Radon inversion problem for holomorphic functions on strictly pseudoconvex domains (Q617756)

Let \(\Omega\) be a bounded, strictly convex domain with boundary of class \(C^2 .\) Let \(X\) be a compact subset of \(\partial\Omega.\) We say that a continuous function \(\gamma : X \times [0,1] \ni (z, t) \to \gamma(z, t)\in\overline{\Omega}\) defines a set of real directions on \(\Omega\) if \(\gamma\) has the following properties: 1. \(\gamma(X \times [0,1))\subset \Omega\). 2. \(\gamma(X \times \{1\})\subset \partial\Omega\). 3. \(\frac{\partial\gamma}{\partial t}(\circ,\circ)\) is a continuous function on \(X \times [0,1]\). 4. There exist constants \(c_1, c_2 >0\) such that \(c_1\|z-\overline{\xi}\|\leq \|\gamma(z, 1)-\gamma(\overline{\xi}, 1)\|\leq c_2\|z-\overline{\xi}\|\) for \(z, \xi \in X\). 5. \(\gamma(\xi,\circ)\) is tangent to \(\partial\Omega\) at \(\gamma(\xi,1)\). Main Theorem. There exists a natural number \(K\) such that, if \(\varepsilon \in (0,1),\) \( T\) is a compact subset of \(\overline{\Omega}\setminus X\) and \(H\) is a continuous, strictly positive function on \(X,\) then there exist holomorphic entire functions \(f_1, \dots, f_K\) such that \(\|f_j\|_T < \varepsilon,\) and one has for \(z \in X\) the following inequality \[ \frac{H(z)}{4}<\max_{j=1,\dots,K}\int_0^1|f_j(\gamma(z,t))|^p\,dt<H(z). \] Several consequences of the above theorem for strictly pseudoconvex domains are given. In particular (Theorem 4.7), for a given lower semicontinuous, strictly positive function \(H\) on \(\partial\Omega\) the author constructs a holomorphic function \(f\in O(\Omega)\) such that \(H(z)=\int_0^1|f_j(\gamma(z,t))|^p\,dt\) for \(\eta\)-almost all \(z\in \partial\Omega\) where \(\eta\) is a given probability measure on \(\partial\Omega.\) As an application of Theorem 4.7, the author gives a description of so called exceptional sets (Theorem 4.8) \(E^p_\Omega(f):= \{\overline{\xi}\in \partial\Omega : \int_0^1|f_j(\gamma(\overline{\xi},t))|^p\,dt = \infty\}.\)