We prove that, for every α > −1, the pull-back measure φ(Aα) of the measure dAα(z) = (α+1)(1− |z| ) dA(z), where A is the normalized area measure on the unit disk D, by every analytic self-map φ : D → D is not only an (α + 2)-Carleson measure, but that the measure of the Carleson windows of size εh is controlled by ε times the measure of the corresponding window of size h. This means that the p...

We characterize the space of multipliers between certain weighted Besov spaces of analytic functions. This extend and give a new proof of a result of Wojtaszczyk about multipliers between Bergman spaces. Introduction. P. Wojtaszczyk [W], using certain factorization theorems due to Maurey and Grothendieck, proved the following results: Let α > 0, 0 < p ≤ 2 ≤ q < ∞ and 1r = 1 p − 1q . (0.1) (Bq, ...

We develop a theory of removable singularities for the weighted Bergman space Aμ(Ω) = {f analytic in Ω : R Ω |f | dμ < ∞}, where μ is a Radon measure on C. The set A is weakly removable for Aμ(Ω \ A) if Aμ(Ω \ A) ⊂ Hol(Ω), and strongly removable for Aμ(Ω \A) if Aμ(Ω \A) = Aμ(Ω). The general theory developed is in many ways similar to the theory of removable singularities for Hardy H spaces, BMO...

For a, b, c, d ≥ 0 with ad − bc > 0, we consider the unilateral weighted shift S(a, b, c, d) with weights αn := √ an+b cn+d (n ≥ 0). Using Schur product techniques, we prove that S(a, b, c, d) is always subnormal; more generally, we establish that for every p ≥ 1, all p-subshifts of S(a, b, c, d) are subnormal. As a consequence, we show that all Bergman-like weighted shifts are subnormal.

Let Ω be a non-compact complex manifold of dimension n, ω = ∂∂Ψ a Kähler form on Ω, and Kα(x, y) the reproducing kernel for the Bergman space Aα of all analytic functions on Ω square-integrable against the measure e−αΨ|ωn|. Under the condition Kα(x, x) = λαe αΨ(x) F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109–1163] was able to establish a quantization procedure on (Ω, ω) which has recently...

It is proved that on any bounded domain in the complex Euclidean space C(n) the Bergman metric is always greater than or equal to the Carathéodory distance. This leads to a number of interesting consequences. Here two such consequences are given. (i) The Bergman metric is complete whenever the Carathéodory distance is complete on a bounded domain. (ii) The Weil-Petersson metric is not uniformly...

Let ϕ be a real-valued plurisubharmonic function on [Formula: see text] whose complex Hessian has uniformly comparable eigenvalues, and let [Formula: see text] be the Fock space induced by ϕ. In this paper, we conclude that the Bergman projection is bounded from the pth Lebesgue space [Formula: see text] to [Formula: see text] for [Formula: see text]. As a remark, we claim that Bergman projecti...

Preface The book is devoted to the spectral theory of commutative C *-algebras of Toeplitz operators on Bergman spaces, and its applications. For each such commutative algebra we construct a unitary operator which reduces each Toeplitz operator from this algebra to a certain multiplication operator, thus also providing its spectral type representation. This gives us a powerful research tool all...