On a problem of Zalcman and its generalizations
The question under consideration in the paper is: Under what conditions the functions which are locally summable in the disk are holomorphic ones? Let be \(D= \{z\in \mathbb{C}\): \(| z|<1\}\); let also \(\gamma_ R=\{z\in \mathbb{C}\): \(| z|= R\}\). Suppose \(f\) to be locally summable functions on \(D\) \((f\in L_{\text{loc}}(D))\) such that for some fixed \(R\in (0,1)\) and almost all \(\tau\in G\) (with respect to the Haar measure) the following integral vanishes \[ \int_{\tau(\gamma_ R)} f(z) dz=0.\tag{1} \] Here \(\tau\) are the set of complex matrices \(\tau= \Bigl({\alpha\over \beta}{\beta\over \alpha}\Bigr)\), \(| \alpha|^ 2- |\beta|=1\) which constitute the group \(G\) of conformal automorphisms of the unit disk. Let be also \(m_ r(f)=\iint_{| z|\leq r} | f(z)|^ 2 dx dy\). Then the main result of the paper reads: Theorem: Let be \(f\in L_{\text{loc}}(D)\) and \(R\in (0,1)\). If for almost all \(\tau\in G\) condition (1) is fulfilled, then: 1) if \(m_ r(f)= o(\ln[1/1- r])\) as \(r\to 1\), then \(f\) coincides almost everywhere with a holomorphic function; 2) there is a nonholomorphic function \(f\in C^ \infty(D)\) satisfying (1), such that \(m_ r(f)= O(\ln[1/1- r])\). Also the other generalizations of the Zalcman's problem in restricted domains are investigated. The paper is closely connected to the previous works of \textit{L. Zalcman} [Arch. Rat. Mech. Anal. 47, 237-254 (1972; Zbl 0251.30047)] and of \textit{M. L. Agranovskij} [Dokl. Akad. Nauk SSR 243, 1353-1356 (1978; Zbl 0416.30001)] (in Russian).
conformal automorphisms of the unit disk
scientific article; zbMATH DE number 567870
scientific article