Abstract:
A Mandelbrojt region is a (not necessarily sectorial) connected open set $D$ in the Riemann surface of the logarithm with 0 in its boundary, symmetric with respect to a direction and with a positive opening (in a precise sense). In 1952, S. Mandelbrojt gave a characterization of the injectivity of the asymptotic Borel mapping for the class of functions admitting uniform $ \M $-asymptotic expansion in these domains, where the control of the remainders is specified in terms of a given weight sequence $\M$ of positive real numbers. His criterion consists of the divergence of an integral where the sequence $\M$ and the boundary of $D$ naturally appears. However, for the closely related Carleman-Roumieu ultraholomorphic class in $D$ consisting of the holomorphic functions with uniformly bounded derivatives in terms of the sequence $ \M $, no such general criterion is known. We will briefly discuss some partial results for unbounded sectors, for convex domains and for bounded domains, and give some hints about our approach in order to fully solve this problem in similar terms as in Mandelbrojt's result.