(Ir-)regularity of canonical projection operators on some weakly pseudoconvex domains
aa r X i v : . [ m a t h . C V ] M a r (IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS ONSOME WEAKLY PSEUDOCONVEX DOMAINS ALESSANDRO MONGUZZI, MARCO M. PELOSO
Abstract.
In this paper we discuss some recent results concerning the regularity and irregu-larity of the Bergman and Szeg˝o projections on some weakly pseudoconvex domains that havethe common feature to possess a nontrivial Nebenh¨ulle.
Introduction
In this note we survey some recent results on the analysis of canonical projection operators,such as the Bergman and Szeg˝o projections, on a family of domains that present some patholog-ical behavior. These domains have the common feature to possess a nontrivial
Nebenh¨ulle , andthey essentially are the worm domain of K. Diederich and J.E. Fornæss, the Hartogs triangle andsome of its variants, and some model worm domains introduced by C. Kiselman and studied,among others by D. Barrett, S. Krantz and the authors of this note.This note is an extended version of a seminar given by the second named author at the Di-partimento di Matematica dell’Universit`a della Basilicata. He wishes to thank such departmentand in particular E. Barletta and S. Dragomir for the kind invitation and the great hospitality.The worm domain W µ was introduced by K. Diederich and J.E. Fornæss in [DF77]. W µ = n ( z , z ) ∈ C : | z − e i log | z | | < − η (log | z | ) o where η is smooth, even, convex, vanishing on [ − µ, µ ], with η ( a ) = 1, and η ′ ( a ) >
0. Theseproperties of η imply that W µ is smooth, bounded and pseudoconvex. Morevover W µ is strictlypseudoconvex at all points ( z , z ) ∈ ∂ W µ with z = 0. The set of points on the boundary A = { (0 , z ) : (cid:12)(cid:12) log | z | (cid:12)(cid:12) ≤ µ } is the critical annulus .In [DF77] the following important features of W µ were shown:(I) W µ has non-trivial Nebenh¨ulle (that is, there exists no neighborhood basis of pseudo-convex domains for W µ ) [Diederich-Fornæss];(II) W µ does not admit any plurisubharmonic defining function (that is, a defining functionthat is plurisubharmonic on the boundary).Concerning (I), by the Hartogs’s extension phenomenon indeed, it follows that if f is holo-morphic in a neighborhood of (cid:8) (0 , z ) : (cid:12)(cid:12) log | z | (cid:12)(cid:12) ≤ π (cid:9) ∪ (cid:8) ( z , z ) : (cid:12)(cid:12) log | z | (cid:12)(cid:12) = π, (cid:12)(cid:12) z − e iµ (cid:12)(cid:12) ≤ (cid:9) , then f is also holomorphic in a neighborhood of the set (cid:8) ( z , z ) : (cid:12)(cid:12) log | z | (cid:12)(cid:12) ≤ π, (cid:12)(cid:12) z − e iµ (cid:12)(cid:12) ≤ (cid:9) . Key words and phrases.
Bergman projection, Bergman kernel, Szeg¨o kernel, Szeg¨o projection, worm domain,Hartogs triangle.
Math Subject Classification
Figure 1.
Representation of W µ in the (Re z , Im z )-plane. Regarding (I) we recall that, given a domain Ω ⊆ C n , a continuous function ϕ : Ω → ( −∞ ,
0) iscalled a bounded exhaustion function if for all c < ∈ R , ϕ − ( −∞ , c ) ∩ ∂ Ω = ∅ . In [DF77] Diederich and Fornæss proved that if Ω ⊆ C n is smooth, bounded and pseudoconvexwith defining function ρ , then there exists τ ∈ (0 ,
1] such that − ( − ρ ) τ is a bounded striclyplurisubharmonic exhaustion function. Such an exponent τ = τ ρ is called a DF -exponent for thedefining function ρ . We setDF(Ω) = sup (cid:8) τ ρ : ρ defining function of Ω (cid:9) , and we call this value the Diederich–Fornæss index of Ω. In [DF77], Diedirich and Fornæssproved that DF( W µ ) ≤ π/ µ . Since its appearance, research on the properties of the worm domains remained dormant fora number of years. We now consider a still open fundamental problem in analysis and geometryof several complex variables: Given D , D bounded, smooth, pseudoconvex domains and abiholomorphic mapping Φ : D → D , does Φ extend smoothly to a diffeormophism of theboundaries? We denote by ∂ D the topological boundary of a given domain D .Given a domain Ω, let A (Ω) = L (Ω) ∩ Hol(Ω) be the Bergman space, and P Ω : L (Ω) → A (Ω) be the Bergman projection. A celebrated theorem by S. Bell and E. Ligocka [BL80], andlater improved by S. Bell [Bel81], says that this is the case if one of the two domains satisfies(say D ) the so-called Condition (R): P D : C ∞ ( D ) → C ∞ ( D ) is bounded . (R)In [Bar92] D. Barrett showed that, writing W in place of W µ for short and denoting theSobolev space on W by W ,s ( W ), P W : W ,s ( W ) W ,s ( W )if s ≥ π/ µ . Hence, P W fails to preserve the L -Sobolev space W ,s ( W ), when s is greater orequal to the reciprocal of the windings of the domain W .Although Barrett’s result constituted a major breakthrough, it did not imply that W failedto satisfy Condition (R). It was M. Christ in [Chr96] to prove that the Neumann operator N on W does not preserve C ∞ ( W ); hence W does not satisfy Condition (R). In fact, by a theorem ofBoas–Straube [BS90], on any given smoothly bounded pseudoconvex domain Ω, N is globally(exactly) regular if and only if P Ω is globally (exactly) regular. We say that P Ω is exactly IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 3 regular if P Ω : W ,s (Ω) → W ,s (Ω) is bounded for all s >
0. We say that P Ω is globally regularif given any s >
0, there exists q = q ( s ) such that P Ω : W ,s + q ( s ) (Ω) → W ,s (Ω) is bounded. Inparticular, if P Ω is globally regular, then P Ω : C ∞ (Ω) → C ∞ (Ω) is bounded, that is, Ω satisfiescondition (R).The problem of the regularity of the Bergman projection on worm domains has been object ofactive and intense research and we mention in particular [KP08b, BS¸12, BEP15, CS¸15, KPS16,KPS19]. We also refer the reader to [KP08a] for a detailed account on the subject.Main goal of this note is to report on recent progress on the analysis of the (ir-)regularityof the boundary analogue of the Bergman projection, that is, the Szeg˝o projection. Given asmoothly bounded domain Ω = { z : ρ ( z ) < } ⊆ C n , the Hardy space H (Ω , dσ ) is defined as H (Ω , dσ ) = (cid:8) f ∈ Hol(Ω) : sup ε> Z ∂ Ω ε | f | dσ ε < ∞ (cid:9) , where Ω ε = { z : ρ ( z ) < − ε } and dσ ε is the induced surface measure on ∂ Ω ε .Then, H (Ω , dσ ) can be identified with a closed subspace of L ( ∂ Ω , dσ ), that we denoteby H ( ∂ Ω , dσ ), where σ is the induced surface measure on ∂ Ω. The Szeg˝o projection is theorthogonal projection S Ω : L ( ∂ Ω , dσ ) → H ( ∂ Ω , dσ ) ;see [Ste70].The note is organized as follows. In Section 1 we recall some further noticeable resultsconcerning the worm domain, some of its generalizations and some ideas involved in the proofsof such results. In Section 2 we discuss the case of Szeg˝o projections on worm domains. Section4 is devoted to the case of another class of domains, the so-called Hartogs triangles. In Section 5we present an interesting problem in the theory of 1-dimensional Bergman spaces that arose inthe study of orthogonal sets in the Bergman space of the truncated worm domain. We concludethis report with some final remarks and open questions.1. Generalizations and some open problems on the worm domain
The worm domain W is still up to today the only known example of a smoothly boundedpseudoconvex domain on which Condition (R) fails. Thus, it is a natural testing ground forthe validity of the extendebility to diffeomorphism to the boundary of biholomorphic mappings.The first class of mappings that one is naturally led to consider are the biholomorphic self-mapsof W , that is, the automorphisms of W , Aut( W ). Clearly, the maps Φ( z , z ) = ( z , e iθ z ) are inAut( W ) and extend smoothly to the boundary. The obvious question is: Are there any others?In [Che93], the author studied the automorphisms group Aut( W ), and claimed that this is thecase. Unfortunately, it is generally accepted that there is a gap in the proof and it has not beenfixed. Thus, the very interesting question of characterizing the automorphism group Aut( W ) isan open and fundamental question.Before going any further, we point out that in [BS¸12] D. Barrett and S. S¸ahuto˘glu constructeda higher dimensional analogue of the worm domain. Let n ≥ z ∈ C n we write z = ( z , z ′ , z n ) ∈ C × C n − × C . For λ, µ > W λ,µ = (cid:8) ( z , z ′ , z n ) ∈ C × C n − × C : (cid:12)(cid:12) z − e iλ log | z n | (cid:12)(cid:12) < − | z ′ | − e η (log | z n | ) (cid:9) , (1) A. MONGUZZI, M. M. PELOSO where e η is a particular , explicit, smooth function which is identically 0 when e − / ≤ | z n | ≤ e µ/ .The function e η is chosen in such a way the domain is smoothly bounded and pseudoconvex, andit is strongly pseudoconvex except at the critical annulus A = (cid:8) ( z , z ′ , z n ) ∈ C × C n − × C ∩ ∂ W λ,µ : z = 0 , z ′ = 0 (cid:9) = (cid:8) (0 , , z n ) ∈ C × C n − × C : e − / ≤ | z n | ≤ e µ/ (cid:9) . Barrett and S. S¸ahuto˘glu proved that the Bergman projection P W λ,µ fails to preserve theSobolev spaces W p,s , with p ∈ [1 , ∞ ) and s ≥
0, hence including the cases p = 2, when s ≥ π λµ + n (cid:16) − p (cid:17) . What is extremely interesting to notice here is that the Bergman projection becomes irregularif either the winding is too “long” (i.e. when µ is large), or is too “fast” (i.e. when λ is large).For simplicity of presentation, we restricted ourselves to the 2-dimensional case, that is, tothe domain W = W µ . However, we point out that the discussion that follows is also valid forthe higher dimensional cases of the domains W λ,µ .Instrumental to Barrett’s proof of the irregularity of P W were two unbounded model wormdomains, that we denote by D µ and D ′ µ , where D µ = n ( z , z ) ∈ C : Re (cid:0) z e − i log | z | (cid:1) > , (cid:12)(cid:12) log | z | (cid:12)(cid:12) < µ o , µ > π, and D ′ µ = n ( z , z ) ∈ C : (cid:12)(cid:12) Im z − log | z | (cid:12)(cid:12) < π , (cid:12)(cid:12) log | z | (cid:12)(cid:12) < µ o . Figure 2.
Representation of D ′ µ in the (Im z , log | z | )-plane. Remarks 1.1.
The following facts are easy to see:(i) the domains D ′ µ and D µ are biholomorphically equivalent via the mapping ϕ : D ′ µ → D µ ϕ ( z , z ) := ( e z , z ) ;(ii) for every z fixed the fiber over z Π( z ) = { z ∈ C : ( z , z ) ∈ D ′ µ } is connected, while the same property does not hold for D µ . IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 5
For a given domain Ω in C that is rotationally invariant in the second variable z , such as W , D µ , D ′ µ , using Fourier expansion in z we can decompose the Bergman space A (Ω) as A (Ω) = M j ∈ Z H j , (2)where H j = (cid:8) F ∈ A : F ( z , e iθ z ) = e ijθ F ( z , z ) (cid:9) . If for every z fixed, the fibers Π( z ) = { z : ( z , z ) ∈ Ω } are connected, then F ∈ H j hasthe form F ( z , z ) = f ( z ) z j , where f is holomorphic. In the case of D ′ µ , the fibers Π( z ) are connected and f is holomorphicon the strip {| Im z | < µ + π/ } . Hence, we may write the kernel K ′ of D ′ µ as K ′ ( ζ, ω ) = ∞ X j = −∞ K ′ j ( ζ , ω ) ζ j ω j and using these observation an explicit computation in 1-dimension, it is possible to computethe Bergman kernel K ′ of D ′ µ quite explicitly. In [Bar92] the kernel K ′− is explicitly computedand it holds K ′− ( ζ, ω ) = 12 π Z R t sinh(2 µt ) sinh(2 πt ) e i ( ζ − ω ) dt . The analysis of the kernels K ′ j ’s for j = − j = − j = − P D ′ µ [ ϕ ′ ( f ◦ ϕ )] = ϕ ′ [( P D µ f ) ◦ ϕ ] , Barrett analyses the kernel K of D µ and, in particular, the ( − P D µ is not exactly regular, that is, P D µ is not a bounded operator P D µ : W s, ( D µ ) → W s, ( D µ ) for s sufficiently large. The same conclusion for the smooth worm W is then obtained via an exhaustion argument. Setting W τ = (cid:8) ( z , z ) ∈ C : ( z τ , z ) ∈ W (cid:9) ,Barrett showed that P W τ f → P D µ f (3)as τ → ∞ . For, if we denote by d τ the dilation in the first variable by τ > d τ ( z , z ) = ( τ z , z ),then d τ ( W ) = W τ and P W τ = T − τ P W T τ . (4)where T τ f ( z , z ) = f ( τ z , z ). From this relation, it is possible to deduce the boundedness of P W τ from the one of P W . Then, passing to the limit as in (3), we would obtain the boundednessof P D µ , hence, a contradiction. In order to prove (3), it is necessary the trivial, but important,remark that given any compact set E ⊆ D µ , there exists τ E > τ ≥ τ E , E ⊆ W τ .Thus, the analysis on the domains D ′ µ and D µ not only provided intuition on the case of thesmooth, bounded worm domain W , but also it was fundamental in proving the result on theirregularity of the Bergman projection on W itself. A. MONGUZZI, M. M. PELOSO
We now briefly comment on Christ’s result [Chr96]. He proved that W does not satisfyCondition (R), by showing that for all s > N satisfies an a priori estimate ||N u || W ,s ≤ C s,j || u || W ,s valid for every u ∈ H j ∩ C ∞ ( W ) such that N u ∈ C ∞ ( W ). (Here the subscript 1 indicates thefact that u is a (0 , N : C ∞ ( W ) → C ∞ ( W ) were bounded, such estimates wouldcontradict the irregularity of P W .We conclude this section by discussing the Diederich–Fornæss index of the worm domain W .In [Liu19a, Liu19b] B. Liu proved that DF( W ) = π/ µ , see also [KLP18].2. Hardy spaces on model worm domains
We now consider another canonical kernel and projection of a domain Ω in C n , the Szeg˝okernel and projection.Let Ω = (cid:8) z ∈ C n : ρ ( z ) < (cid:9) , where ρ is smooth and ∇ ρ = 0 on ∂ Ω. Let Ω ε = { ρ ( z ) < − ε } , and suppose there exists a familyof Borel measures { σ ε } on Ω and supported on ∂ Ω ε ⊆ Ω such σ ε → σ =: σ weakly as ε → f ∈ C (Ω), R f dσ ε → R f dσ as ε →
0. Define the Hardy space H (Ω , dσ ) as H (Ω , dσ ) = n f ∈ Hol(Ω) : sup ε> Z ∂ Ω ε | f ( ζ ) | dσ ε ( ζ ) < ∞ o . Under mild conditions on the family of measures { σ ε } , the Hardy H (Ω , dσ ) is a reproducingkernel Hilbert space and its reproducing kernel is called the Szeg˝o kernel. The classical caseis Ω is a smoothly bounded domain and dσ ε is the induced surface measure on ∂ Ω ε . In thiscase, we simply write H (Ω). It is a classical result (see [Ste70]) that under these assumptions,if f ∈ H (Ω) then f converges non-tangentially to a boundary function e f ∈ L ( ∂ Ω). In fullgenerality, these latter facts have to be shown to hold true.Thus, we may define H ( ∂ Ω , dσ ) = n g ∈ L ( ∂ Ω , dσ ) : g ( ζ ) = lim z → ζ f ( z ) non-tangentially, for some f ∈ H (Ω , dσ ) o . Then, the Szeg˝o projection is the Hilbert space orthogonal projection of L ( ∂ Ω , dσ ) onto its(closed) subspace H ( ∂ Ω , dσ ), the subspace of boundary values of functions in H (Ω , dσ ), S Ω : L ( ∂ Ω , dσ ) → H ( ∂ Ω , dσ ) S Ω g ( ζ ) = lim z → ζ ∈ ∂ Ω Z b Ω g ( ζ ′ ) K ( z, ζ ′ ) dσ ( ζ ′ ) . By definition, the Szeg˝o projection depends on the choice of the measure on the boundary.Another very natural, and thus far little considered, possible choice, is the Fefferman surfacemeasure σ F , see [BL14], or any other surface measure ωdσ , where ω is a continuous positivefunction on ∂ Ω, [LS17]. The surface measure σ F was introduced by C. Fefferman in order toobtain a measure that is biholomorphic invariant . To be precise, suppose Ω and Ω are bounded,smooth, pseudoconvex domains that admit a biholomorphic map ϕ : Ω → Ω that extends toa smooth C ∞ -diffeormophism of the boundary, such as in case one of the two domains satisfiesCondition (R). Then, the mapping Λ( f ) := √ det ϕ ′ ( f ◦ ϕ ) defines an isometric isomorphismΛ : H (Ω , dσ F ) → H (Ω , dσ F ). IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 7
When we consider non-smooth domains, such as D µ and D ′ µ , and more noticeably the polydisk,it is perhaps more natural, and certainly interesting to study Hardy spaces defined by integrationover the so-called distinguished boundary . Given a domain Ω ⊆ C n , we call the distinguishedboundary, and we denote it by d b (Ω),the set d b (Ω) = n ζ ∈ ∂ Ω : sup z ∈ ∂ Ω | f ( z ) | ≤ sup ζ ∈ d b (Ω) | f ( ζ ) | for all f ∈ H ∞ (Ω) o (5)where H ∞ (Ω) is the spaces of holomorphic functions on Ω that are bounded. Then, we considerthe induced measure on d b (Ω) and denote it by dβ .We now describe the main results that we have obtained on the regularity of Szeg˝o projectionson model worm domains. We first consider the case of D ′ µ and the induced surface measure dβ .The following result is in [MP17a]. Theorem 2.1.
The Szeg˝o projection S , initially defined on the dense subspace W s,p ( ∂D ′ µ ) ∩ L ( ∂D ′ µ , dσ ) , extends to a bounded operator S : W s,p ( ∂D ′ µ ) → W s,p ( ∂D ′ µ ) , for < p < ∞ and s ≥ . The proof of such result relies on explicit computations on the boundary of D ′ µ , which can bewritten as union of four pieces that have intersection of null measure. The Szeg˝o projection canbe correspondingly written as sum of 16 different integral operators. For each of these operatorswe apply a decomposition similar to (2) and obtain an explicit expression and thus write themas composition of a bounded Fourier multiplier and an operator of Hilbert-type.It is worth to remark that the boundedness of the corresponding Szeg˝o projection on D µ isstill unexplored and it would be significant to study such (ir)-regularity.We now turn to the case of the Szeg˝o projection on the distinguished boundaries ([Mon16c]).More precisely, denote by d b ( D ′ µ ) and d b ( D µ ) the distinguished boundaries of the domains D ′ µ and D µ , resp., and by S ′ and S the corresponding Szeg˝o projections, resp. We point out thatin this setting, the operators S ′ and S are given by singular integrals over d b ( D ′ µ ) and d b ( D µ ),resp.The case of D ′ µ was considered in [Mon16b], where the main result is the following Theorem 2.2.
The Szeg˝o projection S ′ , initially defined on the dense subspace W s,p ( d b ( D ′ µ ) , dβ ) ∩ L ( d b ( D ′ µ ) , dβ ) , extends to a bounded operator S ′ : W s,p ( d b ( D ′ µ ) , dβ ) → W s,p ( d b ( D ′ µ ) , dβ ) , for < p < ∞ and s ≥ . The proof of this result follows from explicit computations of the Szeg˝o projection of suitablydefined Hardy spaces on the distinguished boundary of D ′ µ . Such a boundary is the union offour different connected components which are mutually disjoint and the Szeg˝o projection turnsout to be a linear combination of bounded Fourier multiplier operators. A detailed analysis ofthe Szeg˝o kernel associated to S ′ is performed in [Mon16a].With a similar proof the analogous result for the Szeg˝o projection S on the distinguishedboundary of D µ is studied and we now describe it with greater details. For ( t, s ) ∈ (0 , π ) × [0 , µ ) A. MONGUZZI, M. M. PELOSO consider the domain D t,s = (cid:8) ( z , z ) ∈ C : (cid:12)(cid:12) arg z − log | z | (cid:12)(cid:12) < t, (cid:12)(cid:12) log | z | (cid:12)(cid:12) < s (cid:9) . Then, the domains { D t,s } t,s constitute a family of approximating domains for D µ . The distin-guished boundary of these domains is given by d b ( D t,s ) = (cid:8) ( z , z ) ∈ C : (cid:12)(cid:12) arg z − log | z | (cid:12)(cid:12) = t, (cid:12)(cid:12) log | z | (cid:12)(cid:12) = s (cid:9) . Consequently, for 1 ≤ p < ∞ , we define the Hardy space H p ( d b ( D β ) , dβ ) defined by H p ( d b ( D β ) , dβ ) = (cid:26) f ∈ Hol( D β ) : k f k pH p ( d b ( D β ) ,dβ ) = sup ( t,s ) ∈ (0 , π × [0 ,µ ) k f k pL p ( d b ( D t,s ) ,dβ ) < ∞ (cid:27) , where, denoting by dβ t,s the induced measure on d b ( D t,s ), k f k pL p ( d b ( D t,s ) ,dβ ) = Z d b ( D t,s ) | f | p dβ t,s = Z ∞ Z π | f (cid:0) re i ( s + t ) , e s e iθ (cid:1) | p e s dθdr + Z ∞ Z π | f (cid:0) re i ( s − t ) , e s e iθ (cid:1) | p e s dθdr + Z ∞ Z π | f (cid:0) re − i ( s + t ) , e − s e iθ (cid:1) | p e − s dθdr + Z ∞ Z π | f (cid:0) re − i ( s − t ) , e − s e iθ (cid:1) | p e − s dθdr . The main results in [MP17b] are the following. The first result provides the sharp interval ofvalues of p for which the Szeg˝o projection S on the distinguished boundary of D µ is bounded.We recall that we set ν = π µ , so that ν = ν µ tends to 0 as µ becomes large. Theorem 2.3.
The Szeg˝o projection S , initially defined on the dense subspace L p ( d b ( D µ ) , dβ ) ∩ L ( d b ( D µ ) , dβ )) , extends to a bounded operator S : L p ( d b ( D µ ) , dβ )) → L p ( d b ( D µ ) , dβ )) if and only if ν < p < − ν . The next result concerns with the sharp boundedness of S on the L -Sobolev spaces on d b ( D µ ). Theorem 2.4.
The Szeg˝o projection S defines a bounded operator S : W s, ( d b ( D µ ) , dβ )) → W s, ( d b ( D µ ) , dβ ) if and only if ≤ s < ν . In the case of Sobolev norms with p = 2 we do not have a complete characterization of themapping properties of S , but we have a partial result. Theorem 2.5.
Let s > and p ∈ (1 , ∞ ) . If the operator S , initially defined on the dense sub-space W s,p ( d b ( D µ ) , dβ )) ∩ L ( d b ( D µ ) , dβ ) , extends to a bounded operator S : W s,p ( d b ( D µ ) , dβ ) → W s,p ( d b ( D µ ) , dβ ) , then − ν β ≤ s + 12 − p ≤ ν β . Assuming p ≥ we obtain the stronger condition ≤ s + 12 − p < ν β . IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 9
The main fact used in the proofs is that we can write the Szeg˝o projection S as a sum ofMellin–Fourier multiplier operators which we now briefly describe. In order to do so we introducesome notation. We set X to denote either R or T , and, accordingly, b X = R , or Z , respectively,where we denote by b the Fourier transform or Fourier series on R and T , resp. Instead, wedenote by F the Fourier transform on R × X , given by F f ( ξ , ξ ) = Z R × X f ( x , x ) e − i ( x ξ + x ξ ) dx dx when f is absolutely integrable. We consider the Fourier multiplier operator given by T m ( f ) = F − (cid:0) m F f (cid:1) when m is a bounded measurable function on R × b X . We say that a bounded function m on R × X is a bounded Fourier multiplier on L p ( R × X ) if T m : L p ( R × X ) → L p ( R × X ) is bounded.Given a function ϕ ∈ C ∞ c (cid:0) (0 , ∞ ) × X (cid:1) we define the operator C p ϕ ( x, y ) = e p ( x ) ϕ ( e x , y ) . It is clear that C p extends to an isometry of L p (cid:0) (0 , ∞ ) × X (cid:1) onto L p (cid:0) (0 , ∞ ) × X (cid:1) .For a, b ∈ R , with 0 < a < b <
1, we denote by S a,b the vertical strip in the complex plane S a,b = (cid:8) z ∈ C : a < Re z < b (cid:9) . Given a bounded measurable function m defined on S a,b × X , when a < p < b we write m p ( ξ , ξ ) = m ( p − iξ , ξ ) . Finally, we define an operator acting on functions defined on (0 , + ∞ ) × X as T m,p = C − p T m p C p . (6)We call such an operator a Mellin–Fourier multiplier operator, the reason for which will soon beclear. A similar class of operators was studied by Rooney [Roo85]. Incidentally, we believe thatthis class of operators is of its own interest. Theorem 2.6.
With the above notation, let m : S a,b × X → C be continuous and such that (i) m ( · , ξ ) ∈ Hol( S a,b ) and bounded in every closed substrip of S a,b , for every ξ ∈ X fixed; (ii) for every q such that a < q < b , m q is a bounded Fourier multiplier on L q ( R × X ) .Then, for a < p < b , T m,p = T m is independent of p and T m : L p ((0 , + ∞ ) × X ) → L p ((0 , + ∞ ) × X ) is bounded. In the course of the proof we show that if m satisfies the hypotheses of the theorem, then C − p T m p C p ( ϕ ) = F − M − (cid:0) m ( M F ϕ ) (cid:1) , (7)where M denotes the Mellin transform in the first variable, that is, M ϕ ( z, ζ ) = Z + ∞ t z − ϕ ( t, ζ ) dt, and F denotes the Fourier transform in the second variable. Equality (7) clearly giustify thefact that the operator T m a Mellin–Fourier multiplier operator: it is a Mellin transformationin the first variable, a Fourier transformation in the second variable, followed by multiplication by m and then the inverses of the Mellin and Fourier transforms. We also point that, if m, ˜ m satisfy the assumptions in the theorem, then T m T ˜ m = T m ˜ m , and thus it is reasonable to callthese operators multipliers .Once we explictly write the Szeg˝o projection S as a linear combination of Mellin–Fouriermultiplier operators, we are able to study its regularity by also exploiting the regularity of S ′ . We recall that, unlike in the case of the Bergman projection, in the Szeg˝o setting, ingeneral, there is no transformation rule for the Szeg˝o projection under biholomorphic mappings.Nonetheless, we are able to prove a transformation rule for the projections S and S ′ . Recallthat D ′ µ and D µ are biholomorphically equivalent via the map ϕ − : D β → D ′ β ( z , z ) (Log( z e − i log | z | ) + i log | z | , z ) , (8)where Log denotes the principal branch of the complex logarithm. Setting ψ p ( z , z ) := e − ip log | z | ( z e − i log | z | ) − p we obtain that S ′ (Λ − f ) = Λ − ( S ) , where Λ f := ψ p ( f ◦ ϕ − ).3. Other results on the regularity of Szeg˝o projections.
Mapping properties of the Szeg˝o projection on other function spaces have been studied forvarious classes of smooth bounded domains and the are several positive results. The Szeg˝oprojection S Ω turns out to be bounded on the Lebesgue–Sobolev spaces W s,p ( ∂ Ω) for 1 < p < ∞ and s ≥ C [NRSW89] and convex domains of finite type in C n [MS97]. The exact regularity of S Ω , thatis, the boundedness S Ω : W s, ( ∂ Ω) → W s, ( ∂ Ω) for every s ≥
0, holds when Ω is a Reinhardtdomain [Boa85, Str86], a domain with partially transverse symmetries [BCS88], a pseudoconvexdomain satisfying Catlin’s property ( P ) [Boa87], a complete Hartogs domain in C [BS89], ora domain with a plurisubharmonic defining function on the boundary [BS91]. We also mentionthat, if Ω is bounded, C and strongly pseudoconvex in C n , the Szeg˝o projection P Ω againextends to bounded operator on L p ( ∂ Ω) for 1 < p < ∞ , [LS16, LS17].There are also examples of domains Ω on which the Szeg˝o projection P Ω is less regular. L.Lanzani and E. M. Stein described the (ir-)regularity of P Ω on Lebesgue spaces in the case ofplanar simply connected domains, [LS04, Thm. 2.1]. In particular they showed that if Ω hasLipschitz boundary, then P Ω : L p ( ∂ Ω) → L p ( ∂ Ω) if and only if p ′ Ω < p < p Ω , where p Ω dependsonly on the Lipschitz constant of ∂ Ω. More recently, S. Munasinghe and Y.E. Zeytuncu providedan example of a piecewise smooth, bounded pseudoconvex domain in C on which the Szeg˝oprojection P Ω is unbounded on L p ( ∂ Ω) for every p = 2 [MZ15]. The same result on tube domainsover irreducible self-dual cones of rank greater than 1 has been known for a number of years,[BB95].In a recent paper [LS19] Lanzani and Stein announced a result concerning the L p continuityof the Szeg˝o projection attached to the smooth worm domain W µ with respect to the inducedsurface measure d σ on ∂ W µ . In particular, they announced that for any p = 2 there is a µ = µ ( p )such that the Szeg˝o projection is not bounded P W µ : L p ( ∂ W µ ) → L p ( ∂ W µ ). IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 11
It is reasonable to think that the culprit of the (ir-)regularity of both the Bergman andSzeg˝o projection on the worm domain W µ is the presence of the critical annulus A = { (0 , z ) : (cid:12)(cid:12) log | z | (cid:12)(cid:12) ≤ µ } in the boundary ∂ W µ ; see, for instance, [BS92]. For this reason, it would beinteresting to study the Szeg˝o projection of W µ with respect to the Fefferman measure dσ F . Infact, as we now see, the Fefferman measure dσ F is given by a smooth density ω times dσ and thedensity ω vanishes identically on the critical annulus A / In detail, given Ω = { z ∈ C n : ρ ( z ) < } ,the Fefferman surface area measure ([Fef79, pg. 259],[BL14]) on ∂ Ω is defined by dσ F = c n n +1 p M ( ρ ) dσ k∇ ρ k (9)where M ( ρ ) is the Fefferman Monge–Amp´ere operator M ( ρ ) = − det (cid:18) ρ ρ k ρ j ρ jk (cid:19) ≤ j,k ≤ n . It can be proved (see also [BL14, Section 2]) that the definition of dσ F does not depend on thedefining function ρ and that there exists a sesqui-holomorphic kernel S ( z, ζ ) such that, for every f ∈ H (Ω), f ( z ) = Z ∂ Ω f ( ζ ) S ( z, ζ ) dσ F ( ζ ) . Hence, Hardy spaces and Szeg˝o projections with respect to the Fefferman measure can be definedand investigated. In the case of the worm domain W µ the defining function is ρ ( z , z ) = | z | − z e − i log | z | ) + η (log | z | ) , therefore, setting R ( z , z ) = Re( iz e − i log | z | ),. M ( ρ )( z , z ) = z − e i log | z | z (cid:0) R ( z , z ) + η ′ (log | z | ) (cid:1) z − e − i log | z | iz e − i log | z | z (cid:0) R ( z , z ) + η ′ (log | z | ) (cid:1) − iz e i log | z | | z | (cid:0) R ( z , z ) + η ′′ (log | z | ) (cid:1) . When we restrict the matrix M ( ρ ) to the critical annulus A we getdet M ( ρ )(0 , z ) = det − e i log | z | − e − i log | z | iz e − i log | z | − iz e i log | z | = 0 . Since the boundary of the domain D µ (similarly, of D ′ µ ) is Levi flat, that is, the Levi formof its defining function is identically zero at every point of bD µ , the density of the Feffermanmeasure on bD µ is identically zero. This can be easily verified by explicitly computing M ( ρ ) for ρ ( z , z ) = Re( z e − i log | z | ) . Thus, the Szeg˝o projection on W µ with respect to the Feffermanarea measure cannot be investigated exploiting the model domains, but it must be directlyapproached. This certainly is an interesting direction for future research. Hartogs triangles
Another class of domains on which it is interesting to test and study the regularity of theBergman and Szeg˝o projection is the one of generalized Hartogs triangles. Given a real parameter γ >
0, the generalized Hartogs triangle H γ is defined as H γ = (cid:8) ( z , z ) ∈ C : | z | γ < | z | < (cid:9) . This family of domains were recently introduced in [Edh16b] and the value γ = 1 correspondsto the classical Hartogs triangle H ([Sha15]). The Hartogs triangle is a simple, but not trivial,model domain on which it is worth to test several conjectures. It turns out that H is a sourceof counterexamples in complex analysis. For instance, as the worm domain W , the Hartogstriangle has non-trivial Nebenh¨ulle. However, unlike W , the domain H is not smooth; on thecontrary, it is highly singular at the point z = z = 0. This pathological geometry affects the L p behavior of the Bergman projection P H and it turns out that P H extends to a bounded operator P H : L p ( H ) → L p ( H ) if and only if p ∈ ( ,
4) ([CZ16a]). This result has been extended to thecase of generalized Hartogs triangle in a series of paper by L. D. Edholm and J. D. McNealand it holds that P H γ extends to a bounded operator L p ( H γ ) → L p ( H γ ) for a restricted rangeof p ∈ (1 , ∞ ) whenever γ ∈ Q , but P H γ is unbounded on L p ( H γ ) for any p = 2 whenever γ isirrational ([Edh16b, EM16, EM17]). The Sobolev (ir-)regulariy of P H γ has been investigated aswell and we refer the reader to the very recent paper [EM20].In addition to the aforementioned papers, we mention also the recent papers [HW19b, HW19a],where weighted L p and endpoint estimates for the classical Hartogs triangle are obtained viadyadic harmonic analysis techniques, and [CZ16b, Che17a, Che17b, Che17c, CKY20], wheresome other generalizations of the Hartogs triangle and the associated weighted Bergman pro-jections are investigated.The definition of a Hardy space H on H , hence the definition of a Szeg˝o projection on H , isnot canonical due to the geometry of the domain. We now recall the definition of a candidateHardy space on the classical Hartogs triangle which is introduced by the first author in a recentpaper [Mon19].Let ν > − D be the unit disc in the complex plane and let us considerthe classical weighted Bergman spaces A ν ( D ) defined as the space of holomorphic functions in D endowed with the norm k f k A ν ( D ) = ( ν + 1) Z D | f ( z ) | (1 − | z | ) ν dz. It is a well-known fact that k f k H ( D ) = lim ν →− + k f k A ν ( D ) where H ( D ) is the Hardy space in D , that is, the space of holomorphic functions in D endowedwith the norm k f k H ( D ) := sup
It would be ideal to be able to obtain the asymptotic expansion of the Bergman and Szeg˝okernels on the worm domain W . A fundamental step in this direction would be to obtain theexplicit expression of Bergman and Szeg˝o kernel on the truncated worm domain W ′ = n ( z , z ) ∈ C : | z − e i log | z | | < , | log | z | | < µ o . (11)Obviously, this domain coincides with W when we select η = χ | t | >µ , and W ′ is bounded, non-smooth, and its boundary contains the same critical annulus A as W . In analogy with the caseof the unit bidisk D , we are led to look for an orthonormal basis of mononials. In the case of W ′ , as well as of W , the following functions resemble the monomials z j z k , j, k ∈ N , where N denotes the set of non-negative integers. We set E η ( z ) = e ηL ( z ) , where L ( z ) = log (cid:0) z e − i log | z | (cid:1) + i log | z | , and log denotes the principal branch of the logarithm, so that E η ( z , z ) = (cid:0) z e − i log | z | (cid:1) η e iη log | z | . Now we define constants γ αβ = h ( α − β ), where h ( z ) = sinh[ µ ( j +1+ iz )] j +1+ iz . The following is Propo-sition 3.1 in [KPS19]. IR-)REGULARITY OF CANONICAL PROJECTION OPERATORS 15
Proposition 5.1.
Let µ > . For α ∈ C and j ∈ Z let F α,j ( z , z ) = E α ( z ) z j . Then F α,j ∈ A ( W ′ µ ) if and only if Re α > − . Moreover, if Re α, Re β > − then h F α,j , F β,j i A ( W ′ µ ) = (2 π ) γ αβ Γ( α + β + 2)Γ( α + 2)Γ( β + 2) . In particular, h F α,j , F β,j i A ( W ′ µ ) = 0 if and only if α − β = 2 kν + i ( j + 1) with k ∈ Z \ { } . (12)Thus, if c > − ℓ ∈ N , and we set H ℓ,j ( z , z ) = E c + νℓ + i ( j +1) / ( z ) z j , (13)the next corollary follows. Corollary 5.2.
Each of the two sets (cid:8) H k,j , j ∈ Z , k ∈ N (cid:9) , and (cid:8) H k +1 ,j , j ∈ Z , k ∈ N (cid:9) , (14) is an orthogonal system in A ( W ′ µ ) . Thus, we are led to consider the following problem. We set ∆ = { ζ : | ζ − | < } and considera set of functions { ζ λ k } , k = 1 , , . . . . We call the M¨untz–Sz´asz problem for the Bergman space the question of determining necessary and sufficient condition for such a set to be a completeset in A (∆), that is, its linear span to be dense in A (∆). The following is Theorem 3.1 in[KPS19], that gives a sufficient condition for the solution of the M¨untz–Sz´asz problem for theBergman space. Theorem 5.3.
For k ∈ N , < a < , c > − and b ∈ R , let λ k = ak + c + ib . Then { ζ λ k } isa complete set in A (∆) . As a consequence we obtain the following density result in A ( W ′ µ ), which is Theorem 3.1 in[KPS19]. Theorem 5.4.
Let µ > π/ . Let H ℓ,j ( z , z ) be as in (13) . Then { H ℓ,j } j ∈ Z , ℓ ∈ N , is a completeset in A ( W ′ µ ) . Notice that the set { H ℓ,j : j ∈ Z , ℓ ∈ N } is the union of the two sets in (14). However,such that set is not an orthogonal set, and we cannot compute the Bergman kernel from suchcomplete set.We now divert a bit from our main course to discuss the question of solving the M¨untz–Sz´aszproblem. This was done in [PS16, PS17], however without finding a complete solution. In [PS17]it is proved that the M¨untz–Sz´asz problem for the Bergman space is equivalent to characterizingthe sets of uniqueness of the Hilbert space of holomorphic functions M ω ( R ) which is the spaceof holomorphic functions on the right half-plane R such that:(H) f ∈ H ( S b ) for every 0 < b < ∞ ;(B) f ∈ L ( R , dω );where H ( S b ) denotes the standard Hardy space on the vertical strip { z ∈ C : 0 < Im z < b } ,and ω is the measure on R ω = + ∞ X n =0 n n ! δ n ( x ) ⊗ dy . Observe thag ω is a translation invariant measure in R . A quite interesting fact is that suchspace M ω ( R ) is closely related to a space of holomorphic functions descovered by T. Kriete andD. Trutt, [KT71, KT75]. The following are the main results in [PS17] on this problem. Theorem 5.5.
Let { z j } ⊆ R , ≤ | z j | → + ∞ . The following properties hold. (i) If { z j } has exponent of convergence and upper density d + < , then { z j } is a zero-setfor M ω ( R ) ∩ Hol( R ) . (ii) If { z j } is a zero-set for M ω ( R ) ∩ Hol( R ) , then lim sup R → + ∞ R X | z j |≤ R Re (cid:0) /z j (cid:1) ≤ π . (15)We observe that part (ii) in the above theorem follows from a generalization of the classicalCarleman’s formula in the right half-plane. Theorem 5.6.
A sequence { z j } of points in R such that Re z j ≥ ε , for some ε > and thatviolates condition (15) , is a set of uniqueness for M ω ( R ) .As a consequence, if { z j } is a sequence as above, the set of powers { ζ z j − } is a complete setin A (∆) . Final Remarks
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