Weighted Bergman spaces induced by doubling weights in the unit ball of $\mathbb{C}^n$

Abstract

The q−Carleson measures for A p ω are characterized in terms of a neat geometric condition involving Carleson block. Some equivalent characterizations for A p ω are obtained by using the radial derivative and admissible approach regions. The boundedness and compactness of Volterra integral operator T g : A p ω → A q ω are also investigated in this paper with 0 < p ≤ q < ∞, where

Bergman projections induced by doubling weights on the unit ball of Cn

“…(i) and (ii) are Lemmas 1.6 and 1.7 in [10], respectively. (iii) was proved in [4]. (iv) can be proved straightly by (i), (ii) and Lemma 1.…”

“…We call S a the Carleson block. See [4] for more information about the Carleson block. As usual, for a measurable set…”

“…For a Banach space or a complete metric space X and a positive Borel measure µ on B, µ is a q−Carleson measure for X means that the identity operator Id : X → L q µ is bounded. When 0 < p ≤ q < ∞ and ω ∈ D, a characterization of q−Carleson measure for A p ω was obtained in [4].…”

“…See [9-13, 15, 16] for more results on A p ω (D) with ω ∈ D. In [4], we extended the Bergman space A p ω (D) to the unit ball B of C n . That is,…”

Toeplitz operators on Bergman spaces induced by doubling weights on the unit ball of $\mathbb{C}^n$

“…In [15], J. Peláez and J. Rättyä introduced a new class function spaces A p ω (D), the weighted Bergman spaces induced by rapidly increasing weights ω in D. That is A p ω (D) = L p (D, ωdA) ∩ H(D), 0 < p < ∞. See [14-18, 20, 21] for more results on A p ω (D) with ω ∈ D. In [2], we extended the Bergman space A p ω (D) with ω ∈ D to the unit ball B of C n . That is A p ω (B) = L p (B, ωdV) ∩ H(B), 0 < p < ∞.…”