For 0 < p < ∞ and α > −1, we let D p α denote the space of those functions f which are analytic in the unit disc D = {z ∈ C: |z| < 1} and satisfy D (1 − |z| 2 ) α |f (z)| p dx dy < ∞. In this paper we characterize the positive Borel measures μ in D such that D p α ⊂ L q (dμ), 0 < p < q < ∞. We also characterize the pointwise multipliers from D p α to D q β (0 < p < q < ∞) if p − 2 < α < p. In particular, we prove that if (2 + α)/p − (β + 2)/q > 0 the only pointwise multiplier from D p α to D q β (0 < p < q < ∞) is the trivial one. This is not longer true for (2 + α)/p − (β + 2)/q 0 and we give a number of explicit examples of functions which are multipliers from D p α to D q β for this range of values.
If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let
$\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu_{n, k})_{n,k\ge
0}$ with entries $\mu_{n, k}=\mu_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $,
$\mu_n$ denotes the moment of orden $n$ of $\mu $. This matrix induces formally
the operator $$\mathcal{H}_\mu (f)(z)=
\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the
space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit
disc $\D $. This is a natural generalization of the classical Hilbert operator.
The action of the operators $H_{\mu }$ on Hardy spaces has been recently
studied. This paper is devoted to study the operators $H_\mu $ acting on
certain conformally invariant spaces of analytic functions on the disc such as
the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.Comment: 24 page
Abstract. We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov, and other authors.
The following space will be useful because of its geometric interpretation:Namely, by a theorem of Smirnov and Riesz, a univalent map F of D onto a simply connected domain W will belong to Lð1; 1Þ if and only if the boundary curve qW is rectifiable ([D1], Theorem 3.12). The result is often formulated in the literature only for Jordan domains W, but a compactness and covering argument allows us to extend it easily to ar-Donaire, Girela and Vukotić, Möbius invariant spaces 44 Brought to you by |
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