Interpolating sequences for the Nevanlinna and Smirnov classes

arXiv (Cornell University), 2014

A sequence which is a finite union of interpolating sequences for H ∞ have turned out to be especially important in the study of Bergman spaces. The Blaschke products B(z) with such zero sequences have been shown to be exactly those such that the multiplication f → f B defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem. We will use D(z, r) for the pseudohyperbolic disk of radius r centered at z, that is, the ball of radius r < 1 in the pseudohyperbolic metric. Let Z = {z k : k = 1, 2, 3,. .. } be a sequence in D without limit points in D. Define the space of sequences l p Z , 0 < p ≤ ∞, to be all those w = (w k) such that

2013

We prove, by using techniques similar to those in [3], that the interpolation space A ρ,Φ contains a copy of the Orlicz sequence space h Φ. Here ρ is a parameter function and Φ is an Orlicz function.

2010

We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined by the norm

We present some inequalities for the Taylor coefficients of a Hardy-Orlicz function, which in the case of the standard Hardy spaces reduce to the inequalities of Hardy and Littlewood. The proofs are new and elementary even in the case of the classical Hardy spaces. We also prove a Marcinkiewicz type theorem that extends a theorem of Kislyakov and Xu. At the end, we prove a Hardy-Prawitz criterion for membership of a univalent function in a Hardy-Orlicz space.

Divulgaciones Matematicas

We prove, by using techniques similar to those in (3), that the inter- polation space A‰;&#39; contains a copy of the Orlicz sequence space h&#39;: Here ‰ is a parameter function and &#39; is an Orlicz function.

Indiana University Mathematics Journal, 2009

We characterize the interpolating sequences for the weighted analytic Besov spaces Bp(s), defined by the norm f p B p (s) = |f (0)| p + D |(1 − |z| 2)f (z)| p (1 − |z| 2) s dA(z) (1 − |z| 2) 2 , 1 < p < ∞ and 0 < s < 1, and for the corresponding multiplier spaces M (Bp(s)).

Journal of Functional Analysis, 1991

Annali di Matematica Pura ed Applicata, 1984

The Journal of Geometric Analysis, 2017

For any p ∈ (0, 1], let H Φ p (R n) be the Musielak-Orlicz Hardy space associated with the Musielak-Orlicz growth function Φ p , defined by setting, for any x ∈ R n and t ∈ [0, ∞), Φ p (x, t) :=                t log (e + t) + [t(1 + |x|) n ] 1−p when n(1/p − 1) N ∪ {0}; t log(e + t) + [t(1 + |x|) n ] 1−p [log(e + |x|)] p when n(1/p − 1) ∈ N ∪ {0}, which is the sharp target space of the bilinear decomposition of the product of the Hardy space H p (R n) and its dual. Moreover, H Φ 1 (R n) is the prototype appearing in the real-variable theory of general Musielak-Orlicz Hardy spaces. In this article, the authors find a new structure of the space H Φ p (R n) by showing that, for any p ∈ (0, 1], H Φ p (R n) = H φ 0 (R n) + H p W p (R n) and, for any p ∈ (0, 1), H Φ p (R n) = H 1 (R n) + H p W p (R n), where H 1 (R n) denotes the classical real Hardy space, H φ 0 (R n) the Orlicz-Hardy space associated with the Orlicz function φ 0 (t) := t/ log(e + t) for any t ∈ [0, ∞) and H p W p (R n) the weighted Hardy space associated with certain weight function W p (x) that is comparable to Φ p (x, 1) for any x ∈ R n. As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.

Hacettepe Journal of Mathematics and Statistics, 2017

We introduce new spaces that are extensions of the Hardy spaces and prove a removable singularity result for holomorphic functions within these spaces. Additionally we provide non-trivial examples.