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Painlevé null sets, dimension and compact embedding of weighted holomorphic spaces

Alexander V. Abanin;  Pham Trong Tien — 2012

Studia Mathematica

We obtain, in terms of associated weights, natural criteria for compact embedding of weighted Banach spaces of holomorphic functions on a wide class of domains in the complex plane. Our study is based on a complete characterization of finite-dimensional weighted spaces and canonical weights for them. In particular, we show that for a domain whose complement is not a Painlevé null set each nontrivial space of holomorphic functions with O-growth condition is infinite-dimensional.

Continuation of holomorphic functions with growth conditions and some of its applications

Alexander V. Abanin;  Pham Trong Tien — 2010

Studia Mathematica

We prove a generalization of the well-known Hörmander theorem on continuation of holomorphic functions with growth conditions from complex planes in ℂ p into the whole ℂ p . We apply this result to construct special families of entire functions playing an important role in convolution equations, interpolation and extension of infinitely differentiable functions from closed sets. These families, in their turn, are used to study optimal or canonical, in a certain sense, weight sequences defining inductive...

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