An Example on s-H-Convexity in $$\pmb {\mathbb {C}^2}$$C2

<jats:p>We construct a bounded domain <jats:inline-formula><jats:alternatives><jats:tex-math>$$\Omega $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>Ω</mml:mi>
                </mml:math></jats:alternatives></jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {C}^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mrow>
                      <mml:mi>C</mml:mi>
                    </mml:mrow>
                    <mml:mn>2</mml:mn>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula> with boundary of class <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {C}^{1,1}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mrow>
                      <mml:mi>C</mml:mi>
                    </mml:mrow>
                    <mml:mrow>
                      <mml:mn>1</mml:mn>
                      <mml:mo>,</mml:mo>
                      <mml:mn>1</mml:mn>
                    </mml:mrow>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\overline{\Omega }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mover>
                    <mml:mi>Ω</mml:mi>
                    <mml:mo>¯</mml:mo>
                  </mml:mover>
                </mml:math></jats:alternatives></jats:inline-formula> has a Stein neighborhood basis, but is <jats:italic>not</jats:italic><jats:italic>s</jats:italic>-H-convex for any real number <jats:inline-formula><jats:alternatives><jats:tex-math>$$s\ge {1}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>s</mml:mi>
                    <mml:mo>≥</mml:mo>
                    <mml:mn>1</mml:mn>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>


Introduction
The notion of s-H-convexity was introduced by Chaumat and Chollet in [1] and goes back to work by Dufresnoy [6]. Given a real number s ≥ 1, a compact set ∅ = K ⊆ C n is called s-H-convex, if there exists a C > 0 with C ≤ 1 such that for all , 0 < ≤ 1, there exists an open pseudoconvex subset of C n satisfying Furthermore, the notion of s-H-convexity is related to the Mergelyan property. Specifically, there exists a k 0 (s, n) > 0 such that O( ) is dense in C k ( ) ∩ O( ) whenever k is an integer ≥ k 0 (s, n) and ⊆ C n is a bounded pseudoconvex domain, satisfying suitable assumptions, whose closure is s-H-convex.
Given these ∂-results and the connection to the Mergelyan property, it becomes desirable to identify sets which are s-H-convex for some s ≥ 1. Specifically, given a bounded (pseudoconvex) domain in C n whose closure admits a Stein neighborhood basis, one can ask under which additional assumptions said closure is necessarily s-H-convex for some s ≥ 1.
To our knowledge, it is unknown whether there exists a bounded (pseudoconvex) domain in C 2 with boundary of class C 2 (or C ∞ ) such that has a Stein neighborhood basis, but is not 1-H-convex. In this paper we show that, if the smoothness assumption on the boundary is relaxed appropriately, there exists a bounded domain whose closure admits a Stein neighborhood basis, but is not s-H-convex for any s ≥ 1. This is achieved by modifying the construction of the classical Diederich-Fornaess worm domain [5]. A precise statement of the main result of this paper goes as follows: Theorem 1.1 There exists a bounded (pseudoconvex) domain = ∅ in C 2 with boundary of class C 1,1 , such that: • has a Stein neighborhood basis, • is not s-H-convex for any real number s ≥ 1.
This paper is organized as follows: in Sect. 2 we introduce some notation, define the domain from Theorem 1.1 and give an informal description of our constructions. In Sect. 3 we show that is not s-H-convex for any s ≥ 1 and in Sect. 4 we construct a Stein neighborhood basis for . Finally, in Sect. 5, we prove the remaining lemmas from Sect. 4.
The we just defined is the set appearing in Theorem 1.1, so we have to show that has the desired properties. We start by collecting some basic properties of in a lemma, whose elementary proof will be omitted: The set is a bounded, connected open subset of C 2 with boundary of class C 1,1 . Furthermore, the boundary of (as a subset of C 2 ) is precisely the set of all points (z, w) ∈ C 2 satisfying z = 0 and ρ(z, w) = 0.
Remark In this paper, we work with the following notion of We end this section with an informal explanation of the intuition behind our constructions A classical worm domain admits a Stein neighborhood basis if the duration of the rotation at maximal radius is less than π . If the duration is exactly π, this fails to be true, as can be seen by refining the classical argument by Diederich and Fornaess [5]. In the case of the domain defined above, we prevent this argument from working by drastically increasing the speed of the round-off, which leads to the boundary regularity dropping to C 1,1 . Using the fact that the function g vanishes to infinite order in 0 ∈ R, one can apply the Kontinuitätssatz for annuli to open pseudoconvex neighborhoods of the closure of to show that is not s-H-convex for any s ≥ 1. The details will be given in Sect. 3.
It is easy to construct a neighborhood basis for (not a Stein one) by taking appropriate worm domains and increasing the radii of the rotating discs without changing the centers. This increase of the radii of course destroys pseudoconvexity. We counteract this by "chopping off" the "bad part," which is done by intersecting with a domain of half planes rotating around 0 in the w-plane. This, however, leads to these sets not being neighborhoods anymore, as can be seen by considering 0 in the w-plane. We finally resolve this issue by moving the center of the rotation from 0 slightly in the direction of −i and slightly slowing down the rotation (symmetrically around the angle π/2), which intuitively speaking amounts to introducing a small tilt. In the wplane, −i represents the "out direction" of , which exists because the duration of the rotation at maximal radius does not exceed π . Since g is positive on R >0 , one actually leaves the closure of , when going from 0 slightly in the direction of −i in the w-plane, which is of course crucial for our construction to work. Since the purpose of the domain of rotating half planes is to help with the pseudoconvexity of the neighborhoods we are constructing, we have to apply these changes to both of the domains we are intersecting. The details will be given in Sect. 4.
By choice of 0 , we can apply the intermediate value theorem to find a zero x ∈ (π, π + α) of H for every ∈ (0, 0 ), which is uniquely determined, since H is strictly decreasing. By direct computation we get Roughly speaking, given some 0 < < 0 and an open pseudoconvex set containing ( ), we need to identify a point contained in said pseudoconvex set that is "far away" from relative to . By inspecting the explicit expression for x , one sees that x − π is much larger than for small enough 0 < 0 . With this in mind, we will identify a point contained in any open pseudoconvex set containing ( ), whose distance to is comparable to x − π . We accomplish this by applying the Kontinuitätssatz for annuli.
The following lemma is the first step of the announced Kontinuitätssatz argument. It deals with the boundaries of the annuli and the "bottom annulus": Lemma 3.1 Given ∈ (0, 0 ), we have: (1) For all φ, π ≤ φ ≤ x , the following set is contained in ( ): (2) The following set is contained in the boundary of and hence in ( ): Proof Property 2 is clear, so we only need to prove Property 1. Let ∈ (0, 0 ), let π ≤ φ ≤ x and consider a point (z, w) = (z, i · sin (φ)) contained in the set from the statement of Property 1. We restrict ourselves to the case |z| 2 = exp (φ), since the other case can be handled analogously.
But then, owing to the choices we made, Making use of the choices made above (in particular that π<x < π + α< π + β< π + π/2 and H ≥ 0 on [π, x ]), we compute Armed with Lemma 3.1, we now finish the Kontinuitätssatz argument: Then, for every φ, π ≤ φ ≤ x , the following set is contained in D: Proof This follows from Lemma 3.1 via the Kontinuitätssatz for annuli.
In view of Lemma 3.2, we need to identify a point contained in F x that is "far away" from . The obvious choice is the following: For all ∈ (0, 0 ) we define The following lemma shows that p is indeed "far away" from Proof Owing to Lemma 2.1, we find a δ > 0 such that (δ), the closure of (δ) in C 2 , is a compact subset of (C \ {0}) × C. So, since ρ is of class C 1 on (C \ {0}) × C (see Sect. 2), there exists an L > 0 such that ρ is Lipschitz continuous with Lipschitz constant L on (δ). This immediately gives the estimate Hence, given ∈ (0, 0 ), we only need to show that ρ( p ) ≥ x − π . Using that x ∈ (π, π + α) and using the defining properties of α, we compute as desired.
We now combine all the previously developed ingredients to achieve the goal of this section.

Proposition 3.4
is not s-H-convex for any real number s ≥ 1.
Proof First note that is indeed compact. Assume for the sake of a contradiction that is s-H-convex for some s ≥ 1. So there exist a constant 0 < C ≤ 1 and a family (D ) 0< ≤1 of open pseudoconvex subsets of C 2 such that i.e., we have For all 0 < < min{ 0 , C} we then get from Lemma 3.2 that which, using Lemma 3.3, directly implies the estimate So, since δ, L, and C are positive constants, we find a constant K > 0 and an 0 < min{ 0 , C} such that (x − π) s < K for all 0 < < .
Using that for all 0 < < 0 , we get that and we arrive at the desired contradiction.

Existence of a Stein Neighborhood Basis
In this section, we construct a Stein neighborhood basis for . We fix an > 0 for the remainder of this section. It suffices to find an open pseudoconvex subset D of C 2 satisfying ⊆ D ⊆ ( ).
We start by defining the domains of "half planes rotating in the w-plane" announced in Sect. 2.

Definition 4.1
For every δ ∈ (0, 1) and every t ∈ [0, 1) we define H (δ) t to be the subset of C 2 consisting of all points (z, w) satisfying z = 0 and Furthermore we will denote the set H (δ) 0 simply as H (δ) .
The expression sin (δπ /(2(1 − δ))) measures by how much the center of the rotation is moved in the direction of −i. In the exponential-term, δ measures how much the rotation is slowed down symmetrically around the angle π/2. The expression sin (δπ / (2(1 − δ))) was chosen specifically to ensure that an appropriate version of Lemma 4.5 (see below) holds true.
We now define the domains of "discs rotating in the w-plane," also announced in Sect. 2. It is important to note that these domains are not pseudoconvex. Definition 4.3 Adopt the notation from 4.2. Then for all δ ∈ (0, 1) and for all η ∈ (0, η 0 ), we define a map ρ δ,η : and we define D (δ,η) to be the subset of C 2 consisting of all points (z, w) satisfying z = 0 and ρ δ,η (z, w) < 0.
It should be noted that D (δ,η) is essentially defined the same way as (resp. a classical worm domain), apart from the fact that S is replaced by S η and that the position of the center and the speed of the rotation have been adjusted slightly (in the same way as above).
We now show that ⊆ D (δ,η) ⊆ ( ) for suitable choices of δ and η. Since D (δ,η) is not pseudoconvex, however, some additional considerations are needed in order to achieve the goal stated in the beginning of this section.
As explained in Sect. 2, we want to intersect the domains of "discs rotating in the w-plane" with suitable domains of "half planes rotating in the w-plane," with the aim of obtaining a pseudoconvex neighborhood of . So we of course need the domains of "half planes rotating in the w-plane" to contain the closure of .
In order to establish this, we need the crucial estimate provided by Lemma 4.5 below. If the function g was replaced by the 0-function in a small neighborhood of 0 ∈ R, then could not possibly have a Stein neighborhood basis as the Kontinuitätssatz for annuli shows. Hence our construction has to make use of the fact that g > 0 on an interval of the form (0, μ) for some small 0 < μ 1. We make use of that fact only once in the entire construction of the Stein neighborhood basis for , namely in the proof of Lemma 4.5, the discovery of which was one of the main obstacles in our construction. In fact, the seemingly arbitrary expression sin (δπ / (2(1 − δ))) featuring in Definition 4.1 was chosen specifically with this lemma in mind.

Lemma 4.5
There exists a 0 < d 1 < 1 such that we have the following estimate for all δ, ψ ∈ R with 0 < δ < d 1 and −β ≤ ψ ≤ π + β: The proof of Lemma 4.5 can be found in Sect. 5. Using this lemma, we can now show that the domains of "half planes rotating in the w-plane" contain the closure of .  ∈ (0, d 1 ), there exists a t δ ∈ (0, 1) such that Proof Let δ ∈ (0, d 1 ). Owing to the compactness of , it suffices to show that ⊆ H (δ) 0 . To this end let (z, w) ∈ . Lemma 2.1 shows that z = 0 and ρ(z, w) ≤ 0. In particular, this implies that ψ:= ln(|z| 2 ) ∈ [−β, π + β] and |w − exp (iψ)| ≤ √ S(ψ). Hence, using that Re(τ ) ≥ −|τ | for all τ ∈ C and writing w = exp (iψ) + (w − exp (iψ)), we get which is > 0 by Lemma 4.5. This shows that (z, w) ∈ H (δ) 0 , as desired. We are now ready to define the Stein neighborhood announced in the beginning of this section. Adopting the notation from Lemmas 4.4 and 4.6, we fix an η ∈ (0, η 1 ( )), a δ> 0 with δ< min{d 1 , d 2 ( , η)} and a t ∈ (0, t δ ) for the remainder of this section. With these fixed choices we now define It is obvious that D is an open subset of C 2 . Furthermore, we have by Lemmas 4.4 and 4.6. Hence, we only have to show that D is pseudoconvex. Pseudoconvexity is a local property of the boundary and we have So, since the boundary bD of D is contained in H   Lemma 4.7 deals with the pseudoconvexity of our chosen domain of "half planes rotating in the w-plane" at certain boundary points. Lemma 4.8 says, roughly speaking, that our chosen domain of "discs rotating in the w-plane" is pseudoconvex at the "good" boundary points, which are precisely those contained in H (δ) 0 . As mentioned previously, pseudoconvexity of D follows from Lemmas 4.7 and 4.8, the proofs of which can be found in Sect. 5; so we have shown that D is pseudoconvex. Hence has a Stein neighborhood basis. Together with Proposition 3.4, this provides a proof for Theorem 1.1.

Remaining Proofs
In this section, we provide the proofs which remain from Sect. 4. We start by proving the crucial estimate, Lemma 4.5.
The first inequality is trivial and the second inequality is obvious from the fact that sin (0) = 0 and sin is concave on [0, π/2].