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Department of Mathematics and Statistics
of University of Helsinki

SOME RECENT ARTICLES
 J.Bonet, W.Lusky, J.Taskinen:
Solid cores and solid hulls of weighted Bergman spaces
Banach J. Math. Analysis. 13, 2 (2019), 468485.
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 nonatomic 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), 893900.
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 ogrowth conditions instead of Ogrowth 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), 52695278
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), 781804.
We extend the results of the Ann.Acad. and Iberoamericanapapers (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), 521530
We adapt the results of the Iberoamericanapaper (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_Xmodules 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^2estimates, 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 codimensionthree 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), 577615.
It is wellknown 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 supnorms.
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.
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