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Let Ω be a C∞-domain in Cn. It is well known that a holomorphic function on Ω behaves twice as well in complex tangential directions (see [GS] and [Kr] for instance). It follows from well known results (see [H], [RS]) that some converse is true for any kind of regular functions when Ω satisfies(P) The real tangent space is generated by the Lie brackets of real and imaginary parts of complex tangent vectorsIn this paper we are interested in the behavior of holomorphic Hardy-Sobolev functions in...
We study the Complex Unconditional Metric Approximation Property for translation invariant spaces of continuous functions on the circle group. We show that although some “tiny” (Sidon) sets do not have this property, there are “big” sets Λ for which has (ℂ-UMAP); though these sets are such that contains functions which are not continuous, we show that there is a linear invariant lifting from these spaces into the Baire class 1 functions.
We show that the Taylor (resp. Bochnak) complexification of the injective (projective) tensor product of any two real Banach spaces is isometrically isomorphic to the injective (projective) tensor product of the Taylor (Bochnak) complexifications of the two spaces.
We give a unified treatment of procedures for complexifying real Banach spaces. These include several approaches used in the past. We obtain best possible results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. These estimates provide generalizations and show sharpness of previously obtained inequalities.
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 and the Roumieu space as intersection and union of spaces and for associated weight sequences, respectively.
The Banach operator ideal of (q,2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces. A key result in this context is a composition formula for such operators due to H. König, J. R. Retherford and N. Tomczak-Jaegermann. Based on abstract interpolation theory, we prove a variant of this result for (E,2)-summing operators, E a symmetric Banach sequence space.
Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator maps the Sobolev-Lorentz space to for some q ≠ n then f must be a locally bilipschitz mapping.
The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.
We characterize the boundedness and compactness of composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk.
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