A note on weighted Bergman spaces and the Cesaro Operator

Abstract

Let B denote the unit ball in ℂn, and dV(z) normalized Lebesgue measure on B. For α > -1, define dVα(z) = (1 - \z\2)αdV(z). Let (B) denote the space of holomorhic functions on B, and for 0 < p < ∞, let p(dVα) denote Lp(dVα) ∩ (B). In this note we characterize p(dVα) as those functions in (B) whose images under the action of a certain set of differential operators lie in Lp(dVα). This is valid for 1 < p < oo. We also show that the Cesàro operator is bounded on p(dVα) for 0 < p < oo. Analogous results are given for the polydisc.

A Characterization of Some Classes of Harmonic Functions

“…The Bergman spaces (with ϕ(x) = x 2 and α = 0), were introduced for the case of analytic functions in the paper [5] and extended in [10] for the case ϕ(x) = x p , p > 0 and α > 0 (see also [6]), and soon after the research field attracted attention of many authors. Some, results on Bergman spaces, can be found, for example, in [2], [3], [4], [7], [22], [23], [24] (see also the references therein).…”

Volterra-Type Operators on Zygmund Spaces

“…The boundedness and compactness of J g and I g between some spaces of analytic functions, as well as their n-dimensional extensions, were investigated in [3][4][5][6][7][8][9][10][11][12][13][14][15][16] (see also the related references therein).…”

A Generalization of a Result of Choa on Analytic Functions With Hadamard Gaps

“…The weighted Bergman space on the unit disk, polydisc or on the unit ball has been investigated recently a great deal, see, for example, [1,2,7,8,9,10,11,12] and the references in there. Motivated by Theorems A and B, in this paper, we investigate analytic functions with Hadamard gaps, which belong to the weighted Bergman space A p α (B), the mixed norm space (see, Section 3) and on the weighted Bergman space on U n (Section 4).…”