Jouni Rättyä

Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Complex linear differential equations of the form with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient a j (z) of (†) belongs to the weighted Bergman space A 1 k-j α where a > 0, for all j = 0,... k - 1, then all solutions are of order of growth at most a, measured according to the Nevanlinna characteristic. In the case when a = 0 all solutions are shown to be not only of order of growth zero, but of bounded characteristic. Conversely, if all solutions are of order of growth at most a > 0, then the coefficient aj (z) is shown to belong to A pj α for all p j ∈ (0, 1 k-j) and j = 0,..., k - 1. Analogous results, when the coefficients belong to certain weighted Hardy spaces, are obtained. The non-homogeneous equation associated to (†) is also briefly discussed.

Linear differential equations with solutions in Hardy spaces

It is shown that there is a positive constant α, depending only on p and k, such that if the analytic coefficients A j (z) of the linear differential equation satisfy and for some δ [{0, 1), then all solutions of (A) belong to the Hardy space H p . This completes in part some earlier results by Pommerenke, and Heittokangas, Korhonen and the author.

Inner functions in the Möbius invariant Besov-type spaces

An analytic function f in the unit disc belongs to F ( p,q,s ), if is uniformly bounded for all a ∈ . Here is the Green function of , and φ a ( z )=( a−z )/(1−ā z ). It is shown that for 0 w |=1 the singular inner function exp(γ( z+w )/( z−w )) belongs to F ( p,q,s ), 0 s ≤1, if and only if . Moreover, it is proved that, if 0 s for some (equivalently for all) p > max{ s ,1− s } if and only if it is a Blaschke product whose zero sequence { z n } satisfies .

Mean growth and geometric zero distribution of solutions of linear differential equations

The aim of this paper is to consider certain conditions on the coefficient A of the differential equation f″ + Af = 0 in the unit disc which place all normal solutions f in the union of Hardy spaces or result in the zero-sequence of each non-trivial solution being uniformly separated. The conditions on the coefficient are given in terms of Carleson measures.

On the boundedness of Bergman projection

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces

Inclusion Relations for New Function Spaces on Riemann Surfaces

A^p_\omega

On α-Polynomial Regular Functions, with Applications to Ordinary Differential Equations

which are useful in the study of this question. In particular, we shall focus on a decomposition norm theorem for radial weights~

Weighted Yosida functions

\omega

Results on meromorphic ϕ-normal functions

with the doubling property

Maximal property of the capacity density Bloch space for on Riemann surfaces

\int_{r}^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds