Holomorphic Hölder‐type spaces and composition operators

Karapetyants, Alexey; Restrepo, Joel E.

doi:10.1002/mma.6675

Cited by 8 publications

(11 citation statements)

References 42 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Proof Denote

{\tilde{κ}}_{ω_{1}, ω_{2}, ϕ} = \sup_{z \in 𝔻} \{| ϕ^{'} (z) | \int_{1 - | ϕ (z) |}^{2} \frac{ω_{1} (t)}{t^{2}} d t \frac{1 - | z |}{ω_{2} (1 - | z |)}\} .

Clearly,

κ_{ω_{1}, ω_{2}, ϕ} \leq C {\overset{κ}{true}}_{ω_{1}, ω_{2}, ϕ}

. It was shown in [previous study, 10 Theorem 5.2] that if ω 2 satisfies the condition (), then

{\overset{κ}{true}}_{ω_{1}, ω_{2}, ϕ} < + \infty

implies the boundedness of the operator

C_{ϕ} : A^{ω_{1}} false(𝔻 false) \to A^{ω_{2}} false(𝔻 false)

. Hence, we simply need to show that

f_{ϕ, ω_{1}} false(z false) \in D L^{ω_{2}} false(𝔻 false)

implies

{\overset{κ}{true}}_{ω_{1}, ω_{2}, ϕ} < + \infty

.…”

Section: Boundedness Of Composition Operator In Generalized Hölder Sp...mentioning

confidence: 91%

“…From the assumptions we know that A 𝜔 1 (D) = B 𝜔 1 (D) with equivalent norms (see, for instance, previous study, 10 Theorem 3.2). Hence, g…”

Section: ) and (22) If Cmentioning

confidence: 99%

“…Here we recall the definitions and some properties of the introduced spaces in previous studies, 12,13 see also other studies. 6,10 Indeed, we recall the generalized Hölder spaces with prescribed modulus of continuity. Definition 2.1.…”

Section: Hölder Type Spaces Of Continuous Functions In the Unit Discmentioning

confidence: 99%

“…This follows easily since

\begin{matrix} C \frac{ω_{2} false(1 - false| z false| false)}{1 - false| z false|} & \geq false| \nabla f_{ϕ, ω_{1}} false(z false) false| \geq false| ϕ^{'} false(z false) false| \int_{0}^{1} \frac{ω_{1} false(t false) d t}{{false(1 - false| ϕ false(z false) false| + t false)}^{2}} \\ \geq false| ϕ^{'} false(z false) false| \int_{1 - false| ϕ false(z false) false|}^{2} \frac{ω_{1} false(t false/ 2 false)}{t^{2}} d t \geq C_{1} false| ϕ^{'} false(z false) false| \int_{1 - false| ϕ false(z false) false|}^{2} \frac{ω_{1} false(t false)}{t^{2}} d t . \end{matrix}

Now we prove the second statement. We denote

g_{a}^{ω_{1}} false(z false) = z \int_{0}^{1} \frac{ω_{1} false(1 - t false)}{1 - t a z} d t, a \in 𝕋, z \in 𝔻 .

By [previous study, 10 Propositions 4.5] the holomorphic functions

g_{a}^{ω_{1}}

belongs to

B^{ω_{1}} false(𝔻 false)

if and only i...…”

Section: Boundedness Of Composition Operator In Generalized Hölder Sp...unclassified

“…We mention the following sources on Hölder (Lipschitz) spaces of holomorphic functions, [3][4][5][6][7][8] see also the books, 1,8,9 and references therein. Let us also mention recent paper 10 in which we studied boundedness and compactness of composition operators in the generalized holomorphic Hölder type space as specified above, and we also devoted a significant part of the article to outline some embeddings between such Hölder type spaces, to discuss properties of modulus of continuity and to construct some useful examples of functions in Hölder type spaces, which are used in the present paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Boundedness of composition operators in holomorphic Hölder type spaces

Karapetyants

Restrepo

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

A new characterization of boundedness of the composition operators in generalized Hölder spaces is obtained in terms that do not use the derivative of the composition operator, but using some averaging construction which represents certain integration of the modulus of continuity involving the symbol of the composition operator. This approach also allows to recover previously known results for the standard weights 𝜔(t) = t 𝛼 with 0 < 𝛼 < 1. Certain further results on characterization of the same spaces are obtained as well.

show abstract

“…Proof Denote