Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Abstract

Let ϕ be an analytic self-map of the unit disk D, H (D) the space of analytic functions on D and g ∈ H (D). The boundedness and compactness of the operator DC ϕ : H ∞ → Z are investigated in this paper.

The product-type operators from logarithmic Bloch spaces to Zygmund-type spaces

“…Inspired by the results [25,30,32,45,47], our aim is to consider the boundedness and compactness of the operators T n ψ 1 ,ψ 2 ,ϕ : B log (or B log,0 ) → Z µ .…”

Weighted Differentiation Composition Operators from Mixed-Norm to Zygmund Spaces

“…By (14), (24), (34), and observe that D n ,u f (w) , D n ,u g (w) , D n ,u h (w) ∈ 0 we know that lim | (z)|→1…”

“…For some recent articles on weighted composition operators on some H ∞ -type spaces, see, for example, [4, 9-11, 26, 29-31, 43] and references therein. If n = 1, u(z) = (z), then D n ,u = DC , which was studied in [6,8,12,22,24,27,42,46,47]. When n = 1, u(z) ≡ 1, then D n ,u = C D, which was studied in [6,27].…”

“…Yang in [47] studied the boundedness and compactness of the operator C D and DC from Q K (p, q) to and ,0 spaces. Liu and Yu in [24] studied the boundedness and compactness of the operator DC between H ∞ to Zygmund spaces. Stević in [32] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Bloch-type space, the little Bloch-type space, the Zygmund space, and the little Zygmund space.…”

“…The products of composition operator and integral-type operators, acting on various spaces of analytic functions, has been recently studied, for example, in [15,16,19,23,42] (see also the related references therein). Motivated by the results [24,32,44,48,49], we consider the boundedness and compactness of the operators D n ,u from H (p, q, ) to the Zygmund space, and the little Zygmund space. For the proof, we need different test functions and some complex calculation skills.…”