Holomorphy tests based on Cauchy's integral formula
In this paper, \(X\) is either \(\mathbb{C}\), the open unit disk \(\mathbb{D}\) or the open upper half-plane \(\mathbb{U}\). \(G\) denotes the group \(M(2)\) of rigid motions of \(\mathbb{C}\) in the first case and in the other cases the full group of conformal automorphisms of \(X\). Cauchy's integral formula gives for rectifiable loops \(\Gamma\) in \(X\) \[ \int_\Gamma{(f\circ\sigma)(z)\over z-a} {dz\over 2\pi i}=\text{Ind}(\Gamma, a)\cdot(f\circ\sigma)(a)\tag{1} \] for all \(f\in H(X)\), \(\sigma\in G\), \(a\in X\backslash\Gamma\). This paper investigates for what Jordan curves \(\Gamma\) and \(a\in X\backslash\Gamma\) the validity of (1) for all \(\sigma\in G\) entails the holomorphy of \(f\) in \(X\). There are strong analogies with the Morera and Pompeiu problems and techniques used in their study are relevant here too. The authors acknowledge a particular debt to the work of \textit{C. A. Berenstein} and \textit{L. Zalcman} [Comment. Math. Helv. 55, 593-621 (1980; Zbl 0452.43012)] and \textit{C. A. Berenstein} and \textit{D. Pascuas} [Isr. J. Math. 86, No. 1-3, 61-106 (1994; Zbl 0827.30001)]. There are methodological differences between the case \(X=\mathbb{C}\) (which is invariant under \(M(2)\)) and the other cases (which are not invariant). That invariance is somehow reflected in the fact that in the planar case there are points, where (1) is automatically satisfied for certain non-holomorphic functions. For example, if \(\Gamma\) is a circle and \(a\) its center, every harmonic function \(f\) satisfies (1), so a conclusion ``\(f\) holomorphic'' is out of the question. Nevertheless, Theorem 1.1 asserts that if \(D\) is a (Euclidean) disk inside \(X\), \(\Gamma=\partial D\), \(a\in X\backslash\Gamma\) is not the center of \(D\) and \(f\in C(X)\) satisfies (1) for all \(\sigma\in G\), then \(f\) is holomorphic in \(X\). In proving Theorem 1.1 for \(X=\mathbb{U}\) the following mean-valued theorem is obtained. Corollary 1.2: Let \(D\) be a closed (Euclidean) disk in \(\mathbb{U}\) with center \(c\) and radius \(r\). Assume that \(f\in C(\mathbb{U})\) satisfies \[ \int_{\partial D}(f\circ\sigma)(z) {|dz|\over 2\pi}= (f\circ\sigma)(c)\tag{2} \] for every \(\sigma\in G\). Then \(f\) is harmonic. It is noteworthy that this is a ``one-radius'' theorem, whereas the known corresponding result in \(\mathbb{C}\) is a ``two-radius'' theorem if \(c=0\), reflecting the non-invariance under biholomorphic mappings of this mean-value property. For \(X=\mathbb{C}\) the principal non-holomorphic harmonic function \(\overline z\) has to be neutralized, so points \(a\) for which \(f(z):=\overline z\) and \(\sigma(z):=z\) satisfy (1) are labelled singular, and only non-singular (=admissible) \(a\) are allowed in the theorems about \(\mathbb{C}\). The authors' multi-parted Theorem (1.4) has \(X=\mathbb{C}\), \(\Gamma\) the boundary of a Jordan region \(\Omega\), and four different smoothness hypotheses on \(\Gamma\) and \(\Omega\) are each shown to be sufficient for the holomorphy of \(f\in C(X)\) to follow from the validity of (1) for all \(\sigma\in M(2)\) (and some admissible \(a\)). The whole aresenal of modern and classical analysis is brought to bear in the proofs: Stokes' theorem, approximation theorems of Keldysh and Runge, Paley-Wiener theorem, Schwartz' spectral synthesis theorem, homogeneous spaces, Haar measure, harmonic measure, distributions, spherical Fourier transforms, Bessel, Legendre and hypergeometric functions, elliptic operators, and ideas from the calculus of variations.
scientific article; zbMATH DE number 912136
scientific article