We study the following integral operators: J g f (z)= z 0 f (ξ)g (ξ)dξ; I g f (z)= z 0 f (ξ)g(ξ)dξ, where g is an analytic function on the open unit disk in the complex plane. The bounded-ness and compactness of J g , I g between the Bergman-type spaces and the α-Bloch spaces are investigated.

In [V. Mathai, K-theory of twisted group C∗-algebras and positive scalar curvature, Contemp. Math. 231 (1999) 203–225], we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz., the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundame...

We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamental result of Friedlander-Suslin, we globalize the Bloch-Lichtenbaum spectral sequence to give a spectral s...

Let φ and ψ be analytic self-maps of the open unit disk D . Using pseudo-hyperbolic distance ρ(φ ,ψ) , we characterize the boundedness and compactness of the differences of generalized composition operators (C φ −Ch ψ ) f (z) = ∫ z 0 [ f ′(φ(ξ ))g(ξ )− f ′(ψ(ξ ))h(ξ )]dξ , z ∈ D between two Bloch-type spaces on D . The results generalize the corresponding results on the single generalized compo...

Suppose / is a holomorphic function on the open unit ball Bn of Cn. For 1 < p < oo and to > 0 an integer, we show that / is in Lp(Bn,dV) (with dV the volume measure) iff all the functions dmf/dza (\a\ = to) are in Lp(Bn,dV). We also prove that / is in the Bloch space of Bn iff all the functions dmf/dza (\a\ = m) are bounded on Bn. The corresponding result for the little Bloch space of Bn is est...