Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

Mark Veraar; Lutz Weis

Displaying similar documents to “Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory”

Interpolation of Cesàro sequence and function spaces

Sergey V. Astashkin, Lech Maligranda (2013)

Studia Mathematica

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The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $C e s_{p} (I)$ is an interpolation space between $C e s_{p ₀} (I)$ and $C e s_{p ₁} (I)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $C e s_{p} [0, 1]$ is not an interpolation space between Ces₁[0,1] and $C e s_{\infty} [0, 1]$ .

$H^{\infty}$ functional calculus in real interpolation spaces, II

Giovanni Dore (2001)

Studia Mathematica

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Let A be a linear closed one-to-one operator in a complex Banach space X, having dense domain and dense range. If A is of type ω (i.e.the spectrum of A is contained in a sector of angle 2ω, symmetric about the real positive axis, and ${| | λ (λ I - A)}^{- 1} | |$ is bounded outside every larger sector), then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the intersection of the domain and the range of the operator itself.

Lagrange approximation in Banach spaces

Lisa Nilsson, Damián Pinasco, Ignacio M. Zalduendo (2015)

Czechoslovak Mathematical Journal

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Starting from Lagrange interpolation of the exponential function $e^{z}$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$ . Given such a representable entire funtion $f : E \to ℂ$ , in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$ , we present a sufficient growth condition on...

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K (t, f; L^{p} (C), L^{\infty} (C))$ to quasicontinuous functions f ∈ QC is equivalent to $K (t, f; L^{p} (C) \cap Q C, L^{\infty} (C) \cap Q C)$ . We apply this result to identify the interpolation space ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .

Several notes on the circumradius condition

Václav Kučera (2016)

Applications of Mathematics

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Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1, p}$ -error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p = \infty$ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how...

$H^{\infty}$ functional calculus in real interpolation spaces

Giovanni Dore (1999)

Studia Mathematica

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Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $∥ λ {(λ I - A)}^{- 1} ∥$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty}$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

Complex interpolation of function spaces with general weights

Douadi Drihem (2023)

Commentationes Mathematicae Universitatis Carolinae

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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_{0}, q_{0}}^{s_{0}} (ω_{0})$ and ${\dot{F}}_{\infty, q_{1}}^{s_{1}} (ω_{1})$ with suitable assumptions on the parameters $s_{0}, s_{1}, p_{0}, q_{0}$ and $q_{1}$ , and the pair of weights $(ω_{0}, ω_{1})$ .

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

The non-pluripolarity of compact sets in complex spaces and the property $(L B^{\infty})$ for the space of germs of holomorphic functions

Le Mau Hai, Tang Van Long (2002)

Studia Mathematica

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The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $(L B^{\infty})$ for the dual space of the space of germs of holomorphic functions on that compact set.

The Lizorkin-Freitag formula for several weighted $L_{p}$ spaces and vector-valued interpolation

Irina Asekritova, Natan Krugljak, Ludmila Nikolova (2005)

Studia Mathematica

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A complete description of the real interpolation space $L = {(L_{p ₀} (ω ₀), . . ., L_{p ₙ} (ω ₙ))}_{θ ⃗, q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces $Ω_{i}$ (i ∈ I) such that L is an $l_{q}$ sum of the restrictions of L to $Ω_{i}$ , and L on each $Ω_{i}$ is a result of interpolation of just two weighted $L_{p}$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

On L₁-subspaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2010)

Studia Mathematica

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We study the spaces $H_{μ} (Ω) = f : Ω \to ℂ h o l o m o r p h i c : \int_{0}^{R} \int_{0}^{2 π} | f (r e^{i φ}) | d φ d μ (r) < \infty$ where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, $H_{μ} (Ω)$ is either isomorphic to l₁ or to ${(\sum \oplus A ₙ)}_{(1)}$ . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.

On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the $_{p}$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among $_{p} (ₙ)$ , the α-Bloch, the Dirichlet $_{p}$ and the little $_{p, 0}$ spaces.

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

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Agler, Lykova and Young introduced a sequence $C_{ν}$ , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition $C_{n - 2}$ is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.

Rigidity of the holomorphic automorphism of the generalized Fock-Bargmann-Hartogs domains

Ting Guo, Zhiming Feng, Enchao Bi (2021)

Czechoslovak Mathematical Journal

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We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n, m}^{p} (μ)$ . The generalized Fock-Bargmann-Hartogs domain is defined by inequality $e^{{μ ∥ z ∥}^{2}} \sum_{j = 1}^{m} {| ω_{j} |}^{2 p} < 1$ , where $(z, ω) \in ℂ^{n} \times ℂ^{m}$ . In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n, m}^{p} (μ)$ becomes a holomorphic automorphism if and only if it keeps the function $\sum_{j = 1}^{m} {| ω_{j} |}^{2 p} e^{{μ ∥ z ∥}^{2}}$ invariant.

Sobolev-Besov spaces of measurable functions

Hans Triebel (2010)

Studia Mathematica

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The paper deals with spaces $L_{p}^{s} (ℝ ⁿ)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B_{p, q}^{s} (ℝ ⁿ)$ of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Subspaces $H_{E}^{p, q, α}$ of mixed-norm spaces $H^{p, q, α}$ of holomorphic functions

M. Jevtić (1985)

Matematički Vesnik

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On the geometry of the space of m-pairs of holomorphic subspaces of the n-dimensional projective space $P_{n}$ ove r an albebra $𝒜$

Е.М. Кузнецова (1986)

Trudy Geometriceskogo Seminara

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