Smoothness of a holomorphic function in a ball and of its modulus on the sphere. (English. Russian original) Zbl 1394.32004
J. Math. Sci., New York 229, No. 5, 568-571 (2018); translation from Zap. Nauchn. Semin. POMI 447, 123-128 (2016).
Let \(B^n\) be the unit ball in \(\mathbb C^n,\) \(n\geq 2.\) Denote by \(H^\alpha(S^n)\) and \(H^\alpha(\overline{B}^n),\) \(0 < \alpha\leq 1,\) where \(S^n = \partial B^n,\) the spaces of functions \(\varphi\) defined on \(S^n\) and \(B^n,\) respectively, such that
\[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\alpha, \quad z,\zeta\in S^n \text{ and } z,\zeta\in \overline{B}^n,\text{ respectively}, \]
with the seminorms \[ ||\varphi||_{\alpha,S^n} \overset{\mathrm{def}}{=}\sup_{z,\zeta\in S^n, z\neq\zeta}\frac{|\varphi(z)-\varphi(\zeta)|}{||z-\zeta||^\alpha}\quad \text{and}\quad ||\varphi||_{\alpha,\overline{B}^n}\overset{\mathrm{def}}{=}\sup_{z,\zeta\in \overline{B}^n, z\neq\zeta}\frac{|\varphi(z)-\varphi(\zeta)|}{||z-\zeta||^\alpha}. \]
The author proves the following theorem.
Theorem 1. Assume that a function \(f\) is holomorphic in the ball \(B^n,\) \(n\geq 2,\) and continuous in the closed ball \(\overline{B}^n.\) Assume, in addition, that \(f(z) \neq 0,\) \(z \in B^n,\) and \(|f| \in H^\alpha(S^n),\) \(0 < \alpha \leq 1.\) Then \(f \in H^\frac{\alpha}{2}(\overline{B}^n).\)
The author also proves an analogue of Theorem 1 for domains of a more general form. Let \(\Omega\) be a bounded convex domain in \( \mathbb C^n\), \(n\geq 2\), with boundary of class \(C^2\).
Denote by \(H^\alpha(\partial\Omega)\) the set of functions \(\varphi\) defined on \(\partial\Omega\) such that \[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\alpha, \quad z,\zeta\in \partial\Omega, \quad 0 < \alpha \leq 1, \] and let \(H^\alpha(\overline{\Omega})\) be the set of functions \(\varphi\) defined on \(\overline{\Omega}\) such that
\[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\beta, \quad z,\zeta\in \partial\Omega,\quad 0 < \beta \leq 1. \]
Theorem 2. Let \(\Omega\) satisfy the conditions above. Consider a function \(f\) that is holomorphic in \(\Omega\) and continuous in \(\overline{\Omega}\), and let \(f(z) \neq 0\), \(z \in \Omega\). If \(|f| \in H^\alpha(\partial\Omega)\), then \(f \in H^\frac{\alpha}{2}(\overline{\Omega}).\)
\[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\alpha, \quad z,\zeta\in S^n \text{ and } z,\zeta\in \overline{B}^n,\text{ respectively}, \]
with the seminorms \[ ||\varphi||_{\alpha,S^n} \overset{\mathrm{def}}{=}\sup_{z,\zeta\in S^n, z\neq\zeta}\frac{|\varphi(z)-\varphi(\zeta)|}{||z-\zeta||^\alpha}\quad \text{and}\quad ||\varphi||_{\alpha,\overline{B}^n}\overset{\mathrm{def}}{=}\sup_{z,\zeta\in \overline{B}^n, z\neq\zeta}\frac{|\varphi(z)-\varphi(\zeta)|}{||z-\zeta||^\alpha}. \]
The author proves the following theorem.
Theorem 1. Assume that a function \(f\) is holomorphic in the ball \(B^n,\) \(n\geq 2,\) and continuous in the closed ball \(\overline{B}^n.\) Assume, in addition, that \(f(z) \neq 0,\) \(z \in B^n,\) and \(|f| \in H^\alpha(S^n),\) \(0 < \alpha \leq 1.\) Then \(f \in H^\frac{\alpha}{2}(\overline{B}^n).\)
The author also proves an analogue of Theorem 1 for domains of a more general form. Let \(\Omega\) be a bounded convex domain in \( \mathbb C^n\), \(n\geq 2\), with boundary of class \(C^2\).
Denote by \(H^\alpha(\partial\Omega)\) the set of functions \(\varphi\) defined on \(\partial\Omega\) such that \[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\alpha, \quad z,\zeta\in \partial\Omega, \quad 0 < \alpha \leq 1, \] and let \(H^\alpha(\overline{\Omega})\) be the set of functions \(\varphi\) defined on \(\overline{\Omega}\) such that
\[ |\varphi(z)-\varphi(\zeta)|\leq C_\varphi ||z-\zeta||^\beta, \quad z,\zeta\in \partial\Omega,\quad 0 < \beta \leq 1. \]
Theorem 2. Let \(\Omega\) satisfy the conditions above. Consider a function \(f\) that is holomorphic in \(\Omega\) and continuous in \(\overline{\Omega}\), and let \(f(z) \neq 0\), \(z \in \Omega\). If \(|f| \in H^\alpha(\partial\Omega)\), then \(f \in H^\frac{\alpha}{2}(\overline{\Omega}).\)
Reviewer: Peter Dovbush (Kishinev)
MSC:
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
References:
| [1] | V. P. Khavin and F. A. Shamoyan, “Analytic functions with a Lipschitzian modulus of the boundary values,” Zap. Nauchn. Semin. LOMI, 19, 237-239 (1970). · Zbl 0247.30025 |
| [2] | S. V. Kislyakov, private communication. |
| [3] | N. A. Shirokov, “Analytic functions smooth up to the boundary,” Lect. Notes Math., 1312, (1988). · Zbl 0656.30029 |
| [4] | U. Rudin, Function Theory in the Unit Ball of Cn [Russian translation], Moscow (1984). · Zbl 0597.32001 |
| [5] | A. Zygmund, Trigonometric Series, vol. 1 [Russian translation], Moscow (1965). · Zbl 0131.06703 |
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