Interpolating Sequences for Besov Spaces

Abstract

We characterise the interpolating sequences for the Besov spaces B p and for their multiplier spaces. We also construct linear operators of interpolation.

Interpolating sequences on analytic Besov type spaces

“…Another result which is relevant to the present paper is that concerning the multiplier space M (B 2 (0)) of the Dirichlet space B 2 (0), whose interpolating sequences were characterized in the very deep paper [19] (see also [3]). The extension to M (B p (0)), 1 < p < ∞, is far from trivial and it is carried out in [6]. In the s = 0 case the Blaschke products play no role, which does not come as a surprise because, for instance, B 2 (0), hence M (B 2 (0)), does not contain infinite Blaschke products.…”

“…Note that, since the real part of the derivative of the reproducing kernels is not always positive, we can not use the generalization of the standard Hilbert space techniques developed in [6] to prove that condition (M) implies conditions (RS) and (UIS). Instead of that, since it is standard that conditions (RS) and (UIS) are equivalent (see, e.g.…”

Function spaces related to the Dirichlet space

“…The equivalence of the first three statements was shown in manuscripts circulated by Marshall and Sundberg [15] and by Bishop [6]. The first published proof is due to Böe [7]. Our contribution is the equivalence of the last statement.…”

On the growth and range of functions in Möbius invariant spaces

“…Using the above-mentioned result of Newman, we know that it is a finite union of exponential sequences. Let us set as before (7) b n (z) = |a n | a n a n − z 1 − a n z ,…”

“…They are also important in view of their relationship to the Bergman projection and Hankel operators on Bergman spaces. These spaces and their operators were studied extensively in the mid 80's and early 90's in [4], [21], [22], [38] and, more recently, in [36], [8], [11], [35], [7], and [9], especially from a geometric point of view.…”