Joint Carleson measure for the difference of composition operators on the polydisks

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

“…Although the converse inequality does not hold we have the following result, which can be found in [9, Theorem 2.8] or [26,Lemma D]. Since Koo and Wang's result is stated for the unit ball setting (see [9]), for the convenience of readers, and in order to offer no doubt of the validity of the result, we give a proof here.…”

“…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.…”

Difference of composition operators between weighted Bergman spaces on the unit ball

“…for short. Given α, β > −1 and 0 < p, q < ∞, the joint pull-back measure ω β,q,ϕ,ψ is defined by (see [8])…”

“…After that, such related problems have been studied between several spaces of analytic functions by many authors. See, for example, [6,14,22] on Hardy spaces and [2,8,9,13,18,19] on weighted Bergman spaces.…”

“…For 0 < p ≤ q < ∞, Saukko [18] obtained some compactness criterion for difference C ϕ − C ψ from A p α (D) to A q β (D). In [8], Koo and Wang gave some characterizations for the boundedness and compactness of the difference of composition operators C ϕ − C ψ : A p α → A p α on the unit ball. It is worth pointing out that the approach in [18, Theorem4.5(i)] does not work as well for the unit ball.…”