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Complex Oscillation Theory of Differential Polynomials

Abdallah El Farissi, Benharrat Belaïdi (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this paper, we investigate the relationship between small functions and differential polynomials g f ( z ) = d 2 f ' ' + d 1 f ' + d 0 f , where d 0 ( z ) , d 1 ( z ) , d 2 ( z ) are entire functions that are not all equal to zero with ρ ( d j ) < 1 ( j = 0 , 1 ...

Composition in ultradifferentiable classes

Armin Rainer, Gerhard Schindl (2014)

Studia Mathematica

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ( ω ) and the Roumieu space ω as intersection and union of spaces ( M ) and M for associated weight sequences, respectively.

Composition operator and Sobolev-Lorentz spaces W L n , q

Stanislav Hencl, Luděk Kleprlík, Jan Malý (2014)

Studia Mathematica

Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator T f : u ↦ u ∘ f maps the Sobolev-Lorentz space W L n , q ( Ω ' ) to W L n , q ( Ω ) for some q ≠ n then f must be a locally bilipschitz mapping.

Composition operators in the Dirichlet series setting

Hervé Queffélec (2007)

Banach Center Publications

In this work, we begin with a survey of composition operators on the Hardy space H² and on the Wiener algebra A⁺ of absolutely convergent Taylor series, with special emphasis on their compactness, or invertibility, or isometric character. The main results are due respectively to J. Shapiro and D.~Newman. In a second part, we present more recent results, due to Gordon and Hedenmalm on the one hand, and to Bayart, the author et al. on the other hand, concerning the analogues of H² and A⁺ in the setting...

Composition operators: N α to the Bloch space to Q β

Jie Xiao (2000)

Studia Mathematica

Let N α ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and Q β are Möbius invariant, but N α is not. We characterize, in function-theoretic terms, when the composition operator C ϕ f = f ◦ ϕ induced by an analytic self-map ϕ of the unit disk defines an operator C ϕ : N α → B , B → Q β , N α → Q β which is bounded resp. compact.

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Luděk Kleprlík (2014)

Open Mathematics

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

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