We prove, for an arithmetic scheme X/S over a discrete valuation ring whose special fiber is a strict normal crossings divisor in X, that the Swan conductor of X/S is equal to the Euler characteristic of the torsion in the logarithmic de Rham complex of X/S. This is a precise logarithmic analog of a theorem by Bloch [1].

We consider the space B logα , of analytic functions on the unit disk D, defined by the requirement ∫ D |f (z)|φ(|z|) dA(z) < ∞, where φ(r) = log(1/(1 − r)) and show that it is a predual of the “log-Bloch” space and the dual of the corresponding little Bloch space. We prove that a function f(z) = ∑ ∞ n=0 anz with an ↓ 0 is in B1logα iff ∑ ∞ n=0 log(n+2)/(n+1) < ∞ and apply this to obtain a crit...

In this article, we study the space $\mathcal B_\mu(B_X,Y)$ of $Y$-valued Bloch-type functions on unit ball $B_X$ an infinite dimensional Hilbert $X$ with $\mu$ is a normal weight and $Y$ Banach space. We also investigate characterizations $\mathcal{WB}_\mu(B_X)$ $Y$-valued, locally bounded, weakly holomorphic associated B_\mu(B_X)$ scalar-valued in sense that $f\in \mathcal{WB}_\mu(B_X)$ if $w...

It is a classical theorem of Sarason that an analytic function bounded mean oscillation (BMOA) vanishing if and only its rotations converge in norm to the original as angle rotation tends zero. In series two papers, Blasco et al. have raised problem characterizing all semigroups holomorphic functions $(\varphi\_{t})$ can replace semigroup Sarason’s theorem. We give complete answer this question...

We present some formulas for the norm, as well essential of a product composition and an integral operator between Bloch-type spaces analytic functions on unit ball, in terms given symbols weights.

and Applied Analysis 3 The following lemma is the crucial criterion for the compactness of Cφ, whose proof is an easy modification of the proof of Proposition 3.11 in 1 . Lemma 2.4. Assume that φ is a holomorphic self-map of D. Then Cφ : Bp → Bq is compact if and only if Cφ is bounded and for any bounded sequence {fm}m∈N in Bp which converges to zero uniformly on compact subsets of D, we have ∥...

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...

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...