We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. Thus, we extend the powerful multisummability theory for finitely many Gevrey levels, developed by J.-P. Ramis, J. Écalle and W. Balser, among others. We provide both the analytical and cohomological approaches, and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized Laplace-like operators.

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We discuss a version ofÉcalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Ω-continuability, where Ω is a discrete filtered set or a dicrete doubly filtered set, and show how to construct a universal Riemann surface X Ω whose holomorphic functions are in one-to-one correspondence with Ωcontinuable functions. We then discuss the Ω-continuability of convolution products and give estimates for iterated convolutions of the formφ 1 * · · · * φ n . This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.

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