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The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity satisfies
are considered on these boundaries. Functions satisfying the Hölder condition of order , , belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise...
Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping and that of , we study various norms for , where is the Toeplitz operator with symbol . In Theorem , given polynomials and we find a symbol such that . We extend some of our results to the polydisc.
The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concerning zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.
When treating spaces of holomorphic functions with growth conditions, one is led to introduce associated weights. In our main theorem we characterize, in terms of the sequence of associated weights, several properties of weighted (LB)-spaces of holomorphic functions on an open subset which play an important role in the projective description problem. A number of relevant examples are provided, and a “new projective description problem” is posed. The proof of our main result can also serve to characterize...
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