Helsingin Yliopisto


Structure of analytic function spaces and sheaves.

Main page

Department of Mathematics and Statistics of University of Helsinki


  • J.Bonet, W.Lusky, J.Taskinen:
    Solid cores and solid hulls of weighted Bergman spaces
    Banach J. Math. Analysis. 13, 2 (2019), 468-485.

    We determine the solid hull for 2 < p < \infty and the solid core for 1 < p < 2 of weighted reflexive Bergman spaces A_v^p of analytic functions functions on the disc and on the entire complex plane. The results are for a very general class of non-atomic positive bounded Borel measures v. We show that the space A_v^p is solid if and only if the monomials form an unconditional (Schauder) basis of this space.

  • J.Bonet, W.Lusky, J.Taskinen:
    Distance formulas on weighted Banach spaces of analytic functions.
    Complex Anal. Operator Th. 13,3 (2019), 893-900.

    We consider weighted H^\infty spaces on the unit disc with a general class of radial weights, and prove a formula for the distance of an element of this space from the subspace H_0^\infty (the smaller space defined using o-growth conditions instead of O-growth conditions.) The result also gives a simplified proof for a distance formula by K.-M. Perfect in Ark.Math (2013).
    Here is a preprint.

  • J.Bonet, W.Lusky, J.Taskinen:
    Monomial basis in Korenblum type spaces of analytic functions.
    Proc.Amer.Math.Soc. 146, 12 (2018), 5269-5278

    By a Korenblum space A^d, d > 0 or d=0, we mean the Fréchet space of analytic functions on the unit disc, which is obtained as the countable intersection of weighted H^\infty spaces with an inreasing sequence of weights (1 - |z|)^b, where b > d. We show that the monomials form a Schauder basis in the space A^d. The same is true for the corresponding (LB)-spaces (inductive limits formed using the same weighted step spaces). We provide sequence space representations for these spaces as Köthe spaces.
    Here is a preprint.

  • J.Bonet, W.Lusky, J.Taskinen:
    Solid hulls and cores of weighted $H^\infty$-spaces.
    Rev. Mat. Compl. 31 (2018), 781-804.

    We extend the results of the Ann.Acad. and Iberoamericana-papers (see below) especially in the case of the disc by presenting a different method for the calculation of the numerical quantities. This leads to a larger class of concrete cases than in the previous paper.
    Here is a preprint.

  • J.Bonet, J.Taskinen:
    Solid hulls of weighted Banach spaces of analytic functions on the unit disc with exponential wieghts.
    Ann.Acad.Sci.Fenn. 43 (2018), 521-530

    We adapt the results of the Iberoamericana-paper (see below) to the case of the unit disc instead of the entire complex plane. It turns out that even in the simplest examples of exponentially decreasing weights (as functions of the boundary distance), the critical radii can only be estimated, contrary to the case of the entire plane, where it was possible to make explicit calculations. This makes the calculation of the solid hulls more difficult, and our concrete examples are more restricted than in the planar case.
    The general result on solid hulls requires the weight to satisfy a condition (b), which is related to a known weight condition, namely the one introduced by W.Lusky and called condition (B) by him.
    Here is a preprint.

  • J.Taskinen, K.Vilonen:
    Cartan theorems for Stein manifolds over a discrete valuation base.
    J.Geometric Anal. 29,1 (2019), 577{615

    We prove Cartan theorems A and B for for coherent A_X-modules when X is a Stein manifold, A_X is the sheaf of functions on X with values in A; moreover, A is a topological discrete valuation ring, which is topologized as a nuclear direct (or inductive) limit of Banach algebras. The results require a combination of several techniques including new vector valued L^2-estimates, which generalize those of Hörmander in the theory of several complex variables. The study is motivated by questions that arouse in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic micro differential systems.
    Here is a preprint.

  • J.Bonet, J.Taskinen:
    Solid hulls of weighted Banach spaces of entire functions.
    Rev.Mat.Iberoamericana 29,1 (2019), 577-615.

    It is well-known that it is often impossible to characterize standard Banach spaces of entire functions in terms of the Taylor coefficients; this is for example true for the function spaces H^\infty_v(C) of entire functions on the complex plane, endowed with weighted sup-norms. The next best thing is to find the strongest growth condition that the coefficients have to satisfy, i.e. to find the solid hull of the given space. In this paper we determine the solid hull for the above mentioned spaces for a large class of rapidly decreasing weights. Our method uses, among other things, the methods of W.Lusky.
    Here is a preprint.

Back to main page .