Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Abstract

Bounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

Difference of composition-differentiation operators from Hardy spaces to weighted Bergman spaces via harmonic analysis

“…authors on several function spaces. See [7,9,10,12,[24][25][26] for example. In 2004, Nieminen and Saksman [16] showed that the compactness of C ϕ − C ψ on H p , with 1 ≤ p < ∞, is independent of p. Very recently, in [2], Choe et al completely characterized compact operators C ϕ − C ψ on H p by using Carleson measures of Bergman spaces.…”