ) = H ( E ∗ p , σ ) for all k ∈ N .In the case of ω p = ds d θ d θ , we have that z − k | b E p = (1 − s ) − k / p e − ik θ ∈ L ( b E p , ω p ) ⇐⇒ k < p . Thus, z − k ∈ H k ( E ∗ p , ω p ) \ H k − ( E ∗ p , ω p ) as long as k ≤ p −
1. For k ≥ p , we may argue,as in the case of σ above, that H k ( E ∗ p , ω p ) = H ( E ∗ p , ω p ).Finally, we show that H k − ( E ∗ p , ω p ) ) z H k ( E ∗ p , ω p ) for any k ∈ N . In view of thestabilization, when k ≥ p , it su ffi ces to show that H p − ( E ∗ p , ω p ) ) z H p − ( E ∗ p , ω p ). Thisis clear since z − ( p − ∈ H p − ( E ∗ p , ω p ), but z − p < L ( b E p , ω p ). For k < p , let f ( s , θ , θ ) = X m ≥ ( m + − k p (cid:16) s p e i θ (cid:17) mp , and h ( s , θ , θ ) = (cid:16) (1 − s ) p e i θ (cid:17) − ( k − f ( s , θ , θ ) . Since, for any fixed r > β ( m + , r ) ∼ ( m + − r as m → ∞ , we have that || f || ω p = X m ≥ ( m + − kp β ( m + , . X m ≥ m − − kp < ∞ . Thus, z k − h = f ∈ H ( E ∗ p , ω p ). Moreover, || h || ω p = X m ≥ ( m + − kp β m + , + p − kp ! . X m ≥ m − − p < ∞ . Thus, h ∈ H k − ( E ∗ p , ω p ). But || z − h || ω p & P m ≥ m − is not finite. Thus, there is no g ∈ H k ( E ∗ p , ω p ) such that z g = h . That is, h ∈ H k − ( E ∗ p , ω p ) \ z H k ( E ∗ p , ω p ). (cid:3) Remark 5.3.
The egg domains E p may be endowed with other natural boundarymeasures. For example, in [3, Def. 43], the authors consider the family of measures n ν τ = f | L | − τ σ o τ ∈ [0 , on b E p , where f is any positive continuous function on b E p ,and | L | = − |∇ ρ | − det ρ z k ρ z j ρ z j z k ≤ j , k ≤
24 A.-K. GALLAGHER, P. GUPTA, L. LANZANI, AND L. VIVAS for any defining function ρ of E p . The measures σ and ω p correspond to ν ( f ≡ ν ( f = |∇ ρ p | / π ), respectively. It is also worth noting that the Fe ff ermanhypersurface measure on b E p is precisely ν / ( f ≡ H k ( E ∗ p , ν τ ). Note,in particular, that the filtration corresponding to the measure ν τ stabilizes at k = ⌈ p (1 − τ ) + τ ⌉ −
1, where ⌈·⌉ is the ceiling function.6. H artogs triangles : an application We construct filtered modules of Hardy spaces for certain power-generalizedHartogs triangles. This family of domains was first introduced in [13, 14]. Specifi-cally, we consider domains of the form H m / n : = { ( z , z ) ∈ C : | z | m < | z | n < } , m , n ∈ N , gcd( m , n ) = , where T = b D × b D is endowed with the product measure σ T : = σ S × σ S . Although H m / n is not a variety-deleted domain, it is a proper holomorphic image of thevariety-deleted domain D × D ∗ via Θ m / n : ( z , z ) ( z n z n , z m ) . Note that Θ m / n maps T to T , and Θ ∗ m / n : f f ◦ Θ m / n induces an isometricisomorphism from L ( T , σ T ) onto a closed subspace of L ( T , σ T ). Thus, we candeduce the Szeg˝o kernels for H m / n from those for D × D ∗ . To do this, we first treatthe case of D × D ∗ in Subsection 6.1. In Subsections 6.2 and 6.3, we treat the case ofthe standard and the power-generalized Hartogs triangles, respectively. As donein Section 2, we omit the measure σ T from the notation for the relevant functionsspaces. Moreover, we use polar coordinates ( θ , θ ) on T .6.1. Hardy spaces on D × D ∗ . We construct the Hardy spaces for D × D ∗ byexecuting the inheritance scheme in Subsection 3.3 for the triple ( D , σ T , { z = } ).To implement the scheme, consider, for k ∈ N , the following subset of O ( D × D ∗ ) ∩C (cid:16) ( D × D ∗ ) ∪ T (cid:17) A k ( D × D ∗ ) = n F : ( D × D ∗ ) ∪ T → C : F ( z , z ) = (cid:16) z − k G ( z , z ) (cid:17) | ( D × D ∗ ) ∪ T for some G ∈ A ( D ) = O ( D ) ∩ C ( D T ) o . For each k ∈ N , set H k ( D × D ∗ ) to be the closure of A k ( D × D ∗ ) | T in L ( T ).As in the case of D and D ∗ , a precise description of these spaces in terms ofFourier series expansions can be given as follows. For k ∈ N ,(6.1) H k ( D × D ∗ ) = X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ e i ( j θ + ℓθ ) ∈ L ( T ) : ˆ f j ,ℓ = , if max { j , ℓ + k } < . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 25
Moreover, the Szeg˝o kernel s k for H k ( D × D ∗ ) can be obtained by applying theCauchy integral formula for D to z k F ( z , z ) for F ∈ A k ( D × D ∗ ). This yields(6.2) s k ( z , w ) = π ) z w ) k (1 − z w )(1 − z w ) , z ∈ D × D ∗ , w ∈ T . We briefly note that in order to verify that H k ( D × D ∗ ) indeed satisfies theminimum criterion for being a Hardy space, we may take X to be H k ( D × D ∗ ) : = n F ∈ O ( D × D ∗ ) : || F || H k ( D × D ∗ ) < ∞ o , where || F || H k ( D × D ∗ ) : = sup < s , r < r k π π Z π Z | F ( se i θ , re i θ ) | d θ d θ , with norm, a constant multiple of, || . || H k ( D × D ∗ ) .6.2. The standard Hartogs triangle.
For the sake of exposition, we first consider, H = H / = { ( z , z ) ∈ C : | z | < | z | < } , for which Θ = Θ / is, in fact, a biholomorphism. This map allows us to de-scribe both a boundary-based construction and an exhaustion-based constructionof Hardy spaces for H . We are primarily interested in the former approach.For k ∈ N , let A k ( H ) = n F ∈ O ( H ) ∩ C ( H ∪ T ) : z k F ( z , z ) is bounded at (0 , o , and H k ( H ) be the closure of A k ( H ) | T in L ( T ). As in the case of D ∗ and D × D ∗ , wecan describe these spaces and their Szeg˝o kernels explicitly. Theorem 6.1.
Let k ∈ N . Then (6.3) H k ( H ) = X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ e i ( j θ + ℓθ ) ∈ L ( T ) : ˆ f j ,ℓ = , if max { j , ℓ + j + k } < . In particular, the filtration { H k ( H ) } k ∈ N does not stabilize. Moreover, (6.4) s k ( z , w ) = π ( z w ) − ( k − ( z w − z w )(1 − z w ) , z ∈ H , w ∈ T , is the Szeg˝o kernel for H k ( H ) .Proof. Fix a k ∈ N . Our proof relies on the fact that Θ ∗ : F | T ( F ◦ Θ ) | T is anisometric isomorphism between A k ( H ) | T and A k ( D × D ∗ ) | T in the L ( T )-norm. Theisometry follows from an integration by substitution argument. For the isomor-phism, note that F ∈ A k ( H ) if and only if the function ( z , z ) z k F ( z z , z ) isholomorphic on D × D ∗ , bounded on a neighborhood of { z = } , and continuous up to T . This is true if and only if z k F ( z z , z ) = G ( z , z ) | D × D ∗ for some G ∈ A ( D ).In other words, F | T ∈ A k ( H ) T if and only if ( Θ ∗ F ) | T ∈ A k ( D × D ∗ ) | T . Now, Θ ∗ extends to an isometry between H k ( H ) and H k ( D × D ∗ ) which, in terms of Fourierexpansions, is given by Θ ∗ : X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ e i ( j θ + ℓθ ) X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ e i ( j θ + ( j + ℓ ) θ ) . The characterization in (6.3) now follows from that of H k ( D × D ∗ ) in (6.1).Finally, for any F ∈ A k ( H ), the reproducing property of the Szeg˝o kernel s D × D ∗ k for H k ( D × D ∗ ) applies to Θ ∗ F ∈ A k ( D × D ∗ ). We obtain that F ( z ) = Z T ( Θ ∗ F )( w ) s D × D ∗ k (cid:16) Θ − ( z ) , Θ − ( w ) (cid:17) d σ T ( w ) , z ∈ H , w ∈ T . Now, a straightforward computation yields the reproducing property of s k asdefined in (6.4). It is also clear that s k ( z , · ) ∈ A k ( H ) for any z ∈ H . (cid:3) We briefly discuss an exhaustion-based construction of Hardy spaces H k ( H ), k ∈ N , for H , which in the case of k = k ∈ N , let H k ( H ) = n F ∈ O ( H ) : || F || H k ( H ) < ∞ o , where || F || H k ( H ) : = sup < s , r < r k π π Z π Z (cid:12)(cid:12)(cid:12) F ( rse i θ , re i θ ) (cid:12)(cid:12)(cid:12) d θ d θ . Rather than establish a direct isometric isomorphism, up to a factor, between H k ( H ) and H k ( H ), we argue that Θ ∗ : F F ◦ Θ is an isometric isomorphismbetween H k ( H ) and H k ( D × D ∗ ). From the proof of Theorem 6.1, we know that Θ ∗ is an isomorphism between A k ( H ) and A k ( D × D ∗ ). Since these spaces aredense in the respective H -spaces, it su ffi ces to show that Θ ∗ is an isometry from( A k ( H ) , || . || H k ( H ) ) to ( A k ( D × D ∗ ) , || . || H k ( D × D ∗ ) ). This is a standard computation, byway of integration by substitution.6.3. The (Rational) Power-Generalized Hartogs Triangles.
We now consider thegeneral case of H m / n . For k ∈ N , define A k ( H m / n ) = n F ∈ O ( H m / n ) ∩ C ( H m / n ∪ T ) : z k F ( z , z ) is bounded at (0 , o . ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 27
Let H k ( H m / n ) be the closure of A k ( H m / n ) | T in L ( T ). As in the case m = n =
1, using Θ ∗ m / n , we see that H k ( H m / n ) = X ( j ,ℓ ) ∈ Z ˆ f j ,ℓ e i ( j θ + ℓθ ) ∈ L ( T ) : ˆ f j ,ℓ = , if max { j , nj + ml + mk } < . Next, we use the map Θ m / n to compute the Szeg˝o kernel for H k ( H m / n ). Theorem 6.2.
Let m , n ∈ N with gcd( m , n ) = . Set P m , n ( a , b ) = m − X r = ( a ) r ( b ) n −⌊ nrm ⌋ , ( a , b ) ∈ C . Then, for k ∈ N , (6.5) s k ( z , w ) = π ( z w ) − k P m , n ( z w , z w )(( z w ) n − ( z w ) m ) (1 − z w ) , z ∈ H m / n , w ∈ T , is the Szeg˝o kernel for H k ( H m / n ) . In order to prove Theorem 6.2, we need the following two lemmas. The proofsare straightforward applications of integration by substitution and partial fractiondecompositions, so they are omitted.
Lemma 6.3.
Suppose f ∈ C ( b D ) . Then (6.6) Z | ζ | = ζ n − f ( ζ n ) d ζ = Z | z | = f ( z ) dz . More generally, if n ∈ N , a ∈ C \ S , and a , ..., a n denote the n th -roots of a (countingmultiplicity). Then n X j = Z | ζ | = f ( ζ n ) ζ − a j d ζ ! = n Z | w | = f ( w ) w − a dw . (6.7) Lemma 6.4.
Let b ∈ C \ { } and b , ..., b m denote the m th -roots of b (counting multiplicty).Then m X ℓ = b n ℓ ( x − b n ℓ )( y − b ℓ ) = mb n + m − P p , q = c p , q x p y q ( x m − b n )( y m − b ) , where (6.8) c p , q = , if np + + q . (mod m) , b − np + + qm , if np + + q ≡ (mod m) . Proof of Theorem 6.2.
For any k ∈ N , s k , as defined in (6.5), satisfies s k ( z , · ) ∈ A k ( H m / n )for all z ∈ H m / n . Thus, it su ffi ces to show that s k has the reproducing property for H k ( H m / n ). Since | z | (cid:12)(cid:12)(cid:12) T ≡
1, by Proposition 3.14, we only need to prove this for k = Θ m / n ( ζ , ζ ) = ( ζ n ζ n , ζ m ) maps D × D ∗ onto H m / n . Given F ∈ A ( H m / n )and z = ( z , z ) ∈ H m / n , let z , ..., z n and z , ..., z m denote the n th -roots and m th -roots of z and z , respectively, so that F ( z , z ) = F ( z n j , z m ℓ ) for any 1 ≤ j ≤ n and1 ≤ ℓ ≤ m . Thus, F ( z , z ) = mn m X ℓ = n X j = ( F ◦ Θ m / n ) (cid:18) z j z ℓ , z ℓ (cid:19) . We apply the Cauchy integral formula for D to ( F ◦ Θ m / n ) ∈ A ( D × D ∗ ) = A ( D ),and obtain the following sequence of arguments.(2 π i ) mnF ( z , z ) = m X ℓ = n X j = " T ( F ◦ Θ m / n )( ζ , ζ ) (cid:16) ζ − z j z ℓ (cid:17) ( ζ − z ℓ ) d ζ d ζ = m X ℓ = Z | ζ | = n X j = Z | ζ | = F ( ζ n ζ n , ζ m ) ζ − z j z ℓ d ζ d ζ ( ζ − z ℓ ) ζ ζ ξ = m X ℓ = Z | ζ | = n X j = Z | ξ | = F ( ξ n , ζ m ) (cid:16) ξ − ζ z j z ℓ (cid:17) d ξ d ζ ( ζ − z ℓ ) (6.7) = n m X ℓ = Z | ζ | = Z | w | = F ( w , ζ m ) (cid:18) w − ζ n z z n ℓ (cid:19) dw d ζ ( ζ − z ℓ ) = n " T F ( w , ζ m ) − w m X ℓ = z n ℓ ( ζ n z w − z n ℓ )( ζ − z ℓ ) dw d ζ . Now, by Lemma 6.4 (with x = ζ n z w , y = ζ and b = z ), we have that(2 π i ) mnF ( z , z ) = mn " T F ( w , ζ m ) − w z n + m − P p , q = c p , q (cid:16) ζ n z w (cid:17) p ζ q ζ mn ( z m w m − z n )( ζ m − z ) dw d ζ = mn " T F ( w , ζ m ) ζ m − m − P p , q = (cid:18) z np + + qm c p , q (cid:19) ( z w ) p ( z ζ m ) ( n + − np + q + m (cid:16) ( z ζ m ) n − z m w m (cid:17) ( ζ m − z ) dw w d ζ , ARDY SPACES FOR A CLASS OF SINGULAR DOMAINS 29 where c p , q are as in (6.8) (with b = z ). Applying (6.6) in the ζ variable, we get(2 π i ) F ( z , z ) = " T F ( w , w ) m − P p , q = e c p , q ( z w ) p ( z w ) n + − np + + qm ( z n w n − z m w m )(1 − z w ) dw w dw w , where e c p , q = z np + + qm c p , q = , if np + + q . , , if np + + q ≡ . This settles our claim, once we observe that m − X p , q = e c p , q ( a ) p ( b ) n + − np + + qm = m − X r = ( a ) r ( b ) n −⌊ nrm ⌋ = P m , n ( a , b ) . (cid:3) In view of our minimum criterion for a Hardy space, we end this subsectionwith an exhaustion-based definition of Hardy spaces for H m / n . For k ∈ N , let H k ( H m / n ) = n F ∈ O ( H m / n ) : || F || H k ( H m / n ) < ∞ o , where || F || H k ( H m / n ) : = sup < s , r < r k π π Z π Z (cid:12)(cid:12)(cid:12)(cid:12) F ( r nm s m e i θ , re i θ ) (cid:12)(cid:12)(cid:12)(cid:12) d θ d θ . L p -regularity of the Szeg ˝o projection. We briefly remark on the L p -mappingproperties of the projection operator induced by the Szeg˝o kernel s k for H k ( H m / n ), k ∈ N , m , n ∈ N , gcd( m , n ) =
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