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The aim of this paper is to establish the theorem of atomic decomposition of weighted Bergman spaces Ap(Ω), where Ω is a domain of finite type in C2. We construct a kernel function H(z,w) which is a reproducing kernel for Ap(Ω) and we prove that the associated integral operator H is bounded in Lp(Ω).
We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space is zero; i.e., . Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study...
The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form for in . Among other things, using the Haar measure on for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.
In this note we establish a vector-valued version of Beurling’s theorem (the Lax-Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the “weak” completion problem in .
We study the duals of the spaces of harmonic functions in the unit ball of with values in a Banach space X, belonging to the Bochner space with weight , denoted by . For 0 < α < p-1 we construct continuous projections onto providing a decomposition . We discuss the conditions on p, α and X for which and , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.
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