  A new construction of linear continuous right inverses for the asymptotic
Borel map is provided in the framework of general Carleman ultraholomorphic
classes in narrow sectors. Such operators were already obtained by V. Thilliez
by means of Whitney extension results for non quasianalytic ultradifferentiable
classes, due to J. Chaumat and A. M. Chollet, but our approach is completely
different, resting on the introduction of a suitable truncated Laplace-type
transform. This technique is better suited for a generalization of these
results to the several variables setting. Moreover, it closely resembles the
classical procedure in the case of Gevrey classes, so indicating the way for
the introduction of a concept of summability which generalizes $k-$summability
theory as developed by J. P. Ramis.
