Advanced Search

Let D be the open unit disc and μ a positive bounded measure on [0,1]. Extending results of Mateljević/Pavlović and Shields/Williams we give Banach-space descriptions of the classes of all harmonic (holomorphic) functions f: D → ℂ satisfying ${ʃ}_0^1({ʃ}_0^{2\pi }|f\left(r{e}^{i\phi }\right){{|}^{p}d\phi )}^{q/p}d\mu \left(r\right)<\infty$.

We show that every separable complex L₁-predual space X is contractively complemented in the CAR-algebra. As an application we deduce that the open unit ball of X is a bounded homogeneous symmetric domain.

Let ${Rₙ}_{n=1}^{\infty }$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an ${ℒ}_p$-space, then both X and A have bases. We apply these results to show that the spaces $C_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset C\left(\right)$ and $L_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset L₁\left(\right)$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $Hv=f:\Omega \to ℂholomorphic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ and investigate some isomorphism classes of $hv=f:\Omega \to ℂharmonic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $Hv\sim l_{\infty }$ or $Hv\sim H_{\infty }$, and at least two possibilities for hv, again $hv\sim l_{\infty }$ and $hv\sim H_{\infty }$. We also discuss many new examples of weights.

It is proved that for any $f,g\in A\left(S\right)$, where $S$ is the Poulsen simplex, so that $f,g$ and $1-f$, $1-g$ are smooth points, there is a rotation of $A\left(S\right)$ carrying $f$ in $g$.

We give necessary and sufficient conditions on the weights v and w such that the differentiation operator D: Hv(Ω) → Hw(Ω) between two weighted spaces of holomorphic functions is bounded and onto. Here Ω = ℂ or Ω = 𝔻. In particular we characterize all weights v such that D: Hv(Ω) → Hw(Ω) is bounded and onto where w(r) = v(r)(1-r) if Ω = 𝔻 and w = v if Ω = ℂ. This leads to a new description of normal weights.

We study Toeplitz operators $T_a$ with radial symbols in weighted Bergman spaces $A_{\mu }^p$, 1 < p < ∞, on the disc. Using a decomposition of $A_{\mu }^p$ into finite-dimensional subspaces the operator $T_a$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_a$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_a$ for a satisfying an assumption on the positivity of certain indefinite...

We study the spaces $H_{\mu }\left(\Omega \right)=f:\Omega \to ℂholomorphic:{\int }_0^R{\int }_0^{2\pi }\left|f\left(r{e}^{i\phi }\right)\right|d\phi d\mu \left(r\right)<\infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{\mu }\left(\Omega \right)$ is either isomorphic to l₁ or to ${\left(\sum \oplus Aₙ\right)}_{\left(1\right)}$. Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

Let v be a standard weight on the upper half-plane , i.e. v: → ]0,∞[ is continuous and satisfies v(w) = v(i Im w), w ∈ , v(it) ≥ v(is) if t ≥ s > 0 and $li{m}_{t\to 0}v\left(it\right)=0$. Put v₁(w) = Im wv(w), w ∈ . We characterize boundedness and surjectivity of the differentiation operator D: Hv() → Hv₁(). For example we show that D is bounded if and only if v is at most of moderate growth. We also study composition operators on Hv().

This work is an introduction to anisotropic spaces of holomorphic functions, which have $\omega$-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding $L_{\omega }^{\infty }$ space. We establish a description of ${\left(A^p\left(\omega \right)\right)}^{*}$ via the Bloch classes for all $0<p\le 1$.

The $\omega$-weighted Besov spaces of holomorphic functions on the unit ball $B^n$ in $C^n$ are introduced as follows. Given a function $\omega$ of regular variation and $0<p<\infty$, a function $f$ holomorphic in $B^n$ is said to belong to the Besov space $B_p\left(\omega \right)$ if $${\parallel f\parallel }_{B_p\left(\omega \right)}^p={\int }_{B^n}{(1-|z|}^2{)}^p{\left|Df\left(z\right)\right|}^p\frac{\omega (1-|z\left|\right)}{{(1-|z|}^2{)}^{n+1}}d\nu \left(z\right)<+\infty ,$$ where $d\nu \left(z\right)$ is the volume measure on $B^n$ and $D$ stands for the fractional derivative of $f$. The holomorphic Besov space is described in the terms of the corresponding $L_p\left(\omega \right)$ space. Some projection theorems and theorems on existence of the inversions of these projections are proved. Also,...