February 13, 2026
Let \(\overline{\Omega}\) be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let \(h:Z\to \overline{\Omega}\) be a fibre bundle of Hölder–Zygmund class \(\Lambda^r\), \(r>0\), which is holomorphic over \(\Omega\). Assuming that the fibre is an Oka manifold, we prove that every continuous section \(f_0:\overline{\Omega}\to Z\) is homotopic to a section \(f_1:\overline{\Omega}\to Z\) of class \(\Lambda^r(\overline{\Omega})\) which is holomorphic on \(\Omega\). We also establish the parametric \(h\)-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder–Zygmund classes on such domains.
In Memory of the 100th Birthday of Professor Lu Qikeng
A complex manifold \(Y\) is said to be an Oka manifold (see [1] and [2]) if maps \(X\to Y\) from any Stein manifold \(X\) satisfy the h-principle, also called the Oka principle. This means in particular that any continuous map \(f_0:X\to Y\) is homotopic to a holomorphic map \(f_1:X\to Y\); if \(f_0\) is holomorphic on a neighbourhood of a compact holomorphically convex subset \(K\) of \(X\) and on a closed complex subvariety \(X'\) of \(X\), then a homotopy \(f_t:X\to Y\) \((t\in [0,1])\) from \(f_0\) to a holomorphic map \(f_1\) can be chosen to be fixed on \(X'\) and to consist of maps which are holomorphic on a neighbourhood of \(K\) and uniformly close to \(f_0\) on \(K\). The analogous results hold for sections of holomorphic fibre bundles with Oka fibres, and for sections of elliptic holomorphic submersions \(Z\to X\) onto a Stein space. For the theory of Oka manifolds and Oka maps, see [2]–[5]. Classical examples of Oka manifolds include complex homogeneous manifolds (see Grauert [6], [7] and [2]) and, more generally, Gromov elliptic manifolds (see [8] and [2]). Oka manifolds \(Y\) are characterised by the approximation property for holomorphic maps \(K\to Y\) from (neighbourhoods of) compact convex sets \(K\subset\mathbb{C}^n\) by entire maps \(\mathbb{C}^n\to Y\) (the convex approximation property, CAP); see [1] and [2]), and by convex relative ellipticity, CRE (see Kusakabe [9] and [4]). It has recently been shown that every projective Oka manifold is elliptic [10], but there exist noncompact Oka manifolds which fail to be elliptic [3], [11]. Modern Oka theory has diverse applications.
In the present paper, we establish the following 1-parametric Oka principle for sections of fibre bundles of Hölder–Zygmund classes \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) (\(r>0\) a real number) with Oka fibres on compact strongly pseudoconvex domains \(\overline{\Omega}\) with Stein interior. The definition of the Banach space \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) is given in Subsect.2.1, and fibre bundles of this class are introduced in Subsect.2.2.
Theorem 1. Assume that \(\Omega\) is a relatively compact, strongly pseudoconvex domain with smooth boundary \(b\Omega\) in a Stein manifold \(X\), \(r>0\), and \(h:Z\to \overline{\Omega}\) is a fibre bundle of Hölder–Zygmund class \(\Lambda^r(\overline{\Omega})\) which is holomorphic on \(\Omega\). Assuming that the fibre of \(h\) is an Oka manifold, every continuous section \(f_0\in \Gamma(\overline{\Omega},Z)\) of \(h\) is homotopic to a section \(f\in \Gamma^r_\mathscr{O}(\overline{\Omega},Z)\) of class \(\Lambda^r(\overline{\Omega})\) which is holomorphic on \(\Omega\). Furthermore, every homotopy of continuous sections \(\{f_t\}_{t\in [0,1]}\in \Gamma(\overline{\Omega},Z)\) with \(f_0, f_1 \in \Gamma^r_\mathscr{O}(\overline{\Omega},Z)\) can be deformed with fixed ends to a homotopy in \(\Gamma^r_\mathscr{O}(\overline{\Omega},Z)\).
Theorem 1 is proved in Sect.6, where we provide a fully parametric version, Theorem 17. The proof also gives an approximation result common in Oka theory: if the given section \(f_0\) in the theorem is holomorphic on a neighbourhood of a compact holomorphically convex subset \(K\subset \Omega\), then the homotopy from \(f_0\) to a holomorphic section \(f_1\) can be chosen to consist of section that are holomorphic on a neighbourhood of \(K\) and approximate \(f_0\) uniformly on \(K\). A stronger approximation theorem, with \(K\subset\overline{\Omega}\) holomorphically convex in \(\overline{\Omega}\), is also possible; see [12] where the analogue of this result is established for fibre bundles \(Z\to \overline{\Omega}\) of class \(\mathscr{A}^r(\overline{\Omega}) =\{f\in \mathscr{C}^r(\overline{\Omega}):f|_\Omega \in\mathscr{O}(\Omega)\}\) with \(r\in \mathbb{Z}_+=\{0,1,2,\ldots\}\). Here, \(\mathscr{O}(\Omega)\) denotes the space of holomorphic functions on \(\Omega\). A motivation for the present generalisation of the mentioned results from [12] is that the spaces \(\Lambda^r\) behave better than the \(\mathscr{C}^k\) or Lipschitz spaces for many operators considered in analysis, as was already noticed by Zygmund in 1945. More explicitly, I was asked by Andrei Teleman in a private communication (January 2026) whether Theorem 1 holds. He intends to apply this result in a project of his.
Although Theorem 1 and the other results of the paper are stated for domains in Stein manifolds, they hold for any compact complex manifold \(\overline{\Omega}\) with Stein interior and smooth strongly pseudoconvex boundary. Indeed, by Ohsawa [13], Heunemann [14], and Catlin [15], such a manifold holomorphically embeds as a smoothly bounded strongly pseudoconvex domain in a Stein manifold.
We refer to Subsect.2.1 for the definition and basic properties of Hölder–Zygmund spaces \(\Lambda^r(\overline{\Omega})\) on a bounded Lipschitz domain \(\Omega\) in a smooth Riemannian manifold. For \(r=k+\alpha\) with \(k\in\mathbb{Z}_+\) and \(0<\alpha<1\), \(\Lambda^r(\overline{\Omega})\) coincides with the Hölder space \(\mathscr{C}^{k,\alpha}(\overline{\Omega})\). For integer values \(r=k+1\in \{1,2,\ldots\}\), the Zygmund space \(\Lambda^{k+1}(\overline{\Omega})\) properly contains the Lipschitz space \(\mathscr{C}^{k,1}(\overline{\Omega})\) of functions whose derivatives up to order \(k\) are Lipschitz continuous on \(\overline{\Omega}\). The main difference is that, in the definition of the \(\Lambda^1\) norm of a function \(f\), one replaces the first difference \(\Delta_h f(x)\) (see 1 ) by the second difference \(\Delta^2_h f(x)\) (see 2 ). The Zygmund class \(\Lambda^{1}\) is a natural substitute for the Lipschitz class \(\mathscr{C}^{0,1}=\mathrm{Lip}^1\) in many contexts. There are continuous strict embeddings among the classical and the Hölder–Zygmund scales on any Lipschitz domain; see 10 . An important fact used in the proof of Theorem 1 is that the canonical (Kohn) solution operator for the \(\overline{\partial}\)-equation is bounded on \(\Lambda^r(\overline{\Omega})\) when \(\Omega\) is a smoothly bounded strongly pseudoconvex domain; see Beals et al.[16]. The precise result that we shall use is stated as Theorem 7.
On the way to Theorem 1, we obtain several other results of independent interest. Given a Lipschitz domain \(\Omega\Subset X\) in a complex manifold \(X\) and a real number \(r>0\), let \[\Lambda^r_\mathscr{O}(\overline{\Omega}) = \{f\in \Lambda^r(\overline{\Omega}): f|_\Omega\in \mathscr{O}(\Omega)\}.\] Similarly we define the mapping spaces \(\mathscr{O}(\Omega,Y)\) and \(\Lambda^r_\mathscr{O}(\overline{\Omega},Y) \subset \Lambda^r(\overline{\Omega},Y)\) for any complex manifold \(Y\). Given a compact set \(K\subset X\), we denote by \(\mathscr{O}(K)\) the space of restrictions to \(K\) of holomorphic functions in open neighbourhoods of \(K\), endowed with the inverse limit topology. The notation \(\mathscr{O}(K,Y)\) is used for the space of maps of this kind to a complex manifold \(Y\).
We have the following approximation theorem.
Theorem 2. Let \(\Omega\) be a relatively compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold \(X\), and let \(Y\) be a complex manifold. Every map in \(\Lambda^r_\mathscr{O}(\overline{\Omega},Y)\), \(r>0\), can be approximated in the \(\Lambda^r(\overline{\Omega},Y)\) topology by maps in \(\mathscr{O}(\overline{\Omega},Y)\).
Theorem 2 is proved in Sect.3; see also the parametric version in Theorem 8. The analogue of Theorem 2 is known for spaces \(\mathscr{A}^r(\overline{\Omega})\) with \(r\in\mathbb{Z}_+\); see [17] and [18]. For approximation of manifold-valued maps in \(\mathscr{A}^r(\overline{\Omega},Y)\), where \(Y\) is a complex manifold and \(r\in\mathbb{Z}_+\), by holomorphic maps from open neighbourhoods of \(\overline{\Omega}\) in \(X\), see [12], [2], and [18]. The last mentioned result in [18] is a more general Mergelyan-type approximation theorem on strongly admissible sets in Stein manifolds.
We also have the following result generalising [19]. The proof in [19] also applies to spaces \(\Lambda^r_\mathscr{O}(\overline{\Omega},Y)\).
Theorem 3. Assume that \(\Omega\) is a relatively compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold and \(Y\) is a complex manifold. For every \(r>0\) the space \(\Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) is a complex Banach manifold. The tangent space \(T_f \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) at a point \(f\in \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) is the space of sections of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) of the complex vector bundle \(f^*TY\to \overline{\Omega}\).
In the proof of Theorem 1, we shall need several results concerning vector bundles \(E\to\overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) on smoothly bounded strongly pseudoconvex Stein domains \(\Omega\); see Sect.4. In particular, we obtain Theorem A for such bundles; see Theorems 9 and 14. The proof is based on the Cartan splitting lemma for maps of Hölder–Zygmund classes to a complex Lie group; see Lemmas 1, 3 and Remark 11. The main technical results used in the proof of Theorem 1 are a splitting lemma (see Lemma 5) and a gluing lemma (see Lemma 6) for sprays of sections of class \(\Lambda^r_\mathscr{O}\).
An application of Theorem 1 is the following Oka principle for vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), proved in Sect.7. The analogous result for principal fibre bundles is Theorem 18.
Theorem 4. Let \(\Omega\) be a relatively compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold. The following hold for every real number \(r>0\).
Every topological complex vector bundle on \(\overline{\Omega}\) is isomorphic to a vector bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\).
If a pair of vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) are isomorphic as topological complex vector bundles, then they are also isomorphic as \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) vector bundles.
This result is classical for vector bundles on Stein spaces; see Grauert [20], with the special case of line bundles due to Oka [21]. (See also Leiterer [22] and [2].) For vector bundles of class \(\mathscr{A}^k(\overline{\Omega})\), \(k\in \mathbb{Z}_+\), see Leiterer [23], [24] and Heunemann [25].
Remark 5. The results of this paper apply to a wider class of mapping spaces. Assume that \(\mathcal{F}\) is a contravariant functor from the category of compact smooth manifolds \(M\) with boundary to the category of Banach algebras of \(\mathbb{C}\)-valued functions on them, satisfying the following conditions:
\(\mathscr{C}^\infty(M) \subset \mathcal{F}(M)\subset \mathscr{C}(M)\) and both inclusions are continuous.
A smooth map \(\Phi:M\to M'\) induces a homomorphism \(\Phi^*:\mathcal{F}(M') \to \mathcal{F}(M)\) of Banach algebras by \(f\mapsto f\circ \Phi\) for \(f\in \mathcal{F}(M')\).
Postcomposition by a smooth function \(\mathbb{C}\to\mathbb{C}\) induces a continuous selfmap of \(\mathcal{F}(M)\).
These properties imply that the topology on \(\mathcal{F}(M)\) can be defined via local charts, and hence the definition of these classes extends to differential forms and other tensor fields on \(M\). Furthermore, we can introduce vector bundles and more general fibre bundles of class \(\mathcal{F}\); see Palais [26], [27] and the references in [19]. When \(M=\overline{\Omega}\) is a compact complex manifold with smoothly boundary, we define \[\mathcal{F}_\mathscr{O}(\overline{\Omega})= \{f\in \mathcal{F}(\overline{\Omega}): f|_\Omega\in\mathscr{O}(\Omega)\}.\] The analogous definition yields the mapping space \(\mathcal{F}_\mathscr{O}(\overline{\Omega},Y)\) for any complex manifold \(Y\). Conditions (a)–(c) on the functor \(\mathcal{F}\) clearly imply the following properties:
A smooth map \(\Phi:\overline{\Omega} \to \overline{\Omega}'\) which is holomorphic on \(\Omega\) induces a homomorphism \(\Phi^*:\mathcal{F}_\mathscr{O}(\overline{\Omega}') \to \mathcal{F}(\overline{\Omega})\) by \(f\mapsto f\circ \Phi\), \(f\in \mathcal{F}_\mathscr{O}(\overline{\Omega}')\).
Postcomposition by an entire function \(g:\mathbb{C}\to\mathbb{C}\) induces a continuous selfmap of \(\mathcal{F}_\mathscr{O}(\overline{\Omega})\).
To conditions (a)–(c) we add the following condition:
In the proofs of our results, the operator \(\overline{\partial}\) is only applied to functions of the form \(\chi f\) with \(\chi\in \mathscr{C}^\infty(\overline{\Omega})\) and \(f\in \mathcal{F}_\mathscr{O}(\overline{\Omega})\). In this case, the derivative \(\overline{\partial}(\chi f)=f\overline{\partial}\chi \in \mathcal{F}_{0,1}(\overline{\Omega})\) is of the same class as \(f\).
Inspections of proofs of our results show that they hold on Banach algebras \(\mathcal{F}_\mathscr{O}(\overline{\Omega})\), and for vector and fibre bundles of this class, when \(\mathcal{F}\) satisfies condition (a)–(d). Examples include Hölder spaces \(\mathscr{C}^{k,\alpha}(\overline{\Omega})\) \((k\in\mathbb{Z}_+,\;0<\alpha<1)\), Hölder–Zygmund spaces \(\Lambda^r(\overline{\Omega})\) \((r>0)\), and Sobolev spaces \(W^{k,p}(\overline{\Omega})\) \((k\in \mathbb{N},\;1\le p<\infty,\;kp>\dim_\mathbb{R}\Omega)\), among others.
In the first subsection, we recall the definition and basic properties of Hölder spaces \(\mathscr{C}^{k,\alpha}\) \((k\in\mathbb{Z}_+,\;0<\alpha\le 1)\) and Hölder–Zygmund spaces \(\Lambda^r\) for any real \(r>0\). We refer to the papers by Gong [28], Wallin [29], and the monographs by Gilbarg and Trudinger [30] and Stein [31] for more information. In the second subsection, we introduce vector bundles and more general fibre bundles of these classes. In the third subsection, we recall the results on the canonical solution to the \(\overline{\partial}\)-equation in the spaces \(\Lambda^r\), which will be used in the paper.
Let \(\Omega\) be a domain in \(\mathbb{R}^n\). Given a function \(f:\Omega\to\mathbb{C}\) and \(h\in \mathbb{R}^n\setminus\{0\}\), the first and the second difference of \(f\) at \(x\in \Omega\) with step \(h\) are defined by \[\begin{align} \tag{1} \Delta_h f(x) &=& f(x+h)-f(x), \quad \;\;\quad\qquad\qquad x+h\in \Omega; \\ \tag{2} \Delta^2_h f(x) &=& f(x+h)+f(x-h)-2f(x),\quad x+h,\;x+2h\in \Omega. \end{align}\] Note that \(\Delta^2_h f(x) =\Delta_h f(x) - \Delta_h f(x-h)\). (Some source use instead the definition \(\Delta^2_h f(x)=\Delta_h \circ\Delta_h f(x) = f(x+2h)+f(x)-2f(x+h)\); this difference is inessential and leads to the same function spaces.) The function \(f\) belongs to \(\mathscr{C}^{0,\alpha}(\Omega,x)\) \((x\in\Omega,\;0<\alpha\le 1)\) if \[[f]_{\alpha,x} := \sup_{h\ne 0,\;x+h\in \Omega} |h|^{-\alpha}|\Delta_h f(x)| = \sup_{y\in\Omega\setminus\{x\}} \frac{|f(y)-f(x)|}{|y-x|^\alpha} < \infty.\] Such \(f\) is said to be Hölder class \(\alpha\) at \(x\). For \(\alpha=1\), \(\mathscr{C}^{0,\alpha}(\Omega,x)=\mathrm{Lip}^1(\Omega,x)\) is the Lipschitz class. The Hölder-\(\alpha\) space on \(\Omega\) is \[\label{eq:Calpha} \mathscr{C}^{0,\alpha}(\Omega) = \bigl\{f:\Omega\to \mathbb{C}: \|f\|_{\mathscr{C}^{(0,\alpha)}(\Omega)} = \sup_{x\in \Omega} |f(x)| + \sup_{x\in \Omega}\, [f]_{\alpha,x}<\infty \bigr\}.\tag{3}\] For \(\alpha=1\) we have the Lipschitz space \(\mathscr{C}^{0,1}(\Omega)=\mathrm{Lip}^1(\Omega)\).
Let \(x=(x_1,\ldots,x_n)\) be the coordinates on \(\mathbb{R}^n\). Given \(\beta=(\beta_1,\ldots,\beta_n)\in\mathbb{Z}_+^n\), set \(|\beta|=\beta_1+\cdots+\beta_n\) and \(D^\beta f=\partial^{|\beta|}f/\partial x_1^{\beta_1} \cdots \partial x_n^{\beta_n}\). For \(k\in \mathbb{Z}_+\) we denote by \(\mathscr{C}^k(\Omega)\) the space of \(k\)-times continuously differentiable functions \(f\) on \(\Omega\) with \[\|f\|_{\mathscr{C}^k(\Omega)} = \sum_{|\beta|\le k} \sup_{x\in \Omega} |D^\beta f(x)| < \infty.\] The Hölder space \(\mathscr{C}^{k,\alpha}(\Omega)\) for \(k\in \mathbb{Z}_+\) and \(0<\alpha\le 1\) is defined by \[\begin{align} \tag{4} \mathscr{C}^{k,\alpha}(\Omega) &=& \Big\{f\in \mathscr{C}^k(\Omega): D^\beta f\in \mathscr{C}^{0,\alpha}(\Omega) \;\;\text{for all \beta\in \mathbb{Z}_+^n with |\beta|\le k}, \\ && \quad \tag{5} \|f\|_{\mathscr{C}^{k,\alpha}(\Omega)} := \|f\|_{\mathscr{C}^{k}(\Omega)} + \sum_{|\beta|=k} \|D^\beta f\|_{\mathscr{C}^{0,\alpha}(\Omega)} <\infty \Big\}. \end{align}\] For \(\alpha=1\) one also writes \(\mathscr{C}^{k,1}(\Omega)= \mathrm{Lip}^{k,1}(\Omega)\).
Assume now that \(\Omega\Subset \mathbb{R}^n\) is a bounded Lipschitz domain. This means that its boundary \(b\Omega\) is locally at each point a Lipschitz graph over an affine hyperplane in \(\mathbb{R}^n\), with the domain lying on one side of the graph. A function \(f:\overline{\Omega}\to\mathbb{C}\) belongs to \(\mathscr{C}^k(\overline{\Omega})\) for some \(k\in\mathbb{Z}_+\) if it is the restriction to \(\overline{\Omega}\) of a function \(\tilde{f}\in \mathscr{C}^k(\mathbb{R}^n)\). (For \(k=0\), this coincides with the usual definition of continuous functions by Tietze’s extension theorem.) Given \(\beta\in \mathbb{Z}_+^n\) with \(|\beta|\le k\), we denote by \(D^\beta f\) the restriction of \(D^\beta \tilde{f}\) to \(\overline{\Omega}\); note that \(D^\beta f\) is independent of the choice of the extension \(\tilde{f}\) if \(\Omega\) is a Lipschitz domain. Given \(0<\alpha\le 1\), the Hölder space \(\mathscr{C}^{k,\alpha}(\overline{\Omega})\) on the closed domain \(\overline{\Omega}\) is defined by \[\mathscr{C}^{k,\alpha}(\overline{\Omega})= \big\{f\in \mathscr{C}^k(\overline{\Omega}): D^\beta f \in \mathscr{C}^{0,\alpha}(\overline{\Omega})\;\; \text{for all \beta\in \mathbb{Z}_+^n with |\beta|\le k} \big\},\] endowed with the norm \(\|f\|_{\mathscr{C}^{k,\alpha}(\Omega)}\) 5 . It follows from definitions that we get the same norm by taking the suprema over \(x\in \overline{\Omega}\), so \(\|f\|_{\mathscr{C}^{k,\alpha}(\Omega)}=\|f\|_{\mathscr{C}^{k,\alpha}(\overline{\Omega})}\).
Proposition 6. If \(\Omega\) is a bounded Lipschitz domain in \(\mathbb{R}^n\) then every function in \(\mathscr{C}^{k,\alpha}(\Omega)\) \((k\in \mathbb{Z}_+,\;0<\alpha\le 1)\) extends to a unique function \(\tilde{f}\in \mathscr{C}^{k,\alpha}(\overline{\Omega})\).
Proof. For \(k=0\) this holds by McShane’s extension theorem [32]. If \(k\ge 1\) and \(f\in \mathscr{C}^{k,\alpha}(\Omega)\), its partial derivatives \(D^\beta f\) of order \(|\beta|=k\) belong to \(\mathscr{C}^{0,\alpha}(\Omega)\), so they extend to functions \(g_\beta=\mathscr{C}^{0,\alpha}(\overline{\Omega})\) by McShane’s theorem. By Whitney’s theorem [33] (see also Malgrange [34]), it follows that \(f\) extends to a function \(\tilde{f}\in \mathscr{C}^k(\mathbb{R}^n)\) satisfying \(D^\beta \tilde{f}= g_\beta\) on \(\overline{\Omega}\) for all \(|\beta|=k\). Thus, \(\tilde{f}|_{\overline{\Omega}}\in \mathscr{C}^{k,\alpha}(\overline{\Omega})\). Clearly, the extension \(\tilde{f}|_{\overline{\Omega}}\) of \(f\) is unique. ◻
We now recall the definition of Hölder–Zygmund spaces \(\Lambda^r(\Omega)\) for any real number \(r>0\). Write \(r=k+\alpha\) with \(k\in\mathbb{Z}_+\) and \(0<\alpha\le 1\). If \(r\) is not an integer (that is, \(\alpha\ne 1\)), set \(\Lambda^r(\Omega)=\mathscr{C}^{k,\alpha}(\Omega)\) and \(\Lambda^r(\overline{\Omega})=\mathscr{C}^{k,\alpha}(\overline{\Omega})\); both spaces are endowed with the norm \(\|f\|_{\Lambda^{k+\alpha}(\Omega)}:=\|f\|_{\mathscr{C}^{k,\alpha}(\Omega)}\) 5 . Assume now that \(r\) is an integer. Recall that the second difference \(\Delta_h^2 f\) is given by 2 . We define \[\begin{align} \tag{6} \Lambda^1(\Omega) &=& \big\{f\in \mathscr{C}(\Omega): \|f\|_{\Lambda^1(\Omega)} = \sup_{x\in\Omega}|f(x)| + \sup_{x,x+h\in\Omega,\, h\ne 0} |h|^{-1} |\Delta_h^2 f(x)| <\infty \big\}, \\ \tag{7} \Lambda^r(\Omega) &=& \big\{f\in \mathscr{C}^{r-1}(\Omega): \|f\|_{\Lambda^r(\Omega)} = \|f\|_{\mathscr{C}^{r-1}(\Omega)} + \sum_{|\beta|=r-1} \|D^\beta f\|_{\Lambda^1(\Omega)} <\infty\big\}, \end{align}\] where \(r>1\) in 7 . (For \(\Omega=\mathbb{R}^n\), see Stein [31].) The space \(\Lambda^1(\mathbb{R}^n)\) is the classical Zygmund space on \(\mathbb{R}^n\); omitting the term \(\|f\|_\infty=\sup_{x\in\Omega}|f(x)|\) in 6 gives the homogeneous Zygmund space. Clearly, \(\Lambda^1(\mathbb{R}^n)\) contains the Lipschitz space \(\mathscr{C}^{0,1}(\mathbb{R}^n) = \mathrm{Lip}^1(\mathbb{R}^n)\) but is not equal to it as shown by [31].
For \(r\in \mathbb{N}\), the Zygmund space \(\Lambda^r(\overline{\Omega})\) on a closed domain \(\overline{\Omega}\) can not be defined in general by considering only values of functions in the points of \(\overline{\Omega}\). Indeed, if \(\Omega\) is strictly convex and \(x,x+h\in b\Omega\) with \(h\ne 0\), then \(x-h\notin \overline{\Omega}\), so the second difference \(\Delta^2_h f(x)\) is not defined. One possible definition is the following; see Wallin [29]. It can be used for any \(r>0\), not only for integers. \[\begin{align} \tag{8} \Lambda^r(\overline{\Omega}) &=& \big\{f=\tilde{f}|_{\overline{\Omega}}: \tilde{f}\in \Lambda^r(\mathbb{R}^n)\big\}, \\ \|f\|_{\Lambda^r(\overline{\Omega})} &=& \tag{9} \inf\big\{ \|\tilde{f}\|_{\Lambda^r(\mathbb{R}^n)}: \tilde{f} \in \Lambda^r(\mathbb{R}^n),\; \tilde{f}|_{\overline{\Omega}} =f\big\}. \end{align}\] One can also use extensions of \(f\) to any domain \(D\subset \mathbb{R}^n\) containing \(\overline{\Omega}\); the resulting norms are comparable. However, it is not clear whether these norms decrease to the interior norm \(\|f\|_{\Lambda^r(\Omega)}\) 7 as \(D\) shrinks to \(\overline{\Omega}\). On any Lipschitz domain \(\Omega\Subset\mathbb{R}^n\), Stein [31] constructed a linear extension operator \(E:\mathscr{C}(\overline{\Omega}) \to \mathscr{C}_0(\mathbb{R}^n)\) to the space of continuous functions with compact support such that \(E:\Lambda^r(\overline{\Omega})\to \Lambda^r(\mathbb{R}^n)\) is bounded for every \(r>0\). (Stein proved that his extension operator is bounded on Sobolev spaces; for Hölder–Zygmund spaces this was shown by Gong [35].)
It turns out that for any bounded Lipschitz domain \(\Omega\) in \(\mathbb{R}^n\), the interior and the exterior definition of the spaces \(\Lambda^r(\overline{\Omega})\) are equivalent. Indeed, Shi and Yao [36] have recently shown that \(\Lambda^r(\Omega)=\{f|_\Omega: f\in \Lambda^r(\mathbb{R}^n)\}\), and the interior norm \(\|f\|_{\Lambda^r(\Omega)}\) 7 is comparable to the exterior norm \(\|f\|_{\Lambda^r(\overline{\Omega})}\) 9 . Their proof uses Rychkov’s universal extension for Besov spaces \(B^r_{p,q}(\Omega)\); see [37] and note that \(\Lambda^r(\Omega)=B^r_{\infty,\infty}(\Omega)\). For noninteger values \(r>0\) and domains \(\Omega\) with smooth boundaries, see also [30] whose proof is based on Seeley’s extension theorem [38].
If \(X\) is a smooth manifold and \(\Omega\) is a Lipschitz domain in \(X\) with compact closure, one defines the spaces \(\Lambda^{r}(\Omega)\) and \(\Lambda^{r}(\overline{\Omega})\) by using a finite system of smooth coordinate charts on \(X\) covering \(\overline{\Omega}\). The norms obtained in this way are comparable to one another. For the details, see Palais [26], [27] and the discussion and references in [39], [19] and [40].
The space \(\Lambda^r(\overline{\Omega})\), \(r>0\), is a commutative unital Banach algebra under pointwise multiplication, with Moser-type estimates for products, compositions, and inverses; see [35], [28], and [39]. In particular, precompositions and postcompositions by smooth maps preserve the Hölder–Zygmund classes. There are continuous strict embeddings among the classical and the Hölder–Zygmund scales on any relatively compact Lipschitz domain: \[\label{eq:scale} \mathscr{C}^{k+1}\subset \mathscr{C}^{k,1} \subset \Lambda^{k+1}\subset \Lambda^{k+\alpha} \subset \Lambda^{k+\beta}\subset \mathscr{C}^k,\quad k\in\mathbb{Z}_+,\;0<\beta<\alpha<1.\tag{10}\]
Since postcompositions of functions in \(\Lambda^r(\overline{\Omega})\) with smooth functions on \(\mathbb{C}\) are again in \(\Lambda^r(\overline{\Omega})\) (and such a postcomposition defines a smooth operator, see [39]), we can also define for any smooth manifold \(Y\) the space \(\Lambda^r(\overline{\Omega},Y)\) of maps \(f:\overline{\Omega}\to Y\) of class \(\Lambda^r\).
Assume that \(X\) and \(Y\) are complex manifolds and \(\Omega\Subset X\) is a Lipschitz domain. Set \[\Lambda^r_\mathscr{O}(\overline{\Omega},Y)= \{f\in \Lambda^r(\overline{\Omega},Y): f\;\text{is holomorphic on}\;\Omega\},\quad r>0.\] In particular, \(\Lambda^r_\mathscr{O}(\overline{\Omega},\mathbb{C}) =\Lambda^r_\mathscr{O}(\overline{\Omega})\).
A holomorphic fibre bundle \(h:Z\to \overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) has total space of the form \(Z= \bigsqcup_{i=1}^m \overline{U}_i \times Y/\!\!\sim\) where the fibre \(Y\) is a complex manifold, the sets \(\overline{U}_i\subset\overline{\Omega}\) and \(\overline{U}_{i,j}=\overline{U}_i\cap \overline{U}_j\) are compact and have Lipschitz boundaries, \(\bigcup_{i=1}^m \overline{U}_i=\overline{\Omega}\), and a point \((x,y)\in \overline{U}_j \times Y\) \((x\in \overline{U}_{i,j})\) is identified by the equivalence relation \(\sim\) with the point \((x,\phi_{i,j}(x,y))\in \overline{U}_i\times Y\), where the map \(\overline{U}_{i,j} \times Y \ni (x,y) \mapsto \phi_{i,j}(x,y)\in Y\) is of class \(\Lambda^r_\mathscr{O}(\overline{U}_{i,j})\) in \(x\), holomorphic in \(y\), and \(\phi_{i,j}(x,\cdotp)\in \mathrm{Aut}(Y)\) for every \(x\in \overline{U}_{i,j}\). A section \(f:\overline{\Omega}\to Z\) of \(h:Z\to \overline{\Omega}\) is given by a collection of maps \(f_i:\overline{U}_i\to Y\) \((i=1,\ldots,m)\) satisfying the compatibility conditions \[f_i(x) = \phi_{i,j}(x, f_j(x)),\quad x\in\overline{U}_{i,j},\;i,j=1,\ldots,m.\] The section \(f\) is of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) if and only if \(f_i\in \Lambda^r_\mathscr{O}(\overline{U}_i,Y)\) for \(i=1,\ldots,m\). We denote by \(\Gamma^r_\mathscr{O}(\overline{\Omega},Z)\) the space of sections of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). If \(Y\) is a smooth manifold, the analogous definition gives fibre bundles \(Z\to \overline{\Omega}\) of class \(\Lambda^r(\overline{\Omega})\).
A vector bundle \(\pi:E\to\overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) and rank \(n\) has fibre \(\mathbb{C}^n\) and transitions maps \[\phi_{i,j}(x,v)=A_{i,j}(x)v, \quad v\in\mathbb{C}^n,\; A_{i,j}\in \Lambda^r_\mathscr{O}(\overline{U}_{i,j},GL_n(\mathbb{C}))\] satisfying the 1-cocycle conditions (with \(I\in GL_n(\mathbb{C})\) the identity matrix): \[A_{i,i}=I,\quad A_{i,j}A_{j,i}=I,\quad A_{i,j}A_{j,k}A_{k,i}=I\quad \text{for all}\;i,j,k.\] A section of \(E\) over \(\overline{\Omega}\) is given by a collection of maps \(f_i\in \Gamma^r_\mathscr{O}(\overline{U}_i,\mathbb{C}^n)\) satisfying \[f_i(x) = A_{i,j}(x) f_j(x),\quad x\in\overline{U}_{i,j},\;i,j=1,\ldots,m.\] Similarly one defines vector bundle morphisms of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), subbundles, quotient bundles, etc.
Let \(p\ge 0\), \(q\ge 1\) be integers, and let \(\Omega\) be a domain in a complex manifold \(X\). The \(\overline{\partial}\)-problem on \(\Omega\) asks for the existence and regularity properties of solutions of the equation \(\overline{\partial} u=f\) for a differential \((p,q)\)-form \(f\) on \(\Omega\) (or on its closure \(\overline{\Omega}\)) satisfying the necessary condition \(\overline{\partial} f=0\). One of the most successful techniques in this field is the \(\overline{\partial}\)–Neumann method introduced in the pioneering works of Kohn [41], [42]; see the books by Folland and Kohn [43] and Chen and Shaw [44], among others.
Let \(\overline{\Omega}\) be a compact complex Hermitian manifold of dimension \(n+1\) with \(\mathscr{C}^\infty\) boundary \(b\Omega\) and Stein interior \(\Omega\) such that at each point of \(b\Omega\) the Levi form has at least \(n+1-q\) positive eigenvalues. Let \(N_q\) denote the Neumann operator for the complex Laplacian \[\Box_q = \overline{\partial}^*_q\overline{\partial}_{q} + \overline{\partial}_{q-1} \overline{\partial}^*_{q-1}, \quad q\ge 1\] acting on \((p,q)\)-forms which satisfy the \(\overline{\partial}\)-Neumann boundary conditions on \(b\Omega\). This means that \(N_q\Box_q = \Box_q N_q\) is the \(L^2\) orthogonal projection onto the range of \(\Box_q\). Then, \(T_q= \overline{\partial}^*_{q-1}N_q\) is Kohn’s canonical solution operator for the \(\overline{\partial}\)-equation \(\overline{\partial}_{q-1}u = f\) with \(f\) a \((p,q)\)-form with \(\overline{\partial}_q f=0\). Denote by \(\Lambda^r_{p,q}(\overline{\Omega})\) the space of \((p,q)\)-forms on \(\overline{\Omega}\) of class \(\Lambda^r(\overline{\Omega})\) for \(r>0\). (Note that \((p,q)\)-forms are sections of the vector bundle \(\Lambda^{p,q}TX|_{\overline{\Omega}}\), so the spaces \(\Lambda^r_{p,q}(\overline{\Omega})\) are naturally defined.) The following is a special case of [16] due to Beals, Greiner and Stanton.
Theorem 7. Let \(\Omega\) be as above. The canonical solution \(T_q=\overline{\partial}_{q-1}^*N_q\) to \(\overline{\partial} u=f\) for \(f\in \Lambda^r_{p,q}(\overline{\Omega})\) satisfying \(\overline{\partial} f=0\) maps \(\Lambda^r_{p,q}(\overline{\Omega})\cap\ker \overline{\partial}\) boundedly to \(\Lambda^{r+1/2}_{p,q-1}(\overline{\Omega})\) for every \(p\ge 0\), \(q\ge 1\), and \(r>0\).
We shall use this result for \(p=0\). The gain of regularity for \(1/2\) implies that the operator \(T_q: \Lambda^r_{p,q}(\overline{\Omega})\cap\ker \overline{\partial} \to \Lambda^{r}_{p,q-1}(\overline{\Omega})\) is compact, a fact which is useful in many applications.
In this section, we prove Theorem 2 and its parametric version, Theorem 8. We shall use the following notion of a Cartan pair; see [2].
Definition 1. A pair \((A,B)\) of compact sets in a complex manifold \(X\) is a Cartan pair if
\(A\), \(B\), \(D=A\cup B\) and \(C=A\cap B\) are closures of smoothly bounded, strongly pseudoconvex domains with Stein interior, and
\(\overline{A\setminus B}\cap \overline{B\setminus A} =\varnothing\).
A Cartan pair \((A,B)\) is special if there is a coordinate neigbourhood of \(B\) in \(X\) in which the sets \(B\) and \(C\) are strongly convex. In this case, \(B\) is said to be a convex bump attached to \(A\).
A more general notion of a Cartan pair in [2] is suitable when considering holomorphic maps on neighbourhoods of the respective sets, which may be any Stein compacts. We shall not need it in this paper since our focus in on mapping spaces on smoothly bounded domains.
Proof of Theorem 2. Since the domain \(\Omega\Subset X\) is smoothly bounded and strongly pseudoconvex, there is a smooth strongly plusubharmonic function \(\rho\) on a neighbourhood \(U\subset X\) of \(\overline{\Omega}\) such that \(\Omega=\{x\in U:\rho(x)<0\}\) and \(d\rho_x\ne 0\) for every point \(x\in b\Omega =\{\rho=0\}\). Pick \(c>0\) such that \(\rho\) has no critical values on the interval \([0,c]\). Set \(A=\overline{\Omega}=\{\rho\le 0\}\) and \(A'=\{\rho\le c\}\). Given an open cover \(\mathcal{U}=\{U_j\}\) of \(\overline{A'\setminus A}\) consisting of holomorphic coordinate charts \(U_j\subset X\), [2] gives compact, smoothly bounded, strongly pseudoconvex domains \[\label{eq:sequenceA} A_0:=A \subset A_1 \subset \cdots \subset A_m=A'\tag{11}\] for some \(m\in\mathbb{N}\) such that for every \(k=0,1,\ldots,m-1\) we have \(A_{k+1}=A_k\cup B_k\), where \((A_k,B_k)\) is a special Cartan pair (see Definition 1) and \(B_k\subset U_j\) for some \(j=j(k)\).
It therefore suffices to show that for every special Cartan pair \((A,B)\) we can approximate any map \(f\in \Lambda^r_{\mathscr{O}} (A,Y)\) as closely as desired in the \(\Lambda^r\) topology by maps \(\tilde{f}\in \Lambda^r_{\mathscr{O}} (D,Y)\) where \(D=A\cup B\); the theorem then follows by a finite induction using the sequence 11 .
We first consider the case of functions, that is, \(Y=\mathbb{C}\). Fix such a pair \((A,B)\) and \(f\in \Lambda^r_\mathscr{O}(A)\). Set \(C=A\cap B\). We can find a holomorphic function \(g\) on a neighbourhood of \(B\) which approximates \(f|_C\) as closely as desired in \(\Lambda^r(C)\). To do this, we proceed as follows. By the assumption, there is a neighbourhood \(W\subset X\) of \(B\) and a holomorphic coordinate map \(\psi:W\to \widetilde{W}\subset\mathbb{C}^n\) \((n=\dim X)\) such that the sets \(\widetilde{C}=\psi(C)\) and \(\widetilde{B}=\psi(B)\) are strongly convex. Choose a point \(p\) in the interior of \(\widetilde{C}\); we may assume that \(p=0\in\mathbb{C}^n\). For \(t\in (0,1)\) the holomorphic map \(\phi_t:W\to W\) defined by \(\phi_t(x)=\psi^{-1}(t \psi(x))\), \(x\in W\), satisfies \(\phi_t(C)\subset \mathring C\). The function \(f_t = f \circ\phi_t\) is then holomorphic on a neighbourhood \(V_t \subset W\) of \(C\) and it approximates \(f|_C\) in \(\Lambda^r(C)\) for \(t\) close to \(1\). Fix \(t\) and pick a compact neigbourhood \(C'\subset V_t\) of \(C\) such that \(\psi(C')\) is convex. By the Oka–Weil theorem, we can approximate \(f_t\) uniformly on \(C'\) by a function \(g\in \mathscr{O}(B)\). By Cauchy estimates, this gives approximation of \(f_t|_C\) by \(g|_C\) in \(\Lambda^r(C)\). If the approximation is close enough in both steps then \(g\) approximates \(f|_C\) to the desired precision in \(\Lambda^r(C)\).
Condition (ii) in Definition 1 ensures the existence of a smooth function \(\chi: X \to[0,1]\) which equals \(1\) on a neighbourhood of \(\overline{A\setminus B}\) and equals \(0\) on a neighbourhood of \(\overline{B \setminus A}\). Set \[u=\chi f+(1-\chi)g\in \Lambda^r(D).\] Note that \(u=f\) on \(\overline{A\setminus B}\), \(u=g\) on \(\overline{B\setminus A}\), \(f-u = (1-\chi) (f-g)\) on \(A\), and hence \[\label{eq:est1} \|f-u\|_{\Lambda^r(A)}= \|(1-\chi) (f-g)\|_{\Lambda^r(A)} \le c_0 \|f-g\|_{\Lambda^r(C)}\tag{12}\] for some \(c_0>0\) depending only on \(\chi\). Furthermore, \(\overline{\partial} u = (f-g)\overline{\partial}\chi \in \Lambda^r_{0,1}(D)\) is a \(\overline{\partial}\)-closed \((0,1)\)-form whose support is disjoint from \(\overline{A\setminus B}\cup \overline{B\setminus A}\). It follows that \[\|\overline{\partial} u\|_{\Lambda^r_{0,1}(D)} \le c_1 \|f-g\|_{\Lambda^r(C)}\] for some \(c_1>0\) depending on \(\chi\). By Theorem 7 there exists \(\tilde{u}\in \Lambda^r(D)\) satisfying \(\overline{\partial}\tilde{u}=\overline{\partial} u\) and \[\label{eq:est3} \|\tilde{u}\|_{\Lambda^r(D)}\le c_2 \|\overline{\partial} u\|_{\Lambda^r(D)} \le c_1c_2 \|f-g\|_{\Lambda^r(C)}\tag{13}\] for some \(c_2>0\) depending only on \(D\). The function \(\tilde{f}=u-\tilde{u} \in \Lambda^r(D)\) satisfies \(\overline{\partial}\tilde{f}=0\), so it is holomorphic on \(\mathring D\). On \(A\), we have \(f-\tilde{f} = (f - u) + \tilde{u}\), and it follows from 12 –13 that \(\|f - \tilde{f}\|_{\Lambda^r(A)}\le (c_0+c_1c_2)\|f-g\|_{\Lambda^r(C)}\), which can be made arbitrarily small by a choice of \(g\). This proves the theorem for functions.
Assume now that \(Y\) is a complex manifold. Every map \(f\in \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) for \(r>0\) is continuous on \(\overline{\Omega}\) and holomorphic on \(\Omega\), so its graph has a Stein neigbourhood in \(X\times Y\) (see [19] or [2]). The proof is then reduced to the case of functions as in [2], using the fact that postcompositions by holomorphic maps preserve the Zygmund spaces. ◻
Theorem 2 has the following extension to the parametric case.
Theorem 8. Assume that \(\Omega\) is a relatively compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold \(X\), \(Y\) is a complex manifold, and \(P\) is a compact Hausdorff space. Every continuous map \(f:P\to \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\), \(r>0\), can be approximated in the \(\Lambda^r(\overline{\Omega},Y)\) topology by continuous maps \(\tilde{f}: P \to \mathscr{O}(\overline{\Omega},Y)\). If in addition \(Q\) is a closed subspace of \(P\) which is a strong neighbourhood deformation retract and \(f|_{Q}:Q \to \mathscr{O}(\overline{\Omega},Y)\), we can choose \(\tilde{f}\) to agree with \(f\) on \(Q\).
Proof. Denote by \(pr_X:X\times Y\to X\) the projection onto the first factor. Fix a point \(p_0\in P\). The map \(f(p_0)\in \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) is continuous on \(\overline{\Omega}\), so its graph over \(\overline{\Omega}\) has an open Stein neighbourhood \(V_0 \subset X\times Y\). Furthermore, \(V_0\) can be chosen to be fibrewise biholomorphic (with respect to the projection \(pr_X\)) to a domain with convex fibres in a holomorphic vector bundle \(E_0 \to U_0\) over an open neighbourhood \(U_0\subset X\) of \(\overline{\Omega}\) (see [19] or [2]). With respect to such a biholomorphism, the notion of a convex combination of points in the fibres of \(pr_X:V_0\to X\) is well defined. By Theorem 2 there is a holomorphic map \(\tilde{f}(p_0) \in \mathscr{O}(U_0,Y)\) on a neighbourhood \(U_0\subset X\) of \(\overline{\Omega}\) which approximates \(f(p_0)\) in \(\Lambda^r(\overline{\Omega},Y)\) to a desired precision and whose graph is contained in \(V_0\). For \(p\in P\) close to \(p_0\), the graph of \(f(p)\in \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) lies in \(V_0\) and \(\tilde{f}(p_0)|_{\overline{\Omega}}\) is an approximant to \(f(p)\). Repeating the same procedure at other points of \(P\) gives a finite open covering \(\mathcal{P}=\{P_i\}_{i=0}^m\) of \(P\) and for each \(i=0,\ldots,m\) a pair of open Stein domains \(U_i\subset X\), \(V_i\subset X\times Y\) and a map \(\tilde{f}_i\in \mathscr{O}(U_i,Y)\) such that the graph of every map \(f(p)\), \(p\in U_i\), and also of \(\tilde{f}_i\), lies in \(V_i\), and \(f(p)\) is close to \(\tilde{f}_i\) to a desired precision in \(\Lambda^r(\overline{\Omega},Y)\) for all \(p\in U_i\). In view of the fibrewise convex structure of every domain \(V_i\), we can use the method of successive patching (see [2] for the details) to obtain a map \(\tilde{f}:P\to \mathscr{O}(\overline{\Omega},Y)\) approximating \(f\) to a desired precision. For the last statement in the theorem, we precompose \(f\) by a continuous map \(\psi:P\to P\) which maps a small neighbourhood \(Q'_0\subset P\) of \(Q\) to itself, it retracts a neighbourhood \(Q_0\subset Q'_0\) of \(Q\) onto \(Q\), and it equals the identity map on \(P\setminus Q'_0\). (Such \(\psi\) exists since \(Q\) is a strong deformation neighbourhood retract in \(P\).) This yields a continuous map \(f_0=f\circ \psi : P\to \Lambda^r_\mathscr{O}(\overline{\Omega},Y)\) which approximates \(f\), it agrees with \(f\) on \(Q\), and such that \(f_0|_{Q_0}:Q_0 \to \mathscr{O}(\overline{\Omega},Y)\). Applying the above procedure to \(f_0\) yields a map \(\tilde{f}:P\to \mathscr{O}(\overline{\Omega},Y)\) approximating \(f\) which agrees with \(f\) on \(Q\). ◻
The proofs of Theorems 2 and 8 also apply to any mapping space as in Remark 5.
The notion of a vector bundle of Zygmund class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), \(r>0\), was introduced in Subsect.2.2. In this section, we prove the following version of Cartan’s Theorem A for such vector bundles.
Theorem 9. Let \(\overline{\Omega}\) be a compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold \(X\), and let \(\pi:E\to \overline{\Omega}\) be a vector bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) for some \(r>0\). There exist finitely many sections in \(\Gamma^r_\mathscr{O}(\overline{\Omega},E)\) spanning every fibre \(E_x:=\pi^{-1}(x)\), \(x\in \overline{\Omega}\).
The classical Theorem A of Henry Cartan gives such a statement for holomorphic vector bundles on finite dimensional Stein spaces. For vector bundles of class \(\mathscr{A}^r(\overline{\Omega})\) with \(r\in\mathbb{Z}_+\), and for more general coherent analytic sheaves of this class, the analogue of Theorem 9 is due to Leiterer [24] for domains in Euclidean spaces and Heumenann [45] in general.
Remark 10. An equivalent statement of Theorem 9 is that every vector bundle \(E\to \overline{\Omega}\) as in the theorem admits a vector bundle epimorphism \(\Phi:\overline{\Omega}\times \mathbb{C}^m \to E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Indeed, if \(\xi_1,\ldots,\xi_m:\overline{\Omega}\to E\) are sections of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) spanning every fibre of \(E\), then the map \(\Phi:\overline{\Omega}\times \mathbb{C}^m \to E\) given by \[\label{eq:Phi} \Phi(x,z_1,\ldots,z_m)= \sum_{i=1}^m z_i\xi_i(x) \in E_x=\pi^{-1}(x), \quad x\in\overline{\Omega},\;z=(z_,\ldots,z_m)\in\mathbb{C}^m\tag{14}\] is a vector bundle epimorphism of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Conversely, given such an epimorphism, the images of standard basis sections of the trivial bundle generate each fibre of \(E\).
We also have the following embedding result for vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\).
Corollary 1. Given a vector bundle \(\pi:E\to \overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) as in Theorem 9, there exists a vector bundle embedding \(E\hookrightarrow \overline{\Omega}\times \mathbb{C}^m\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) for some \(m\in\mathbb{N}\).
Proof. Theorem 9 gives a vector bundle epimorphism \(\Phi: \overline{\Omega} \times \mathbb{C}^m \to E^*\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Its dual \(\Phi^*:(E^*)^*=E \to (\overline{\Omega} \times \mathbb{C}^m)^* \cong \overline{\Omega} \times \mathbb{C}^m\) is a vector bundle embedding of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). ◻
Denote by \(M_n=M_n(\mathbb{C})\) the space of complex \(n\times n\) matrices and by \(GL_n=GL_n(\mathbb{C})\subset M_n\) the general linear group of rank \(n\) over \(\mathbb{C}\). By \(I\in GL_n\) we denote the identity matrix. In the proof of Theorem 9 we shall use the following version of Cartan’s lemma. See also the parametric version, Lemma 3, and [46] or [47] for the classical Cartan lemma.
Lemma 1. Let \((A,B)\) be a Cartan pair in a complex manifold \(X\) (see Definition 1) such that \(C:=A\cap B\) is holomorphically convex in \(B\). Given a map \(\gamma\in \Lambda^r_\mathscr{O}(C,GL_n)\) which is homotopic to the constant map \(C\ni x\mapsto I\in GL_n\) by a path in \(\mathscr{C}(C,GL_n)\) and a number \(\epsilon>0\), there are maps \(\alpha \in \Lambda^r_\mathscr{O}(A,GL_n)\) and \(\beta \in \Lambda^r_\mathscr{O}(B,GL_n)\) such that \(\|\alpha-I\|_{\Lambda^r(A)} <\epsilon\) and \[\label{eq:split} \gamma = \alpha^{-1} \cdotp \beta\;\;\;\text{holds on C}.\tag{15}\]
Proof. We first construct the product splitting 15 for \(\gamma\) close to \(I\in GL_n\). Write \(\gamma=I+c\) with \(c\in \Lambda^r_\mathscr{O}(C,M_n)\). We have the following solution to the Cousin problem in \(\Lambda^r_\mathscr{O}\).
Lemma 2. There are bounded linear operators \(\mathcal{A}:\Lambda^r_\mathscr{O}(C)\to \Lambda^r_\mathscr{O}(A)\), \(\mathcal{B}:\Lambda^r_\mathscr{O}(C)\to \Lambda^r_\mathscr{O}(B)\) with \[\label{eq:AplusB} \mathcal{A}c + \mathcal{B}c =c\;\;\;\text{for all}\;c\in \Lambda^r_\mathscr{O}(C).\tag{16}\]
Proof. Set \(D=A\cup B\). Condition (ii) in Definition 1 implies that there is a smooth function \(\chi: X \to[0,1]\) which equals \(1\) on a neighbourhood of \(\overline{A\setminus B}\) and equals \(0\) on a neighbourhood of \(\overline{B \setminus A}\). Given \(c\in \Lambda^r_\mathscr{O}(C)\), we have \(c\chi \in \Lambda^r(A)\), \(c(1-\chi)\in \Lambda^r(B)\), and \(\overline{\partial}(c \chi)=c\, \overline{\partial}\chi \in \Lambda^r_{0,1}(D)\) is a closed \((0,1)\)-form on \(D\) of class \(\Lambda^r(D)\) supported on \(C\). Let \[T=\overline{\partial}^* N: \{\omega\in \Lambda^r_{0,1}(D):\overline{\partial}\omega=0\} \to \Lambda^r(D)\] denote the (bounded, linear) canonical solution operator for the \(\overline{\partial}\)-equation for \((0,1)\)-forms of class \(\Lambda^r(D)\); see Theorem 7. The linear operators on \(\Lambda^r_\mathscr{O}(C)\) defined by \[\mathcal{A}c= c \, \chi - T(c\, \overline{\partial}\chi)\in \Lambda^r_\mathscr{O}(A), \quad \mathcal{B}c= c(1-\chi) + T(c\, \overline{\partial}\chi) \in \Lambda^r_\mathscr{O}(B)\] for all \(c\in \Lambda^r_\mathscr{O}(C)\) then satisfy the lemma. ◻
Applying Lemma 2 componentwise gives bounded linear operators \[\mathcal{A}:\Lambda^r_\mathscr{O}(C,M_n) \to \Lambda^r_\mathscr{O}(A,M_n), \quad \mathcal{B}:\Lambda^r_\mathscr{O}(C,M_n) \to \Lambda^r_\mathscr{O}(B,M_n)\] satisfying \(\mathcal{A}+\mathcal{B}=\mathrm{Id}\) on \(\Lambda^r_\mathscr{O}(C,M_n)\); see 16 . We define a map \(\Phi:U\to \Lambda^r_\mathscr{O}(C,GL_n)\) on a small neighbourhood \(U\subset \Lambda^r_\mathscr{O}(C,M_n)\) of \(c=0\) by \[\Phi(c) = (I-\mathcal{A}c)^{-1} (I+\mathcal{B}c) \in \Lambda^r_\mathscr{O}(C,GL_n), \quad c\in U.\] Clearly, \(\Phi\) is smooth and satisfies \(\Phi(0)=I\) and \(d\Phi_0 = \mathcal{A}+\mathcal{B}=\mathrm{Id}\). If follows that \(\Phi\) has a smooth right inverse \(\Psi\) on a neigbourhood \(V\subset \Lambda^r_\mathscr{O}(C,GL_n)\) of \(I\) such that \(\Psi(I)=0\) and \[(\Phi\circ \Psi)(\gamma) = \left(I-\mathcal{A}\circ \Psi(\gamma)\right)^{-1} \left(I+\mathcal{B}\circ\Psi(\gamma)\right) = \gamma \quad \text{for all}\;\gamma\in V.\] The smooth operators \(\widetilde{\mathcal{A}}:V\to \Lambda^r_\mathscr{O}(A,GL_n)\), \(\widetilde{\mathcal{B}}:V\to \Lambda^r_\mathscr{O}(B,GL_n)\) defined by \[\widetilde{\mathcal{A}}= I - \mathcal{A}\circ\Psi,\qquad \widetilde{\mathcal{B}}= I + \mathcal{B}\circ\Psi\] then provide a splitting \(\gamma=(\widetilde{\mathcal{A}}\gamma)^{-1} (\widetilde{\mathcal{B}}\gamma)\) for \(\gamma\in V\) (see 15 ) satisfying \[\label{eq:estAB} \|\widetilde{\mathcal{A}}\gamma-I\|_{\Lambda^r(A)} \le const \|\gamma-I\|_{\Lambda^r(C)}, \quad \|\widetilde{\mathcal{B}}\gamma-I\|_{\Lambda^r(B)} \le const \|\gamma-I\|_{\Lambda^r(C)}\tag{17}\] for some \(const>0\) depending only on \((A,B)\) and \(r\).
This proves the lemma for maps \(\gamma\in \Lambda^r_\mathscr{O}(C,GL_n)\) near the constant map \(C\ni x\mapsto I\). The general case is obtained as follows. By Theorem 2 we can approximate \(\gamma\) as closely as desired in \(\Lambda^r_\mathscr{O}(C)\) by a holomorphic map \({\gamma\,}' :U\to GL_n\) from an open neighbourhood \(U\subset X\) of \(C\). Since \(GL_n\) is an Oka manifold (every complex homogeneous manifold is an Oka manifold by Grauert [6], [7]; see also [2]), \(\gamma\) is homotopic to the constant map, and \(C\) if holomorphically convex in \(B\), the Oka principle [2] shows that \({\gamma\,}'\) can be approximated uniformly on a compact neighbourhood \(C'\subset U\) of \(C\) by holomorphic maps \(\tilde{\gamma}\in \mathscr{O}(B,GL_n)\). Then, \(\gamma= (\gamma {\tilde{\gamma}}^{-1}) \tilde{\gamma}\) on \(C\), and \(\gamma {\tilde{\gamma}}^{-1}\) is close to \(I\) in \(\Lambda^r(C,GL_n)\). By the first part, we have \(\gamma {\tilde{\gamma}}^{-1}=\alpha^{-1}\tilde{\beta}\) with \(\alpha \in \Lambda^r_\mathscr{O}(A,GL_n)\) close to \(I\) and \(\tilde{\beta} \in \Lambda^r_\mathscr{O}(B,GL_n)\). Setting \(\beta=\tilde{\beta} \tilde{\gamma}\in \Lambda^r_\mathscr{O}(B,GL_n)\) gives \(\gamma=\alpha^{-1}\beta\) as in 15 . ◻
Lemma 1 has the following generalisation to the parametric case.
Lemma 3. Assume that \(X\) and \((A,B)\) are as in Lemma 1. Given a compact Hausdorff space \(P\), a closed subspace \(Q\subset P\) which is a strong neighbourhood deformation retract, and a continuous map \(\gamma : P\to \Lambda^r_\mathscr{O}(C,GL_n)\) such that \(\gamma(p)=I\) for \(p\in Q\) and there is a homotopy \(\gamma_t:P\to \mathscr{C}(C,GL_n)\) \((t\in [0,1])\) which is fixed on \(Q\) such that \(\gamma_0=I\) and \(\gamma_1=\gamma\), there are continuous maps \(\alpha:P \to \Lambda^r_\mathscr{O}(A,GL_n)\), \(\beta:P\to \Lambda^r_\mathscr{O}(B,GL_n)\) such that \(\|\alpha-I\|_{\Lambda^r(A)}\) is arbitrarily small, \(\alpha|_Q=\beta|_Q=I\), and \(\gamma = \alpha^{-1} \cdotp \beta\) holds on \(C\).
Proof. By Theorem 8 and the argument in the last part of the proof of Lemma 1, we can reduce to the case when \(\gamma\) is close to the constant map \(P\to I\in GL_n\). Since the splitting 16 in Lemma 2 is given by bounded linear operators, it also applies to the parametric case. The proof of the first part of Lemma 1 then carries over verbatim. ◻
Remark 11. Lemmas 1 and 3 also hold, with the same proofs, for maps with values in any complex Lie group \(G\) in place of \(GL_n(\mathbb{C})\). In this case, we replace the matrix algebra \(M_n\) (which is the Lie algebra of \(GL_n(\mathbb{C})\)) by the Lie algebra of \(G\). Furthermore, the analogous results hold with the spaces \(\Lambda^r_\mathscr{O}(\overline{\Omega},G)\) replaced by any space \(\mathcal{F}_\mathscr{O}(\overline{\Omega},G)\) described in Remark 5.
The main step in the proof of Theorem 9 is given by the following lemma.
Lemma 4. Assume that \((A,B)\) is a special Cartan pair (see Definition 1). Let \(D=A\cup B\) and \(\pi:E\to D\) be a vector bundle of class \(\Lambda_\mathscr{O}^r(D)\), \(r>0\), such that \(E|_B\) is \(\Lambda_\mathscr{O}^r(B)\)-isomorphic to a trivial bundle. Then, every vector bundle epimorphism \(\Phi:A\times\mathbb{C}^n \to E|_A\) of class \(\Lambda_\mathscr{O}^r(A)\) can be approximated in \(\Lambda^r(A)\) by vector bundle epimorphisms \(\widetilde{\Phi}:D\times\mathbb{C}^n \to E\) of class \(\Lambda_\mathscr{O}^r(D)\).
Proof. Let \(e_1,\ldots,e_n\) be sections of \(E|_A\) of class \(\Lambda^r_\mathscr{O}(A)\) which are \(\Phi\)-images of the standard basis sections of \(A\times\mathbb{C}^n\). Also, let \(g_1,\ldots,g_m\) with \(m=\mathrm{rank}E\) be basis sections of class \(\Lambda^r_\mathscr{O}(B)\) of the trivial bundle \(E|_B\cong B\times\mathbb{C}^m\). On \(C\), we have \(e_i=\sum_{j=1}^m \gamma_{i,j}g_j\) (\(i=1,\ldots,n\)) with \(\gamma_{i,j}\in\Lambda^r_\mathscr{O}(C)\), and the \(n\times m\) matrix \(\Gamma'(x)=(\gamma_{i,j}(x))\) has rank \(m\) at every point \(x\in C\). We claim that there is an \(n\times n\) matrix function \[\Gamma=(\gamma_{i,j})_{i,j=1}^n =(\Gamma',\Gamma'') \in \Lambda^r_\mathscr{O}(C,GL_n)\] whose left hand side \(n\times m\) submatrix equals \(\Gamma'\). This is equivalent to saying that the subbundle \(E'\) of \(C\times\mathbb{C}^n\), spanned by the columns of \(\Gamma'\), has a trivial complementary subbundle \(E''\subset C\times\mathbb{C}^n\) of class \(\Lambda^r_\mathscr{O}(C)\). The existence of a complementary subbundle \(\widetilde{E}''\) of class \(\mathscr{A}(C)\) (continuous on \(C\) and holomorphic on \(\mathring C\)) follows from [25]. Since \(C\) is contractible, \(\widetilde{E}''\) is trivial as a bundle of class \(\mathscr{A}(C)\) by [25]. Let \(\widetilde{\Gamma}''\) be an \(n\times (n-m)\) matrix of class \(\mathscr{A}(C)\) spanned by the basis sections of \(\widetilde{E}''\). Since \(C\) is convex, we can approximate \(\widetilde{\Gamma}''\) uniformly on \(C\) by a matrix \(\Gamma''\) of class \(\Lambda^r_\mathscr{O}(C)\). If the approximation is close enough then \(\Gamma=(\Gamma',\Gamma'') \in \Lambda^r_\mathscr{O}(C,GL_n)\). Set \(g_{m+1}=0,\ldots,g_n=0\). It follows that \((e_1,e_2,\ldots,e_n)^t = \Gamma (g_1,g_2,\ldots,g_n)^t,\) where the superscript \(t\) denotes the transpose. By Lemma 1 we have \(\Gamma=\alpha^{-1}\beta\) where \(\alpha\in \Lambda^r_\mathscr{O}(A,GL_n)\) is close to \(I\) and \(\beta \in \Lambda^r_\mathscr{O}(B,GL_n)\). Then, \(\alpha (e_1,e_2,\ldots,e_n)^t = \beta (g_1,g_2,\ldots,g_n)^t\) holds on \(C\). The resulting sections of \(E\to D\) of class \(\Lambda^r_\mathscr{O}(D)\) determine a vector bundle epimorphism \(\widetilde{\Phi}:D\times \mathbb{C}^n\to E\) of class \(\Lambda^r_\mathscr{O}(D)\) (see 14 ) which approximates \(\Phi\) in \(\Lambda^r_\mathscr{O}(A)\). ◻
Proof of Theorem 9. Pick a smooth strongly plurisubharmonic function \(\rho\) on a neighbourhood \(U\) of \(\overline{\Omega}\) such that \(\Omega=\{x\in U:\rho(x)<0\}\) and \(d\rho_x\ne 0\) for every point \(x\in b\Omega =\{\rho=0\}\). Choose \(c<0\) such that \(\rho\) has no critical values on \([c,0]\) and set \(A_0=\{\rho\le c\}\). By [2], given an open cover \(\mathcal{U}=\{U_j\}\) of \(\overline{\Omega\setminus A_0}=\{c\le \rho\le 0\}\) consisting of holomorphic coordinate charts \(U_j\subset X\), there are compact, smoothly bounded, strongly pseudoconvex domains \(A_0\subset A_1 \subset \cdots \subset A_{k_0}=\overline{\Omega}\) for some \(k_0\in\mathbb{N}\) such that for every \(k=0,1,\ldots,k_0-1\) we have \(A_{k+1}=A_k\cup B_k\), where \((A_k,B_k)\) is a special Cartan pair (see Definition 1) and \(B_k\subset U_j\) for some \(j=j(k)\). We choose the sets \(U_j\) small enough so that the restricted bundle \(E|_{B_k}\) is trivial for \(k=0,1,\ldots,k_0-1\). Since \(E\) is holomorphic over \(\Omega\), there is a holomorphic vector bundle epimorphism \(\Phi_0:A_0\times \mathbb{C}^n \to E|_{A_0}\) for some \(n\in \mathbb{N}\). We inductively apply Lemma 4 to find vector bundle epimorphisms \(\Phi_k:A_k\times \mathbb{C}^n \to E|_{A_k}\) of class \(\Lambda^r_\mathscr{O}(A_k)\) for \(k=1,\ldots,k_0\) such that \(\Phi_{k+1}\) approximates \(\Phi_k\) on \(A_k\) for every \(k=0,1,\ldots,k_0-1\). Then, \(\Phi:=\Phi_{k_0}:\overline{\Omega} \times \mathbb{C}^n\to E\) satisfies the conclusion of the theorem. ◻
Remark 13. In the sequel, we shall be using the notion of a continuous family of holomorphic vector bundles on a smoothly bounded domain \(\Omega\Subset X\) which are continuous or better on \(\overline{\Omega}\); see Leiterer [22] (where the parameter space is \([0,1]\)) or the statement of his stability theorem [22] for the general case. This means that, locally in the parameter, the family is defined by a continuous family of 1-cocycles on the same finite open covering of \(\overline{\Omega}\). The stability theorem says that, under suitable cohomological conditions (i), (ii) on the endomorphism bundle \(\mathrm{Ad}(E)\) of a vector bundle \(E\to \overline{\Omega}\) of class \(\mathscr{A}(\overline{\Omega})\), all nearby vector bundles of the same class are isomorphic to \(E\) with continuous dependence on the parameter. This follows from an implicit function theorem in Banach spaces [22]. The aforementioned cohomological conditions in [22] are satisfied if for every continuous \(E\)-valued \((0,q)\)-form \(\phi\) on \(\overline{\Omega}\) \((q>0)\) the equation \(\overline{\partial}\psi=\phi\) can be solved with a continuous \(\psi\) on \(\overline{\Omega}\). This holds in particular if \(\Omega\) is strongly pseudoconvex. By Theorems 7 and 15, the same conclusions hold in Hölder–Zygmund spaces, so [22] also holds for vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). For later reference we state this explicitly.
Theorem 12. Assume that \(\overline{\Omega}\) is a compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold \(X\), \(P\) is a topological space, and \(E_p\to \overline{\Omega}\) \((p\in P)\) is a continuous family of vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), \(r>0\). Then for each \(p_0\in P\) there are a neighbourhood \(P_0\subset P\) and a continuous family of vector bundle isomorphisms \(\phi_p:E_p\to E_{p_0}\) \((p\in P_0)\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\).
We have the following parametric version of Theorem 9.
Theorem 14. Assume that \(\overline{\Omega}\) is a compact, smoothly bounded, strongly pseudoconvex domain in a Stein manifold \(X\), \(P\) is a compact Hausdorff space, and \(E_p\to \overline{\Omega}\) \((p\in P)\) is a continuous family of vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), \(r>0\). There exist an integer \(N\in \mathbb{N}\) and a continuous family of vector bundle epimorphisms \(\Phi_p: \overline{\Omega}\times \mathbb{C}^N \to E_p\), \(p\in P\), of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\).
Proof. By Theorem 12, every point \(p_0\in P\) has a neighbourhood \(P_0\subset P\) and a continuous family of vector bundle isomorphisms \(\phi_p:E_p\to E_{p_0}\) \((p\in P_0)\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). A vector bundle epimorphism \(\Psi_{p_0}: \overline{\Omega}\times \mathbb{C}^{n_0} \to E_{p_0}\), given by Theorem 9, extends to a continuous family of such epimorphisms \(\Psi_p=\phi_p^{-1}\circ \Psi_{p_0}: \overline{\Omega}\times \mathbb{C}^{n_0} \to E_{p}\) for \(p\in P_0\). Since \(P\) is compact, we obtain finitely many pairs of open subsets \(P'_i \Subset P_i\subset P\) and continuous families of vector bundle epimorphism \(\Psi^i_p:\overline{\Omega}\times \mathbb{C}^{n_i}\to E_p\) (\(p\in P_i\), \(i=1,\ldots,m\)) such that \(\bigcup_{i=1}^m P'_i=P\) . For every \(i=1,\ldots,m\) let \(\chi_i:P\to [0,1]\) be a continuous function such that \(\chi_i=1\) on \(P'_i\) and \(\mathrm{supp}\chi_i\subset P_i\). Let \(\Phi^i_p: \overline{\Omega}\times \mathbb{C}^{n_i}\to E_p\) be defined by \(\Phi^i_p(x,z)= \Psi^i_p(x,\chi_i(p)z)\). Take \(N=\sum_{i=1}^m n_i\). The family \(\Phi_p=\bigoplus_{i=1}^m \Phi^i_p: \overline{\Omega}\times \mathbb{C}^{N}\to E_p\) for \(p\in P\) clearly satisfies the theorem. ◻
Next, we show that every complex vector subbundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) is complemented.
Theorem 15. Let \(\pi:E\to \overline{\Omega}\) be a complex vector bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). For every complex vector subbundle \(E'\subset E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) there is a complementary to \(E'\) complex vector subbundle \(E''\subset E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) such that \(E\cong E'\oplus E''\).
Proof. We first consider the case when \(E= \overline{\Omega}\times\mathbb{C}^n\) is a trivial bundle. By [45] there is a vector subbundle \(\widetilde{E}\subset E\) of class \(\mathscr{A}(\overline{\Omega})\) complementary to \(E'\). To complete the proof, it suffices to approximate \(\widetilde{E}\) sufficiently closely by a subbundle \(E''\subset E=\overline{\Omega}\times\mathbb{C}^n\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). We follow the idea in [48]. Let \(\widetilde{L}: \overline{\Omega} \to M_n=\mathrm{Lin}_\mathbb{C}(\mathbb{C}^n,\mathbb{C}^n)\) be the unique map such that \(\widetilde{L}(x):\mathbb{C}^n\to\mathbb{C}^n\) is the projection onto \(\widetilde{E}_x\) with kernel \(E'_x\) for every \(x\in \overline{\Omega}\). Clearly, \(\widetilde{L}\) is of class \(\mathscr{A}(\overline{\Omega})\), and the eigenvalues of \(\widetilde{L}(x)\) are \(0\) and \(1\) for every \(x\in \overline{\Omega}\). Let \(C=\{z\in \mathbb{C}: |z-1|=1/2\}\). We can approximate \(\widetilde{L}\) uniformly on \(\overline{\Omega}\) by a holomorphic map \(L:U \to M_n\) on a neighbourhood \(U\) of \(\overline{\Omega}\). Assuming that the approximation is close enough, \(L(x)\) has no eigenvalues on the curve \(C\) for \(x\in\overline{\Omega}\), we have \(\mathbb{C}^n=V_{x,+} \oplus V_{x,-}\) where \(V_{x,+}\) resp.\(V_{x,-}\) are the \(L(x)\)-invariant subspaces of \(\mathbb{C}^n\) spanned by the generalised eigenvectors of \(L(x)\) inside resp.outside of \(C\), and \(V_{x,+}\) is close to the subspace \(\widetilde{E}_x=\widetilde{L}(x)(\mathbb{C}^n) \subset\mathbb{C}^n\). The map \[P(L(x))= \frac{1}{2\pi \mathfrak{i}} \int_{\zeta\in C} \left(\zeta I-L(x) \right)^{-1}\, d\zeta \in M_n\] is the projection of \(\mathbb{C}^n\) onto \(V_{x,+}\) with kernel \(V_{x,-}\) (see [49]), it depends holomorphically on \(x\) in a neighbourhood of \(\overline{\Omega}\), and it approximates \(L(x)\) uniformly on \(x\in \overline{\Omega}\). Assuming that the approximations are close enough, the subbundle \(E''\subset E\) with fibres \(E''_x=V_{x,+}=P(L(x))(\mathbb{C}^n)\), \(x\in \overline{\Omega}\), is complementary to \(E'\), and \(E''\) is holomorphic on a neighbourhood of \(\overline{\Omega}\).
For a general vector bundle \(E\to\overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), we apply Theorem 9 to find a vector bundle epimorphism \(\Phi:\overline{\Omega}\times\mathbb{C}^N\to E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). The preimage \(\Phi^{-1}(E')\) is a vector subbundle of \(\overline{\Omega}\times\mathbb{C}^N\) class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), so it has a complementary subbundle \(\widetilde{E}''\subset \overline{\Omega}\times\mathbb{C}^N\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Its image \(E''=\Phi(\widetilde{E}'')\) is then a subbundle of \(E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) which is complementary to \(E'\). ◻
We also have the following parametric version of Theorem 15.
Theorem 16. Let \(\Omega\) be as Theorem 15, \(P\) a compact Hausdorff space, and \(\pi_p:E_p\to \overline{\Omega}\) a continuous family of vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). (See Remark 13.) Given a continuous family \(E'_p\subset E_p\) \((p\in P)\) of vector subbundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), there is a continuous family \(E''_p\subset E_p\) \((p\in P)\) of complementary vector subbundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) such that \[\label{eq:directsum} E_p\cong E'_p\oplus E''_p\;\;\text{holds for all p\in P.}\tag{18}\] If in addition \(Q\) is a closed subspace of \(P\) which is a strong neighbourhood deformation retract and \(E''_p\subset E_p\) \((p\in Q)\) is a continuous family of complementary to \(E'_p\) vector subbundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\), then the family \(E''_p\) can be extended to all values \(p\in P\) so that 18 holds.
Proof. Assume first that the family of vector bundles \(E_p=E\to \overline{\Omega}\) is constant, independent of \(p\in P\). Consider the short exact sequence of vector bundle homomorphisms \[\label{eq:SES} 0\longrightarrow E'_p \longrightarrow E \stackrel{\phi_p}{\longrightarrow} E/E'_p \longrightarrow 0,\tag{19}\] where \(\phi_p\) is the natural quotient projection for every \(p\in P\). Theorem 15 gives for every \(p\in P\) a complementary to \(E'_p\) subbundle \(E''_p\subset E\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Such a subbundle is the image of a unique vector bundle monomorphism \(\sigma_p: E/E'_p\to E\) of the same class satisfying \(\phi_p\circ \sigma_p=\mathrm{Id}\) on \(E/E'_p\), and vice versa. Such \(\sigma_p\) is called a splitting of the sequence 19 . Since \(\overline{\Omega}\) is compact and the subbundles \(E'_p\) of \(E\) vary continuously with \(p\), a complementary to \(E'_{p_0}\) subbundle \(E''_{p_0}\subset E\) is also complementary to \(E'_p\) for all \(p\in P\) near \(p_0\), so it gives a continuous family of splittings \(\sigma_p:E/E'_p\to E\). Furthermore, a convex linear combination of splittings of \(\phi_p\) in 19 is again a splitting. Hence, the proof follows by using a continuous partition of unity on the parameter space \(P\).
The case of a continuously variable family \(E_p\to \overline{\Omega}\), \(p\in P\), can be reduced to the special case as follows. By Theorem 12, every point \(p_0\in P\) has a neighbourhood \(P_0\subset P\) and a continuous family of vector bundle isomorphisms \(\phi_p:E_p\to E_{p_0}\) \((p\in P_0)\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). Hence, the family \(E_p\) is locally constant and our proof applies locally on \(P\). It remains to patch the locally defined continuous families of splittings by a continuous partition of unity on \(P\). ◻
Let \(X\) be a complex manifold. Given a compact smoothly bounded subset \(K\) of \(X\) and an open convex domain \(0\in W\subset \mathbb{C}^N\) (the precise choice of \(W\) is not important; for convenience we may choose a ball or a polydisc), we shall consider maps \(\gamma : K\times W\to K\times \mathbb{C}^N\) of the form \[\label{eq:gamma} \gamma(x,w) = \bigl(x,\psi(x,w)\bigl),\quad x\in K, \;w\in W\tag{20}\] and of class \(\Lambda^r_\mathscr{O}(K\times W)\). The domain \(W\) will have the role of a parameter space and will be allowed to shrink during the construction. Let \(\mathrm{Id}(x,w)=(x,w)\) denote the identity map on \(X\times \mathbb{C}^N\).
The following splitting lemma is an analogue of [2], which pertains to function spaces \(\mathscr{A}^r(K)\) with \(r\in \mathbb{Z}_+\). It was first proved in [12] and [19].
Lemma 5. Let \((A,B)\) be a Cartain pair in a Stein manifold \(X\) (see Def.1) and set \(C=A\cap B\), \(D=A\cup B\). Given a convex domain \(0\in W\subset\mathbb{C}^N\) and a number \(\epsilon \in (0,1)\), there is a number \(\delta>0\) satisfying the following. For every map \(\gamma : C\times W \to C\times \mathbb{C}^N\) of the form 20 and of class \(\Lambda^r_\mathscr{O}(C\times W)\) satisfying \(\|\gamma-\mathrm{Id}\|_{\Lambda^r(C\times W)}<\delta\) there exist maps \[\label{eq:alphabeta} \alpha_\gamma : A \times \epsilon W \to A\times \mathbb{C}^N, \qquad \beta_\gamma : B \times \epsilon W \to B \times \mathbb{C}^N\tag{21}\] of the form 20 and class \(\Lambda^r_\mathscr{O}(A\times \epsilon W)\) and \(\Lambda^r_\mathscr{O}(B\times \epsilon W)\), respectively, depending smoothly on \(\gamma\), such that \(\alpha_{\mathrm{Id}}=\mathrm{Id}\), \(\beta_{\mathrm{Id}}=\mathrm{Id}\), and \[\label{eq:splitspray} \gamma\circ\alpha_\gamma = \beta_\gamma \quad \text{holds on C\times \epsilon W.}\tag{22}\] If \(\gamma\) agrees with \(\mathrm{Id}\) to order \(k\in\mathbb{N}\) along \(w=0\) then so do \(\alpha_\gamma\) and \(\beta_\gamma\). The analogous result holds with a continuous dependence of maps on a parameter \(p\in P\) in a compact Hausdorff space.
Proof. We follow [2], taking into account [2] and Theorem 7 in the present paper. In particular, the use of [2] is replaced by Lemma 2. We recall the proof and adjust it to the Hölder–Zygmund classes.
Given a number \(\epsilon\in (0,1]\), consider the Banach space \(C_{\epsilon}\) consisting of all continuous maps \(\psi=(\psi_1,\ldots,\psi_N):C\times \epsilon W \to \mathbb{C}^N\) which are holomorphic in \(\mathring C\times \epsilon W\) and satisfy \[\psi(\cdotp,w)\in \Lambda^r(C,\mathbb{C}^N)\;\; \text{for all w\in \epsilon W}, \qquad \|\psi\|_{\epsilon} = \sup_{w\in \epsilon W} \sum_{i=1}^N \|\psi_i(\cdotp,w)\|_{\Lambda^r (C)} < \infty.\] Similarly, \(A_{\epsilon}\) and \(B_{\epsilon}\) denote Banach spaces of the same kind associated to the sets \(A\) and \(B\), respectively. We may identify \(\psi\in C_\epsilon\) with the bounded holomorphic map \(\epsilon W\ni w \mapsto \psi(\cdotp,w)\in \Lambda^r_\mathscr{O}(C,\mathbb{C}^N).\) Note that \(\Lambda^r_\mathscr{O}(C\times \epsilon W) \subset C_\epsilon\) and the inclusion is continuous. Furthermore, if \(0<\epsilon'<\epsilon\) then the restriction map \(C_\epsilon\ni \psi\mapsto \psi|_{C\times \epsilon'W} \in \Lambda^r_\mathscr{O}(C\times \epsilon' W)\) is continuous.
Let \(\mathcal{A}:\Lambda^r_\mathscr{O}(C)\to \Lambda^r_\mathscr{O}(A)\) and \(\mathcal{B}:\Lambda^r_\mathscr{O}(C)\to \Lambda^r_\mathscr{O}(B)\) be the bounded linear operators in Lemma 2, satisfying \(\mathcal{A}c - \mathcal{B}c=c\) for all \(c\in \Lambda^r_\mathscr{O}(C)\) (see 16 ). We extend them to maps \(c\in C_\epsilon\) by \((\mathcal{A}c)(\cdotp,w) = \mathcal{A}(c(\cdotp,w))\) for each \(w\in\epsilon W\), and likewise for \(\mathcal{B}\). This gives bounded linear operators \[\mathcal{A}:C_\epsilon \to A_\epsilon,\quad \mathcal{B}:C_\epsilon \to B_\epsilon, \quad \mathcal{A}c - \mathcal{B}c=c \;\; \text{for every c\in C_\epsilon}.\]
We are given a map \(\gamma(x,w)=(x,\psi(x,w))\) of the form 20 , with \(\psi\in C_{1}\) close to \(\psi_0(x,w)=w\), and a number \(0<\epsilon <1\). Given \(c\in C_{\epsilon}\) near \(0\), we define \[\label{xiyscmqb} \Phi(\psi,c)(x,w) = \psi\bigl(x,w+(\mathcal{A}c)(x,w)\bigr) - \bigl(w+(\mathcal{B}c)(x,w)\bigr), \quad x\in C,\;w\in \epsilon W.\tag{23}\] Since \(\mathcal{A}:C_\epsilon \to A_\epsilon\) is a bounded linear operator, \(\Phi(\psi,c)\) is well defined for any \(c\in C_{\epsilon}\) in a neighbourhood of \(0\). We claim that \((\psi,c)\mapsto \Phi(\psi,c)\) is a smooth map from an open neighborhood of \((\psi_0,0)\) in the Banach space \(C_{1} \times C_{\epsilon}\) to \(C_{\epsilon}\). To see this, note that \(\Phi\) is affine linear in \(\psi\), and it is continuous in \(c\in C_{\epsilon}\) as a map to \(C_{\epsilon}\). Indeed, the function \(\psi\) is of the form \[\psi(x,w)= \sum_{j \in \mathbb{Z}_+^N} \psi_j(x) w^j,\quad x\in C,\;w\in W\subset\mathbb{C}^N,\] with \(\psi_j\in \Lambda^r_\mathscr{O}(C)\) for \(j\in \mathbb{Z}_+^N\). Choose a number \(\epsilon_1\in (\epsilon,1)\). For \(c\in C_{\epsilon}\) sufficiently near \(0\), the map \[C\times \epsilon W \ni (x,w) \mapsto w+(\mathcal{A}c)(x,w) \in \mathbb{C}^N\] belongs to \(C_\epsilon\) and has range compactly contained in \(C\times \epsilon_1 W\). For such \(c\), the composition \((x,w) \mapsto \psi(x,w+(\mathcal{A}c)(x,w))\) lies in \(C_{\epsilon}\) (since it is a convergent power series in \(w\) with coefficients in \(\Lambda^r(C)\)), and it depends continuously on \(c\) as an element of \(C_{\epsilon}\). Furthermore, the partial differential of \(\Phi(\psi,c)\) with respect to the second variable \(c\) equals \[\partial_c\, \Phi(\psi,c_0) c(x,w) = \partial_w\, \psi\bigl(x, w+(\mathcal{A}c_0)(x,w)\bigr) \cdotp (\mathcal{A}c)(x,w) - (\mathcal{B}c)(x,w).\] This map is again affine linear in \(\psi\) and continuous in all variables. A similar argument applies to higher order differentials of \(\Phi\). Note that \[\Phi(\psi_0,c)= \mathcal{A}(c)-\mathcal{B}(c)=c,\quad c\in C_\epsilon,\] and hence \(\partial_c\, \Phi(\psi_0,0)\) is the identity map on \(C_\epsilon\). The implicit function theorem gives a smooth map \(\psi \to \mathcal{C}(\psi) \in C_{\epsilon}\), defined in a neighborhood of \(\psi_0\) in \(C_1\), such that \[\Phi(\psi,\mathcal{C}(\psi))=0 \quad {\rm and} \quad \mathcal{C}(\psi_0)=0.\] The maps \(a_\psi\) and \(b_\psi\) defined by \[\label{eq:apsi} a_\psi(x,w) = w + \mathcal{A}\circ \mathcal{C}(\psi)(x,w), \quad b_\psi(x,w) = w + \mathcal{B}\circ \mathcal{C}(\psi)(x,w)\tag{24}\] then satisfy \(a_\psi\in A_{\epsilon}\), \(a_{\psi_0}=\psi_0\), \(b_\psi\in B_\epsilon\), \(b_{\psi_0}=\psi_0\), and \[\psi\bigl(x,a_\psi(x,w)\bigr) = b_\psi(x,w), \quad (x,w)\in C \times \epsilon W.\] The associated maps \[\alpha_\gamma (x,w) = \bigl(x,a_\psi(x,w)\bigr), \quad \beta_\gamma (x,w) = \bigl(x,b_\psi(x,w)\bigr)\] depend smoothly on \(\gamma\) and satisfy 22 .
It remains to prove the last claim in the lemma. (This proof was omitted in [2] on the grounds of being trivial. It looks less trivial now, so we use this opportunity to include a proof.) Assuming that the map \(\psi(x,w)\) is tangent to \(\psi_0(x,w)=w\) to order \(k\ge 1\) along \(w=0\), we must show that the same holds for the maps \(a_\psi\) and \(b_\psi\) in 24 . For simplicity, we consider the case when \(w\in \mathbb{C}\) is a scalar variable; a similar argument applies in general and we leave it to the reader. We consider the Taylor expansions of our functions along \(w=0\), beginning with the case \(k=1\). Thus, \[\psi(x,w)=h_1(x) w + O(|w|^2),\quad h_1\in \Lambda^r(C).\] Write \[c(x,w) = \sum_{j=0}^\infty c_j(x)w^j, \quad (\mathcal{A}c)(x,w) = \sum_{j=0}^\infty a_j(x)w^j, \quad (\mathcal{B}c)(x,w) = \sum_{j=0}^\infty b_j(x)w^j,\] where \(c_j\in \Lambda^r_\mathscr{O}(C)\), \(a_j=\mathcal{A}c_j \in \Lambda^r_\mathscr{O}(A)\), and \(b_j=\mathcal{B}c_j \in \Lambda^r_\mathscr{O}(B)\) for all \(j=0,1,\ldots\). By the definition of \(\Phi\) in 14 , we have that \[\begin{align} \Phi(\psi,c)(x,w) &=& \psi\bigl(x,w+(\mathcal{A}c)(x,w)\bigr) - \bigl(w+(\mathcal{B}c)(x,w)\bigr) \\ &=& h_1(x) \bigl(w+(\mathcal{A}c)(x,w)\bigr) - w - (\mathcal{B}c)(x,w) + O(|w|^2)\\ &=& h_1(x) a_0(x) - b_0(x) + O(|w|) =0. \end{align}\] The equation \(\Phi(\psi,c)=0\) gives \(h_1a_0 -b_0=0\). From \(\mathcal{A}c-\mathcal{B}c=c\) we get \(a_0-b_0=c_0\), and hence \(a_0(1-h_1)=c_0\) on \(C\). Note that \(\|a_0\|_{\Lambda^r(A)} \le const \|c_0\|_{\Lambda^r(C)}\). If \(\psi(x,w)\) is close to \(w\) then \(h_1\) is close to \(1\), which makes \(\|a_0(1-h_1)\|_{\Lambda^r(C)}\) smaller than \(\|c_0\|_{\Lambda^r(C)}\). In view of \(a_0(1-h_1)=c_0\) this forces \(c_0=0\), and hence also \(a_0=0\) and \(b_0=0\), thereby proving the claim for \(k=1\).
Assume now that \(k>1\) and \[\psi(x,w)=w+ h_k(x) w^k + O(|w|^{k+1}),\quad h_k\in\Lambda^r_\mathscr{O}(C,\mathbb{C}^N).\] The equation \(\Phi(\psi,c)=0\) gives \[\begin{align} \Phi(\psi,c)(x,w) &=& \psi\bigl(x,w+(\mathcal{A}c)(x,w)\bigr) - \bigl(w+(\mathcal{B}c)(x,w)\bigr) \\ &=& w+(\mathcal{A}c)(x,w) + h_k(x)(w+(\mathcal{A}c)(x,w))^k - w - (\mathcal{B}c)(x,w) + O(|w|^{k+1}) \\ &=& c(x,w) + h_k(x)(w+(\mathcal{A}c)(x,w))^k + O(|w|^{k+1}) =0. \end{align}\] By the proof for \(k=1\) we have \(|(\mathcal{A}c)(x,w)|=O(|w|)\). Therefore, \((w+(\mathcal{A}c)(x,w))^k=O(|w|^k)\) and the above identity implies \(c(x,w)=O(|w|^k)\). Thus, \(c_j=0\) for \(j=1,\ldots, k-1\), which also implies \(a_j=0\) and \(b_j=0\) for \(j=1,\ldots, k-1\). ◻
Assume now that \(D\) is a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold \(X\) and \(h:Z\to D\) is a topological fibre bundle which is holomorphic on \(\mathring D=D\setminus bD\). Let \(Y\) the denote the fibre of \(h\), a complex manifold. For \(x\in D\) write \(Z_x=h^{-1}(x)\cong Y\). The set \[VT_z(Z) = T_z Z_{\pi(z)},\quad z\in Z\] is called the vertical tangent space to \(Z\) at \(z\), and the vector bundle \(VT(Z)\to Z\) with fibres \(VT_z(Z)\) is the vertical tangent bundle of \((Z,h)\). When \(h\) is differentiable in the variable \(x\in D\), we have \(VT_z(Z) = \ker dh_z, \;z\in Z.\) If the bundle \(h:Z\to D\) is of Hölder–Zygmund class \(\Lambda^r_\mathscr{O}(D)\) (see Subsect.2.2) then \(VT(Z)\) is of local class \(\Lambda^r_\mathscr{O}(Z)\), and for every section \(f:D\to Z\) of class \(\Lambda^r_\mathscr{O}(D)\) the pullback bundle \(f^*VT(Z)\to D\) is a vector bundle of class \(\Lambda^r_\mathscr{O}(D)\).
We recall the notion of (local) dominating sprays of sections; see [2] for sprays over open domains and [2] for sprays over compact domains in a complex manifold.
Definition 2. A holomorphic spray of sections of \(h:Z\to D\) is a continuous map \(f : D\times W\to Z\) which is holomorphic on \(\mathring D\times W\), where \(0\in W\subset \mathbb{C}^N\) is an open convex set, such that \[\label{eq:spray} h(f(x,w)) = x\quad \text{for x \in D and w\in W}.\tag{25}\] The section \(f_0=f(\cdotp,0):D\to Z\) is called the core of \(f\). The spray \(f\) is dominating on a subset \(K \subset D\) if the vertical derivative of \(f\) at \(w=0\), given by \[\partial_w|_{w=0} f(x,w) : T_0 \mathbb{C}^N\cong \mathbb{C}^N \longrightarrow VT_{f(x,0)} Z,\] is surjective for all \(x\in K\). We say that \(f\) is dominating if this holds on \(K=D\).
We shall consider fibre bundles \(h:Z\to D\) and sprays \(f:D\times W\to Z\) of classes \(\Lambda^r_\mathscr{O}(D)\) for \(r>0\). The main ingredient in the proof of Theorem 1 is the following gluing lemma for such sprays. Its analogue in spaces \(\mathscr{A}^r(D)\), \(r\in\mathbb{Z}_+\), was first proved in [50]. See also the more concise proofs in [12] and [2].
Lemma 6 (Gluing sprays of class \(\Lambda^r_\mathscr{O}\)). Let \((A,B)\) be a Cartain pair in a Stein manifold (see Def.1) and set \(C=A\cap B\), \(D=A\cup B\). Assume that \(h:Z\to D\) is a fibre bundle of class \(\Lambda^r_\mathscr{O}(D)\). Given a convex domain \(0\in W_0\subset\mathbb{C}^N\) and a spray of sections \(f:A\times W_0\to Z\) of class \(\Lambda^r_\mathscr{O}(A\times W_0)\) which is dominating on \(C\), there is a ball \(W\subset\mathbb{C}^N\) with \(0\in W \subset W_0\) satisfying the following conditions.
For every holomorphic spray of sections \(g : B \times W_0 \to Z\) of class \(\Lambda^r_\mathscr{O}(B \times W_0)\) which is sufficiently close to \(f\) in \(\Lambda^r(C \times W_0)\) there exists a spray of sections \(f' : D \times W \to Z\) of class \(\Lambda^r_\mathscr{O}(D\times W)\), close to \(f\) in \(\Lambda^r(A\times W)\) (depending on the \(\Lambda^r\) distance between \(f\) and \(g\) on \(C\times W_0\)), whose core \(f'_0\) is homotopic to \(f_0=f(\cdotp,0)\) on \(A\) and is homotopic to \(g_0=g(\cdotp,0)\) on \(B\).
If \(f\) and \(g\) agree to order \(k\in \mathbb{N}\) along \(C\times\{0\}\), then \(f'\) can be chosen to agree to order \(k\) with \(f\) along \(A\times\{0\}\) and with \(g\) along \(B \times\{0\}\).
The analogous result holds for families of sprays depending continuously on a parameter \(p\in P\) in a compact Hausdorff space. If in addition the sprays \(f,g\) agree over \(C\) for values of \(p\) in a closed subset \(Q\subset P\) which is a strong neighbourhood deformation retract, then \(f'\) can be chosen to agree with \(f\) and \(g\) on \(A\) and \(B\), respectively, for \(p\in Q\).
Sketch of proof. The proof is analogous to that of [2], which applies to function spaces \(\mathscr{A}(D)\). As noted in [2], the same proof applies in any Banach spaces on which there is a linear bounded solution operator for the \(\overline{\partial}\)-equation on the level of \((0,1)\)-forms. By Theorem 7, this holds for the Zygmund spaces \(\Lambda^r\) with \(r>0\). Nevertheless, some steps must be adjusted using results in Sections 3 and 4. We indicate the necessary changes.
The first step is to find a ball \(0\in W\Subset W_0\) and a map \(\gamma:C\times W\to C\times \mathbb{C}^N\) of the form \[\gamma(x,w) = (x,w + c(x,w)), \quad x\in C,\;w\in W,\] with \(c\in \Lambda^r_\mathscr{O}(C\times W)\) close to \(0\) (depending on the \(\Lambda^r(C\times W_0)\) distance between \(f\) and \(g\)), such that \[\label{eq:fggama} f = g\circ \gamma \quad \text{holds on C \times W}.\tag{26}\] (For maps of class \(\mathscr{A}^r\) with \(r\in \mathbb{Z}_+\), a solution is given by [2].) In the process of constructing \(\gamma\), we must split the trivial bundle \(C\times \mathbb{C}^N\) by the subbundle \(E'=\ker \partial_w f|_{w=0}\), the kernel of the vertical derivative of the spray \(f\) at \(w=0\). In our case, \(E'\) is of class \(\Lambda^r_\mathscr{O}(C)\), and a splitting is furnished by Theorem 15 in the basic case and Theorem 16 in the parametric case. The proof of [2] then applies without further changes.
In the second step, we pick a number \(0<\epsilon<1\) and apply Lemma 5 to split \(\gamma\) in the form \[\gamma\circ\alpha = \beta \;\;\text{on C\times \epsilon W},\] with \(\alpha\) and \(\beta\) as in 21 . From this and 26 it follows that \[\label{eq:falphagbeta} f\circ \alpha = g\circ \beta \;\;\text{holds on C\times \epsilon W}.\tag{27}\] Hence, the two sides of the above equation amalgamate to a spray \(f':D\times \epsilon W\to Z|_D\) satisfying the lemma. The proof in the parametric case follows the same scheme. ◻
We begin by explaining the proof of the basic (nonparametric) case of Theorem 1. The parametric version in Theorem 17 includes the 1-parametric case, stated in Theorem 1.
Thus, our task is to prove that any continuous section \(f_0\in \Gamma(\overline{\Omega},Z)\) of the fibre bundle \(h:Z\to \overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) with Oka fibre is homotopic to a section \(f\in \Gamma^r_\mathscr{O}(\overline{\Omega},Z)\).
Pick a smooth strongly plurisubharmonic function \(\rho\) on a neighbourhood \(U\subset X\) of \(\overline{\Omega}\) such that \(\Omega=\{x\in U:\rho(x)<0\}\) and \(d\rho_x\ne 0\) for every point \(x\in b\Omega =\{\rho=0\}\). Pick \(c<0\) such that \(\rho\) has no critical values on the interval \([c,0]\). Set \(A_0=\{\rho\le c\}\) and \(A'=\{\rho\le 0\}=\overline{\Omega}\). Given an open cover \(\mathcal{U}=\{U_j\}\) of \(\overline{A'\setminus A}\) consisting of holomorphic coordinate charts \(U_j\subset X\), [2] gives compact, smoothly bounded, strongly pseudoconvex domains \[\label{eq:sequenceAbis} A_0 \subset A_1 \subset \cdots \subset A_m=A'=\overline{\Omega}\tag{28}\] for some \(m\in\mathbb{N}\) such that for every \(k=0,1,\ldots,m-1\) we have \(A_{k+1}=A_k\cup B_k\), where \((A_k,B_k)\) is a special Cartan pair (see Definition 1) and \(B_k\subset U_j\) for some \(j=j(k)\). Hence, we may assume that the bundle \(Z\to \overline{\Omega}\) is trivial over \(B_k\) for all \(k=0,1,\ldots,m-1\).
By the Oka principle on open Stein manifolds [2], we may assume that \(f_0\) is holomorphic on a neighbourhood of \(A_0\). It suffices to show that for every special Cartan pair \((A,B)\) in \(\overline{\Omega}\) such that the bundle \(Z\to \overline{\Omega}\) is trivial over \(B\), we can approximate a section \(f\in \Gamma^r_{\mathscr{O}}(A,Z)\) as closely as desired by sections \(\tilde{f}\in \Gamma^r_{\mathscr{O}}(D,Z)\), where \(D=A\cup B\). The theorem then follows by a finite induction, using the sequence 28 and starting with the section \(f_0\) on \(A_0\). The existence of a homotopy from \(f_0\) to \(f=f_m\) is obvious since there is no change of topology from \(A_0\) to \(A'=\overline{\Omega}\).
Fix a special Cartan pair \((A,B)\) and a section \(f_0\in \Gamma^r_{\mathscr{O}}(A,Z)\). The proof proceeds in three steps.
We embed \(f_0\) as the core of a dominating spray of sections \(F:A\times W\to Z\) of class \(\Lambda^r_{\mathscr{O}}(A\times W,Z)\) (see Definition 2), where \(0\in W\subset \mathbb{C}^N\) is a ball. The construction of \(F\) is explained in the sequel.
Set \(C=A\cap B\) and fix a number \(\delta\in (0,1)\). Since the sets \(C\subset B\) are convex in a local holomorphic coordinate on a neighbourhood of \(B\), the bundle \(h\) is trivial over \(B\), and the fibre \(Y\) of \(h\) is an Oka manifold, we can approximate \(F\) as closely as desired in \(\Lambda^r(C\times \delta W)\) by holomorphic sprays of sections \(G:B\times \delta W \to Z\). (See the proof of Theorem 2 for the details.)
Assuming that the approximation in the previous step is close enough, we can apply Lemma 6 to glue \(F\) and \(G\) into a spray \(\widetilde{F}\in \Gamma^r_\mathscr{O}(D\times \delta' W)\) for some \(\delta'\in (0,\delta)\) which approximates \(F\) in \(\Lambda^r(A\times \delta' W)\). The section \(\tilde{f}:= \widetilde{F}(\cdotp,0) \in \Gamma^r_{\mathscr{O}}(D,Z)\) then approximates \(f\) in \(\Gamma^r_{\mathscr{O}}(A,Z)\).
This completes the proof, modulo the construction of a dominating spray in step (1). For this, we follow the proof of [2] (the original reference is [12]), where a result of this kind was proved for classes \(\mathscr{A}^r\) with \(r\in\mathbb{Z}_+\). Here is a sketch.
By Remark 10, there is a vector bundle epimorphism \(L:A\times \mathbb{C}^N\to f_0^* VT(Z)\) of class \(\Lambda^r_\mathscr{O}(A\times\mathbb{C}^N)\) for some \(N\in \mathbb{N}\).
Let \(L\) be as above. There is a dominating spray \(F:A\times W\to Z\) of class \(\Lambda^r_{\mathscr{O}}(A\times W,Z)\), where \(0\in W\subset \mathbb{C}^N\) is a ball, such that \(F(\cdotp,0)=f_0\) and \[\label{eq:L} \partial_w|_{w=0} F(x,w) = L(x,\cdotp): \mathbb{C}^N \to VT_{f(x,0)} Z \;\;\text{for all x\in A}.\tag{29}\]
To see this, choose a sequence of compact, smoothly bounded, strongly pseudoconvex domains \[A_0 \subset A_1 \subset \cdots \subset A_m=A\] as in 28 such \(A_0\) is contained in the interior of \(A\) and for every \(k=0,1,\ldots,m-1\) we have \(A_{k+1}=A_k\cup B_k\), where \((A_k,B_k)\) is a special Cartan pair and the bundle \(Z\) is trivial over \(B_k\). The existence of a holomorphic spray \(F_0:A_0\times W_0\to Z|_{A_0}\) with the core \(F_0(\cdotp,0)=f_0|_{A_0}\) and satisfying 29 is standard; see [2]. We inductively find sprays \(F_k:A_k\times W_k \to Z|_{A_k}\) of classes \(\Lambda^r_{\mathscr{O}}(A_k\times W_k,Z)\) with the core \(f_0\) and satisfying 29 for \(x\in A_k\) \((k=1,\ldots,m)\), where \(W_0\supset W_1\supset\cdots\supset W_m\) are balls in \(\mathbb{C}^N\) centred at \(0\). Every induction step is of the same kind and proceeds as follows.
We first approximate \(F_k\) over \(C_k=A_k\cap B_k\) by a spray \(G_k:B_k\times V_k \to Z|_{B_k}\) of class \(\Lambda^r_{\mathscr{O}}(B_k\times V_k,Z)\) satisfying 29 for points \(x\in B_k\), where \(0\in V_k\subset W_k\) is a smaller ball; see [2] and note that its proof also applies in classes \(\Lambda^r\). In particular, \(F_k\) and \(G_k\) agree along \(C\times \{0\}\) to the second order. Next, we apply Lemma 6 to glue \(F_k\) and \(G_k\) into a spray \(F_{k+1}\) over \(A_{k+1}=A_k\cup B_k\) with the core \(f_0\) and satisfying condition 29 for all points \(x\in A_{k+1}\). This is done by first finding a map \(\gamma_k\) of the form 20 and of class \(\Lambda^r_\mathscr{O}(C_k\times V_k)\) which agrees with the identity to the second order along \(C_k\times\{0\}\) and satisfies \(F_k\circ \gamma_k = G_k\) on \(C_k\times V_k\); see 26 . (The ball \(V_k\) is allowed to shrink. The construction of such a map \(\gamma_k\) was explained in the proof of Lemma 6.) Next, we apply Lemma 5 to find sprays \(\alpha_k\) and \(\beta_k\) over \(A_k\) and \(B_k\) such that \(\alpha_k \circ \gamma_k =\beta_k\) (see 22 ), and \(\alpha_k\) and \(\beta_k\) agree with the identity to the second order along \(A_k\times \{0\}\) and \(B_k\times \{0\}\), respectively. As in 27 , this yields the spray \(F_{k+1}\) over \(A_{k+1}=A_k\cup B_k\), defined by the identity \[F_k \circ \alpha_k = G_k \circ \beta_k \;\; \text{on C_k\times \epsilon V_k}, \quad 0<\epsilon<1.\] By the construction, \(F_{k+1}\) has the core \(f_0\) and satisfies condition 29 on the set \(A_{k+1}\). Taking \(W_{k+1}=\epsilon V_k\) completes the induction step and proves the Claim.
This completes the proof of the basic case of Theorem 1.
We now state a general parametric version of Theorem 1. Recall that \(\Gamma(\overline{\Omega},Z)\) denotes the space of continuous sections of a topological fibre bundle \(h:Z\to \overline{\Omega}\). If the bundle is of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) then \(\Gamma^r_\mathscr{O}(\overline{\Omega},Z)\) denotes the space of sections of the same class.
Theorem 17. Assume that \(\Omega\) is as in Theorem 1, \(r>0\), \(h:Z\to \overline{\Omega}\) is a fibre bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) with Oka fibre, \(P\) is a compact Hausdorff space, and \(Q\) is a closed subset of \(P\) which is a strong neighbourhood deformation retract. Let \(f:P\to \Gamma(\overline{\Omega},Z)\) be a continuous map such that \(f|_Q:Q\to \Gamma^r_\mathscr{O}(\overline{\Omega},Z)\). Then there is a homotopy \(f_s:P\to \Gamma(\overline{\Omega},Z)\), \(s\in [0,1]\), which is fixed on \(Q\) such that \(f_0=f\) and \(f_1:P\to \Gamma^r_\mathscr{O}(\overline{\Omega},Z)\).
The proof follows the same scheme as that of Theorem 1, using the parametric versions of the tools in Sections 4 and 5. We leave the details to the reader.
In this section, we prove Theorem 4. We follow [2], which is due to Grauert [20]. See also Cartan’s exposition of Grauert’s Oka principle in [51].
Proof of Theorem 4. A topological vector bundle \(E \to \overline{\Omega}\) of rank \(m\) is the pullback \(f^*\mathbb{U}\) by a continuous map \(f\) from \(\overline{\Omega}\) to a suitable Grassmannian \(G(m,N)\) (consisting of complex \(m\)-planes in \(\mathbb{C}^N\)) of the universal bundle \(\mathbb{U}\to G(m,N)\) of rank \(m\). (We take \(N\) big enough such that \(E\) embeds as a topological vector subbundle of the trivial bundle \(\overline{\Omega} \times \mathbb{C}^N\).) Since \(G(m,N)\) is a complex homogeneous manifold, and hence an Oka manifold by Grauert [7], Theorem 1 shows that \(f\) is homotopic to a map \(F:\overline{\Omega} \to G(m,N)\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\). The pullback \(F^*\mathbb{U}\to \overline{\Omega}\) is then a vector bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) which is topologically isomorphic to \(E\cong f^*\mathbb{U}\). This proves part (i) of the theorem.
To prove the second statement, let \(E\to \overline{\Omega}\) and \(E' \to \overline{\Omega}\) be vector bundles of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) and rank \(m\). There are an open cover \(\{U_j\}\) of \({\overline{\Omega}}\) by smoothly bounded domains and vector bundle isomorphisms \[\theta_j: E|_{\overline{U}_j} \stackrel{\cong}{\longrightarrow} \overline{U}_j\times\mathbb{C}^m, \qquad \theta'_j: E'|_{\overline{U}_j} \stackrel{\cong}{\longrightarrow} \overline{U}_j\times\mathbb{C}^m\] of class \(\Lambda^r_\mathscr{O}(\overline{U}_j)\). Set \(U_{i,j}=U_i\cap U_j\). Let \[g_{i,j} : \overline{U}_{i,j} \to GL_m(\mathbb{C}), \qquad g'_{i,j} : \overline{U}_{i,j} \to GL_m(\mathbb{C})\] denote the fibrewise holomorphic transition maps of class \(\Lambda^r_\mathscr{O}(\overline{U}_{i,j})\) so that \[\theta_i \circ\theta_j^{-1}(x,v)= \bigl(x,g_{i,j}(x)v\bigr), \quad x \in \overline{U}_{i,j},\;v\in\mathbb{C}^m,\] and likewise for \(E'\). A complex vector bundle isomorphism \(\Phi : E \to E'\) is given by a collection of complex vector bundle isomorphisms \(\Phi_j: \overline{U}_j\times\mathbb{C}^m\to \overline{U}_j\times \mathbb{C}^m\) of the form \[\Phi_j(x,v)=\bigl(x,\phi_j(x)v\bigr), \quad x\in \overline{U}_j, \;v\in \mathbb{C}^m,\] with \(\phi_j(x)\in GL_m(\mathbb{C})\) for \(x\in \overline{U}_j\), satisfying the compatibility conditions \[\label{eq:bundle-P} \phi_i = g'_{i,j}\phi_j g_{i,j}^{-1} = g'_{i,j} \phi_j g_{j,i} \quad {\rm on}\;\overline{U}_{i,j}.\tag{30}\] Let \(h:Z\to {\overline{\Omega}}\) denote the fibre bundle of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) with fibre \(G=GL_m(\mathbb{C})\) and transition maps 30 . This means that \(Z|_{\overline{U}_j}\cong \overline{U}_j\times G\) for each \(j\), an element \((x,v) \in \overline{U}_j\times G\) for \(x\in \overline{U}_{i,j}\) is identified with \((x,v')\in \overline{U}_i\times G\) where \(v'= g'_{i,j}(x)\, v \, g_{j,i}(x)\), and no other identifications are made. A collection of maps \(\phi_j : \overline{U}_j\to G\) satisfying conditions 30 is then a section \({\overline{\Omega}}\to Z\).
This shows that complex vector bundle isomorphisms \(E\to E'\) correspond to sections of \(Z\to {\overline{\Omega}}\), with isomorphisms of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) corresponding to sections of the same class. Hence, part (ii) follows from Theorem 1. ◻
Similarly one can prove the following analogue of [2] due to Grauert [20]. In the proof, we use the analogue of Lemma 1 for an arbitrary complex Lie group \(G\); see Remark 11.
Theorem 18. Let \(\Omega\) be as in Theorem 4. For every complex Lie group \(G\), the isomorphism classes of principal \(G\) bundles on \(\overline{\Omega}\) of class \(\Lambda^r_\mathscr{O}(\overline{\Omega})\) are in bijective correspondence with the topological isomorphism classes of principal \(G\) bundles.
Remark 19. Grassmann manifolds \(G(m,N)\) play a major role in the theory of complex vector bundles as the classifying spaces. Being projective, they are Kähler manifolds. An explicit formula for a Kähler metric on \(G(m,N)\) was given by Lu Qi-Keng in 1963, see [52], [53].
Acknowledgements. Research was supported by the European Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291 and N1-0237 from ARIS, Republic of Slovenia.
I wish to thank Andrei Teleman for asking the question which led to the main results of the paper (private communication, January 2026), and for helpful communication regarding the Hölder–Zygmund spaces. I also thank Xianghong Gong for sharing with me his expertise on Hölder–Zygmund spaces. A part of the background work on the paper originates in my joint papers [12], [50] with Drinovec Drnovšek, whom I thank for this indirect contribution. A part of the work was done during my visit to Adelaide University in February 2026. I wish to thank Finnur Lárusson for the invitation and the mentioned institution for its hospitality.