Irregularity of the Bergman Projection on Smooth Unbounded Worm Domains

In this work, we consider smooth unbounded worm domains Zλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}_\lambda $$\end{document} in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^2$$\end{document} and show that the Bergman projection, densely defined on the Sobolev spaces Hs,p(Zλ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{s,p}({\mathcal {Z}}_\lambda ),$$\end{document}p∈(1,∞),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,\infty ),$$\end{document}s≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ge 0,$$\end{document} does not extend to a bounded operator Pλ:Hs,p(Zλ)→Hs,p(Zλ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\lambda :H^{s,p}({\mathcal {Z}}_\lambda )\rightarrow H^{s,p}({\mathcal {Z}}_\lambda )$$\end{document} when s>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>0$$\end{document} or p≠2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ne 2.$$\end{document} The same irregularity was known in the case of the non-smooth unbounded worm. This improved result shows that the irregularity of the projection is not a consequence of the irregularity of the boundary but instead of the infinite windings of the worm domain.


Introduction
Let φ be a non-negative smooth function on R such that ‚ φ is convex ‚ φ ´1p0q " p´8, 0s.Notice that φ 1 ptq ą 0 for t ą 0 and that there exists a ą 0 such that φpaq " 1.For λ ą 0 we set Then, Z λ is smooth, unbounded and pseudoconvex (see Theorem 1.1 below).Moreover, tZ λ u λą0 is a nested family of domains whose union is the unbounded non-smooth worm The domain W was studied in [KPS16], where three main facts were proved (see the enumerated list below).For p P r1, 8s and s ě 0, given any domain Ω, denote by H s,p " H s,p pΩq the standard Sobolev space on Ω.When s " k is an integer, H s,p consists of functions with k-derivatives in L p pΩq, and for noninteger s, H s,p can be defined by interpolation, see Section 2. For p P r1, 8s, let A p pΩq :" L p pΩq X HolpΩq denote the Bergman space.In [KPS16] it was proved that: (i) the space A 2 pWq ‰ t0u, so that the Bergman projection P : L 2 pWq Ñ A 2 pWq is a non-trivial orthogonal projector; (ii) the operator P , initially defined on a dense subspace of L p pWq, extends to a bounded operator P : L p pWq Ñ L p pWq (if and) only if p " 2; (iii) the operator P , initially defined on a dense subspace of H s,2 pWq, extends to a bounded operator P : H s,2 pWq Ñ H s,2 pWq (if and) only if s " 0.
The goal of this paper is to show that also in the case of the unbounded smooth worms Z λ , λ ą 0, the Bergman projection P λ on Z λ cannot be extended to a bounded operator P λ : H s,p pZ λ q Ñ H s,p pZ λ q when s ą 0 or p ‰ 2. Observe that, since Z λ Ď W for all λ ą 0, (i) above implies that A 2 pZ λ q and hence P λ are non-trivial.We now state our main results.
Theorem 1.2.Let λ ą 0, Z λ be defined as above, and let P λ denote the Bergman projection on Z λ .If P λ , initially defined on the dense subspace pL 2 X H s,p qpZ λ q, p P p1, 8q and s ě 0, extends to a bounded operator P λ : H s,p pZ λ q Ñ H s,p pZ λ q then necessarily s " 0 and p " 2.
The problem of the regularity of the Bergman projection on worm domains has been an object of active and intense research.In the seminal paper [Bar92] D. Barrett considered the smoothly bounded worm domain where η is smooth, non-negative, convex, η ´1p0q " r´µ, µs, and such that W µ is smooth, bounded and pseudoconvex, see e.g.[Kis91, Proposition 2.1].Barrett showed that the Bergman projection on W µ does not preserve the Sobolev space H s,2 pW µ q if s ě π{µ, whereas in [KP08b] it was then shown that the Bergman projection on W µ does not preserve L p if ˇˇ1 2 ´1 p ˇˇě π µ .We further mention in particular [KP08b, BS ¸12, BEP15, ČS ¸18, KPS16,BDP17].We also refer the reader to [KP08a] for an expository account of the subject, and to [KPS19, PS16,PS17] for some interesting connections between Bergman spaces on worm domains and the Müntz-Szász problem for the Bergman space in one complex dimension.
In the next section we prove Theorem 1.1, whereas in Section 3 we introduce the tools that we need to deal with Sobolev spaces on smoothly bounded domains.In Section 4 we prove Theorem 1.2 and in the final Section 5 we discuss some open problems and future work.

The unbounded smooth worm
Consider the domains Z λ .It is clear that they are unbounded, that Since Z λ Ď W, where W is as in (2), [KPS16, Proposition 2.3] gives that A 2 pZ λ q is infinite dimensional.Similar calculations also show that also the spaces A p pZ λ q are infinite dimensional, p P p0, 8s.Explicitly, for α P C, for z " pz z , z 2 q P W, let where logpzq denotes the principal branch of the logarithm on Czp´8, 0s.Then L, E α P HolpWq by [KPS16, Lemma 2.2].Moreover, for j P Z, m P N, c P R, c ą log 2, α " Re α `ip j 2 `1 p q, setting The argument to show that Z λ is smooth and pseudoconvex is standard, but we repeat it for the sake of completeness.Letting ρ denote the defining function of Z λ we observe that Let pz 1 , z 2 q P bZ λ be such that B z1 ρpz 1 , z 2 q " 0. Then z 1 " e i log |z2| 2 so that φplogpλ|z 2 |q ´2q " 1.The assumptions on φ imply that B z2 ρ ‰ 0 at such points.Thus Z λ is smooth, and it is clearly unbounded since it contains points pz 1 , z 2 q with |z 2 | arbitrarily large.
In order to show that Z λ is pseudoconvex, arguing as in [Kis91], we observe that locally a branch of arg z 2 is defined and that the local defining function e arg z 2 2 ρ equals The first two terms are plurisubharmonic, while the third one satisfies the differential inequality since φ is smooth and convex.Hence, Z λ is pseudoconvex.Moreover, the defining function is strictly plurisubharmonic at every boundary point where z 1 ‰ 0. Next, at pz 1 , z 2 q P bZ λ , the complex tangent space is spanned by the vector and 2 Repz 1 e ´i log |z2| 2 q " |z 1 | 2 `φplogpλ|z 2 |q ´2q on the boundary, the Levi form is given by It follows that the boundary points tp0, z 2 q : |z 2 | ě 1{λu are of weak pseudoconvexity.This proves Theorem 1.1.

Sobolev spaces on smoothly bounded domains and on Z λ
In this section we collect the results on Sobolev spaces on smoothly bounded domains and prove a few properties that we shall need later.We begin by recalling the definition and a few standard results from the theory of function spaces on smoothly bounded domains, see e.g.[Tri83,Chapter 3] and [LM61].In what follows the space H s,p pR d q is defined by means of the Fourier transform F on R d and D1 pΩq is the dual of the space C 8 c pΩq of smooth functions with compact support in Ω. Namely, H s,p pR d q " ) .
Definition 3.1.Let Ω be a smoothly bounded domain in R d , s ě 0 and p P p1, 8q.We define H s,p pΩq " We also denote by H s,p 0 pΩq the closure of C 8 c pΩq in the H s,p pΩq-norm.Then, for s ă 0 and p P p1, 8q, we define H s,p pΩq as the dual of H ´s,p 1 0 pΩq, where p 1 " p{pp ´1q is the exponent conjugate to p. 1 When s " k is a non-negative integer the space H k,p pΩq has a natural characterization.On the space C 8 pΩq consider the norm }ψ} W k,p pΩq :" and define W k,p pΩq as the closure of C 8 pΩq with respect to this norm.Then W k,p pΩq is isomorphic to H k,p pΩq, with equivalence of norms, see e.g.[Tri83].
Using the complex interpolation method we have that, when s ą 0, where θ P p0, 1q and s " k `θ, cf.[Tri83] or [LM61], so that rW k,p pΩq, W k`1,p pΩqs θ is isomorphic as Banach space to H s,p pΩq, s " k `θ.For the complex interpolation method we refer to [BL76].
Since Ω is a bounded, smooth domain, the multiplier operator f Þ Ñ χ Ω f is bounded on H s,p pR d q when 0 ď s ă 1 p , p P p1, 8q.This fact in turn implies the key property that C 8 c pΩq is dense in H s,p pΩq when 0 ď s ă 1 p -see [Tri83,Theorem 3.4.3].We now prove a result that is probably well known, but for which we do not know a precise reference.Lemma 3.2.For ´1{p 1 ă s ă 1{p, the spaces H s,p pΩq and H ´s,p 1 pΩq are mutually dual with respect to the L 2 pΩq pairing of duality.
Proof.Observe that, by duality, we may assume that 0 ď s ă 1{p.Since H s,p pΩq " H s,p 0 pΩq in the given range, H ´s,p 1 pΩq " `Hs,p pΩqq ˚with the L 2 -pairing of duality.
Conversely, let ℓ P `H´s,p 1 pΩq ˘˚.Since the multiplication f Þ Ñ χ Ω f is bounded on H s,p pR d q, H s,p pΩq can be identified with the subspace of H s,p pR d q of functions vanishing on Ω c .Therefore also H ´s,p 1 pΩq can be identified with the elements of `Hs,p pR d q ˘˚" H ´s,p 1 pR d q that annihilate functions of H s,p pR d q vanishing on Ω c .Therefore, by the Hahn-Banach theorem, there exists L P `H´s,p 1 pR d q ˘˚" H s,p pR d q with the same norm, that agrees with ℓ on H ´s,p 1 pΩq.Hence there exists F P H s,p pR d q such that ℓpuq " ş Ω F u " ş Ω pχ Ω F qu, where χ Ω F P H s,p pΩq; that is, `H´s,p 1 pΩq ˘˚" H s,p pΩq.l Next we need an extension of a result by E. Ligocka, namely [Lig87, Theorem 2].We denote by H s,p har pΩq the subspace of H s,p pΩq consisting of harmonic functions.Let ̺ : R d Ñ R be a smooth defining function (see [Kra01]) for Ω and let L p har pΩ, |̺| q q be the subspace of L p pΩ, |̺| q dmq consisting of harmonic functions on Ω, p P p1, 8q.In [Lig87, Theorem 2], Ligocka proved that, for s ě 0, p P p1, 8q, har pΩq and H ´s,p 1 har pΩq are mutually dual with respect to the L 2 pΩq-inner product; (ii) H ´s,p 1 har pΩq is isomorphically equivalent (as a Banach space) to L p 1 har pΩ, |̺| sp 1 q.We shall need the following extension of (ii).
Proof.As mentioned, the case s ď 0 is proved in [Lig87, Theorem 2].Next, let 0 ă s ă 1{p.If f P L p har pΩ, |̺| ´sp q and g P L p 1 har pΩ, |̺| sp 1 q we have that ˇˇˇż We now define Sobolev spaces on the smooth unbounded domains Z λ .Definition 3.4.For k a non-negative integer and p P p1, 8q, define the space (of test functions) where D z :" pB z1 , B z1 ; B z2 , B z2 q.We define H k,p pZ λ q as the closure of T pZ λ q with respect to the norm } ¨}H k,p pZ λ q .For s " k `θ with 0 ă θ ă 1, we define H s,p pZ λ q, p P p1, 8q, by complex interpolation, as H s,p pZ λ q :" rH k,p pZ λ q, H k`1,p pZ λ qs θ .
Finally, we point out the following fact that we will need later.
Remark 3.5.Let µpλq " log λ 2 , and consider the domain W µpλq as defined in (3), where η is given by ηptq " φpt ´log λ 2 q `φp´t ´log λ 2 q, so that W µpλq Ď Z λ .Observe then that the restriction operator H s,p pZ λ q Q f Þ Ñ f |W µpλq P H s,p pW µpλq q is well defined and norm decreasing when s " k is a non-negative integer and p P p1, 8q, and then, by interpolation, also when s ě 0 and p P p1, 8q.Analogously, for all λ 1 ą λ, }f } H s,p pW µpλq q ď }f } H s,p pZ λ 1 q .

Irregularity of the Bergman projection
The proof of Theorem 1.2 will combine some new ideas with Barrett's arguments [Bar92] and results from [KPS16].We first extend [KPS16, Corollary 5.5] to the case of the Sobolev spaces H s,p pW µ q.
Proposition 4.1.Let W be the unbounded non-smooth worm, K w its Bergman kernel at w P W, and W µ be the smoothly bounded worm as in (3).Suppose p P p1, 8q.Then the following properties hold: the open segments of end points p0, 0q and p1, 1q and p0, 0q and p 1 2 , 0q, resp., in Figure 1), then K w R H s,p pW µ q; (ii) if s " 2 p ´1 and p P p1, 2q (the open segment of end points p 1 2 , 0q, p1, 1q in Figure 1), then }K w } H s,p pWµq Ñ 8 as µ Ñ 8.
We now recall some notation from [KPS16, Corollary 5.5].We let Spe i log |z2| 2 , εq denote the angular sector in the z 1 -plane , εq " z 1 " re ipt`log |z2| 2 q : |t| ă δ, 0 ă r ă ε ( , with 0 ă δ ă π{2.For ε ą 0 sufficiently small, the set is contained in W µ .Then, from [KPS16, (5.8) and p. 1180], for w P W and z P G µ we have the estimate where C w does not depend on µ.Therefore, arguing as in [KPS16, Corollary 5.5], for s ă 1{p we have (i) Suppose then that s P `2 p ´1, 1 p ˘. From Lemma 3.3 K w P H s,p pW µ q if and only if K w P L p p|̺| ´sp q.From (7) it then follows that K w R H s,p pW µ q when s P p 2 p ´1, 1 p q.We now use the natural embedding H s,p pW µ q Ď H s 1 ,p pW µ q when 0 ď s 1 ď s (see [Tri83,Theorem 3.3.1]).It follows that K w R H s,p pW µ q for all p, s such that p P p1, 8q and s ą 2 p ´1.This proves (i).(ii) We look at the estimate in (7) when s " 2 p ´1 (notice that s ă 1{p in this case) and observe that Clearly, if p ‰ 2, the right hand side above tends to 8 if µ Ñ 8.The rest of the proof will show that the same is true for }K w } H s,p pWµq .We observe in passing that, on the other hand, if p " 2, the right hand side above remains bounded (actually, it tends to 0) when µ Ñ 8, in accordance to the fact that }K w } L 2 pWµq ď }K w } L 2 pWq ă 8.
In order to prove Theorem 1.2 we need two preliminary lemmas.We denote by }T } pX,Xq the operator norm of T : X Ñ X. Lemma 4.2.For λ, λ 1 ą 0, the domain Z λ is biholomorphic to Z λ 1 .Moreover, the Bergman projection P λ induces a bounded operator on L p pZ λ q for some λ ą 0 if and only if P λ induces a bounded operator on L p pZ λ q for every λ ą 0 and in this case }P λ } pL p pZ λ q,L p pZ λ qq is independent of λ.
Proof.In order to show that the domains Z λ are all biholomorphic to each other, it suffices to observe that for all r, λ ą 0, is a biholomorphic map, since Φ λ P GLp2, Cq and Φ λ pZ r q " Z λr .Moreover, det Φ 1 λ " e ´i log λ 2 {λ and T λ,p f :" pdet Φ 1 λ q 2{p f ˝Φλ is an isometric isomorphism T λ,p : L p pZ λr q Ñ L p pZ r q, }T λ,p f } L p pZr q " }f } L p pZ λr q , (12) that also gives an isometric isomorphism T λ,p : A p pZ λr q Ñ A p pZ r q when restricted to A p pZ λr q, p P r1, 8s.
Recalling the transformation rule for the Bergman projections P r pdet Φ 1 λ f ˝Φλ q " det Φ 1 λ pP λr f q ˝Φλ for every f P L 2 pZ λ q, since det Φ 1 λ is constant, it follows that P r pf ˝Φλ q " pP λr f q ˝Φλ , for all f P L 2 pZ λ q and λ ą 0. This implies that (also when p ‰ 2) P r pT λ,p f q " T λ,p pP λr f q for all f P pL 2 X L p qpZ λr q.Since pL 2 X L p qpZ λr q is dense in L p pZ λr q and T λ,p pL 2 X L p qpZ λr q is dense in L p pZ r q, for f P L p pZ λr q we have }P λr f } L p pZ λr q " }T λ,p P λr f } L p pZrq " }P r pT λ,p f q} L p pZrq .
Since T λ,p : L p pZ λr q Ñ L p pZ r q is an isometric isomorphism, the equality of the operator norms of P λ easily follows.
Lemma 4.3.Let s ą 0, p P p1, 8q and suppose P λ induces a bounded operator on H s,p pZ λ q for some λ ą 0.Then, P λ 1 induces a bounded operator on H s,p pZ λ 1 q for all λ 1 ą λ and, setting N s,p pλq " }P λ } pH s,p pZ λ q,H s,p pZ λ qq , we have N s,p pλ 1 q ď N s,p pλq. ( for all λ 1 ą λ. Proof.For r ą 0, let T r,p be as in the proof of Lemma 4.2.We argue as in [Bar92].Recalling that D z " pB z1 , B z1 ; B z2 , B z2 q, if α " pa 1 , b 1 ; a 2 , b 2 q is a given multi-index, we have that D α z pf ˝Φr qpzq " e ipb1´a1q log r 2 r ´pa2`b2q pD α z f qpΦ r pzqq.Therefore, for λ ą 0, r ą 1, and k a positive integer, using (12), we have Next observe that, using the transformation rule and a change of variables, for z P Z r , K λ pz, w 1 qf pΦ r pw 1 qq dV pw 1 q " D α z pP λ pf ˝Φr qqpzq, so that T r,p pD α z P rλ f q " D α z pP λ T r,p f q.
Therefore, assuming that P λ is bounded on H s.p pZ λ q, for r ą 1, using both the fact that T r,p : L p pZ rλ q Ñ L p pZ λ q is an isometry and (14) we have Therefore N k,p prλq ď N k,p pλq for all integers k and r ą 1.Thus, by interpolation, for all s ą 0 and r ą 1, (13) follows.
Proof of Theorem 1.2.Step 1. Suppose that P λ0 is bounded on L p pZ λ0 q for some λ 0 ą 0 and some p P p1, 8q.Hence P λ is bounded on L p pZ λ q for all λ ą λ 0 by Lemma 4.2.Fix f P C 8 c pWq where W is the non-smooth unbounded worm and suppose that supp f Ď Z λ for all λ ě λ 0 .For all such λ's, denoting by χ λ the characteristic function of Z λ , for some constant C 1 independent of λ.In the second-before-last inequality, we have used Lemma 4.2.Then, there exist a sequence tλ n u, λ n Ñ 8 as n Ñ 8, and h P pL 2 X L p qpWq such that χ λn P λn f Ñ h in the weak-˚topology, as n Ñ 8.It is easy to see that h P HolpWq arguing as follows.Let ψ be smooth, and compactly supported in W.Then, denoting by dV the Lebesgue volume form, for j " 1, 2 we have Hence, B zj h " 0, j " 1, 2 and therefore h is holomorphic.We claim that h " P f , where P denotes the Bergman projection on W. It suffices to show that f ´h K A 2 pWq.To this end, let g P A 2 pWq.Then ż since the restriction of g to Z λ belongs to A 2 pZ λ q for all λ ą 0 as well.
Seeking a contradiction, we suppose p ‰ 2 and remark that This implies that P : L p pWq Ñ L p pWq is bounded, contradicting [KPS16, Theorem 1.1].Therefore, P λ0 cannot be bounded on L p pZ λ0 q for p P p1, 8q and p ‰ 2. We also observe that, by interpolation with the case p " 2, P λ0 cannot be bounded on L 1 and L 8 either.
Step 2. In order to prove the irregularity of P λ in the Sobolev scale, we first show that P λ is densely defined by showing that pL 2 X H s,p qpZ λ q is dense in H s,p pZ λ q.Let ϕ P C 8 c pC 2 q, ϕ " 1 on the ball Bp0, 1q and set ϕ ε p ¨q " ϕpε¨q.Given f P H s,p pZ λ q, let f pεq :" f ϕ ε .It is easy to check that f pεq P pL 2 XH s,p qpZ λ q and that f pεq Ñ f as ε Ñ 0 `in H s,p pZ λ q.
Step 3. Let us show that it suffices to consider the case s P p0, 1{pq (the region T 1 Y T 3 in Figure 1).Suppose we have a bounded extension P λ : H s,p pZ λ q Ñ H s,p pZ λ q for some s ě 1{p and p P p1, 8q (the region R in Figure 1) .Interpolating with L 2 pZ λ q, we obtain a bounded extension P λ : H s θ ,p θ pZ λ q Ñ H s θ ,p θ pZ λ q, where θ P p0, 1q, s θ " θs, 1 p θ " θ p `1´θ 2 .By taking θ small enough we obtain that 0 ă s θ ă 1{p θ .
Step 4. We show that, if p P p1, 8q, s P p0, 1{pq, and P λ : H s,p pZ λ q Ñ H s,p pZ λ q is bounded, then K w P H s,p pW µpλq q and K w P H ´s,p 1 pW µpλq q.
Lemma 4.3 gives bounded extensions P λ 1 : H s,p pZ λ 1 q Ñ H s,p pZ λ 1 q for all λ 1 ą λ as well as }P λ 1 } pH s,p pZ λ 1 q,H s,p pZ λ 1 qq ď N s,p pλq for all λ 1 ą λ.Fix w P W and let K w " Kp¨, wq denote the Bergman kernel of W at w.If we choose ϕ w P C 8 c supported in a ball centered at w within W, with radial symmetry and with ş ϕ w " 1, then P ϕ w " K w .Then, for all λ 1 ą λ large enough for supp ϕ w Ď Z λ , using Remark 3.5 we have that }P λ 1 ϕ w } H s,p pW µpλq q ď }P λ 1 ϕ w } H s,p pZ λ 1 q ď N s,p pλq}ϕ w } H s,p pZ λ 1 q " N s,p pλq}ϕ w } H s,p pC 2 q .
Therefore tP λ 1 ϕ w u λ 1 is a family of functions contained in the ball of radius N s,p pλq}ϕ w } H s,p pC 2 q centered at the origin in H s,p pW µpλq q.Since we are assuming 0 ă s ă 1{p, using Lemma 3.2 and the Hahn-Banach theorem we have that tP λ 1 ϕ w u λ 1 ąλ admits a subsequence weak-˚converging to a function h in H s,p pW µpλq q.Recalling that H s,p pW µpλq q is the dual of H ´s,p 1 pW µpλq q with respect to the L 2 pW µpλq q inner product, this implies that for all g P C 8 c pW µpλq q we have ż as n Ñ 8.

Arguing as in
Step 1, we have that (up to refinements) χ λ 1 n P λ 1 n ϕ w converges to P ϕ w " K w in the weak-˚topology of L 2 pW µppλq q.Thus ż for all g P C 8 c pW µpλq q.This implies that h " K w on W µpλq , whence K w P H s,p pW µpλq q.In order to prove that K w P H ´s,p 1 pW µpλq q, we use Lemma 3.2.For all λ 1 ą λ we have c pW µpλq q, }ψ} H s,p pW µpλq q ď 1 + ď }ϕ w } H ´s,p 1 pC 2 q }P λ 1 ψ} H s,p pZ λ 1 q ď N s,p pλq}ϕ w } H ´s,p 1 pC 2 q .
We now argue as before and conclude that K w P H ´s,p 1 pW µpλq q.We split the remaining part of the argument into three steps: one concerning the region T 1 , one concerning the region T 3 and one concerning the line segment separating them, see Figure 1.
Step 5. We assume p P p1, 8q and s ą maxp 2 p ´1, 0q and P λ : H s,p pZ λ q Ñ H s,p pZ λ q is bounded.By Step 4, we have that K w P H s,p pW µpλq q.Then, Proposition 4.1 immediately gives a contradiction.
Step 6.We assume that p P p1, 2q, 0 ă s ă 2 p ´1 and P λ : H s,p pZ λ q Ñ H s,p pZ λ q is bounded.Notice that 2 p ´1 ă 1 p , so that, by Step 4, we obtain that K w P H ´s,p 1 pW µpλq q, where ´s ą 1 ´2 p " 2 p 1 ´1.But, again, this is false by Proposition 4.1 (i) and we have reached a contradiction.Hence, the projector P λ does not extend to a bounded operator P λ : H s,p pZ λ q Ñ H s,p pZ λ q.
Step 7. Finally, let p P p1, 2q and s " 2 p ´1 and suppose that P λ : H s,p pZ λ q Ñ H s,p pZ λ q is bounded.Again, Lemma 4.3 gives that P λ 1 : H s,p pZ λ 1 q Ñ H s,p pZ λ 1 q is bounded and }P λ 1 } pH s,p pZ λ 1 q,H s,p pZ λ 1 qq ď N s,p pλq for all λ 1 ą λ.Take any µ ą π and let λ 1 sufficiently large so that W µ Ď Z λ 1 .Let ϕ w P C 8 c pZ λ 1 q be as in Step 4. We have shown that there exists a sequence tP λn ϕ w u such that P λn ϕ w Ñ K w weak-˚in H s,p pZ λ 1 q as λ n Ñ 8, so that }K w } H s,p pWµq ď }K w } H s,p pZ λ 1 q ď lim nÑ8 }P λn } pH s,p pZ λn q,H s,p pZ λn qq }ϕ w } H s,p pC 2 q ď CN s,p pλq independent of µ.This contradicts Proposition 4.1 (ii) and the proof is complete.

Final remarks and open questions
We wish to conclude by indicating a number of open problems.First of all, we recall that the exact range of regularity on the Lebesgue Sobolev spaces H s,p of the Bergman projection on the smoothly bounded domain W µ is not known.Clearly, in order to prove a positive result, one needs to have precise information on the Bergman kernel itself.In fact, also the precise behaviour of the kernel near the critical annulus A " tp0, z 2 q : e ´µ{2 ă |z 2 | ă e µ{2 u on the boundary of W µ remains to be understood.
The equivalence of the regularity of the Bergman projections on p0, qq-forms and the Neumann operator N , proved in [BS90], was later exploited by M. Christ [Chr96] to show that P µ does not preseve C 8 pW µ q.These results heavily relied on the boundedness of the domain W µ .We believe that the Neumann operator N on Z λ is as irregular as the Bergman projection P λ , but this problem has not been addressed and (to the best of our knowledge) is open.
Finally, we mention the boundary analogue of this problem, namely the study of the behaviour of the Szegő projection on Z λ .Given a smooth domain Ω " tz : ρpzq ă 0u Ď C n , the Hardy space H 2 pΩ, dσq is defined as H 2 pΩ, dσq " f P HolpΩq : sup where Ω ε " tz : ρpzq ă ´εu and dσ ε is the induced surface measure on BΩ ε .Then H 2 pΩ, dσq can be identified with a closed subspace of L 2 pBΩ, dσq, that we denote by H 2 pBΩ, dσq, where σ is the induced surface measure on BΩ.The Szegő projection is the orthogonal projection S Ω : L 2 pBΩ, dσq Ñ H 2 pBΩ, dσq ; see [Ste70] for the case of bounded domains.The regularity of S Ω when Ω is a (model) worm domain was studied in a series of papers [Mon16a, Mon16c, Mon16b, MP17a, MP17b, LS21].In particular, in [LS21] it was announced that S Wµ does not preserve L p pBW µ q when ˇˇ1 2 ´1 p ˇˇě π µ , in analogy to the case of the Bergman projection.L. Lanzani and E. Stein also studied the L p -regularity of the Szegő and other projections on the boundary on bounded domains under minimal smoothness conditions [LS14,LS17], whereas a definition of Hardy spaces and associated Szegő projection for singular domains was studied, for instance, in [Mon21,GGLV21].It is certaintly of interest to consider the case of the Szegő projection also in the case of the domains Z λ .
Declarations.Data sharing not applicable to this article as no datasets were generated or analysed during the current study.The authors have no relevant financial or non-financial interests to disclose.