Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Andreas Defant; Marius Junge

Displaying similar documents to “Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities”

Absolutely p-summing operators and Banach spaces containing all $l_{p}^{n}$ uniformly complemented

Andreas Defant (1990)

Studia Mathematica

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Boundedness of classical operators on classical Lorentz spaces

E. Sawyer (1990)

Studia Mathematica

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Type and cotype numbers of operators on Banach spaces

Albrecht Pietsch (1990)

Studia Mathematica

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Inequalities between absolutely (p,q)-summing norms

B. Carl (1981)

Studia Mathematica

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Two-sided estimates for the approximation numbers of Hardy-type operators in $L^{\infty}$ and L¹

W. Evans, D. Harris, J. Lang (1998)

Studia Mathematica

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In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})$ defined by $(T f) (x) ≔ v (x) ʃ_{0}^{\infty} u (t) f (t) d t$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

Fourier analysis, Schur multipliers on $S^{p}$ and non-commutative Λ(p)-sets

Asma Harcharras (1999)

Studia Mathematica

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This work deals with various questions concerning Fourier multipliers on $L^{p}$ , Schur multipliers on the Schatten class $S^{p}$ as well as their completely bounded versions when $L^{p}$ and $S^{p}$ are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning...

On operator ideals related to (p,σ)-absolutely continuous operators

J. López Molina, E. Sánchez Pérez (2000)

Studia Mathematica

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We study tensor norms and operator ideals related to the ideal $P_{p, σ}$ , 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_{p, σ}$ (in the sense of Defant and Floret), we characterize the ${(α^{'})}^{t}$ -nuclear and ${(α^{'})}^{t}$ - integral operators by factorizations by means of the composition of the inclusion map $L^{r} (μ) \to L^{1} (μ) + L^{p} (μ)$ with a diagonal operator $B_{w} : L^{\infty} (μ) \to L^{r} (μ)$ , where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components...

Construction of standard exact sequences of power series spaces

Markus Poppenberg, Dietmar Vogt (1995)

Studia Mathematica

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The following result is proved: Let $Λ_{R}^{p} (α)$ denote a power series space of infinite or of finite type, and equip $Λ_{R}^{p} (α)$ with its canonical fundamental system of norms, R ∈ 0,∞, 1 ≤ p < ∞. Then a tamely exact sequence (⁎) $0 \to Λ_{R}^{p} (α) \to Λ_{R}^{p} (α) \to Λ_{R}^{p} {(α)}^{ℕ} \to 0$ exists iff α is strongly stable, i.e. $l i m_{n} α_{2 n} / α_{n} = 1$ , and a linear-tamely exact sequence (*) exists iff α is uniformly stable, i.e. there is A such that $l i m s u p_{n} α_{K n} / α_{n} \leq A < \infty$ for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence (*) exists iff α is stable,...

A characterization of probability measures by f-moments

K. Urbanik (1996)

Studia Mathematica

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Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments $ʃ_{0}^{\infty} ƒ (x) μ^{* n} (d x)$ (n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and ${(- 1)}^{n} ƒ^{(n + 1)} (x)$ is completely monotone for some nonnegative integer n. The purpose...

Two-sided estimates of the approximation numbers of certain Volterra integral operators

D. Edmunds, W. Evans, D. Harris (1997)

Studia Mathematica

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We consider the Volterra integral operator $T : L^{p} (ℝ^{+}) \to L^{p} (ℝ^{+})$ defined by $(T f) (x) = v (x) ʃ_{0}^{x} u (t) f (t) d t$ . Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_{n} (T)$ of T are established when 1 < p < ∞. When p = 2 these yield $l i m_{n \to \infty} n a_{n} (T) = π^{- 1} ʃ_{0}^{\infty} | u (t) v (t) | d t$ . We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (an(T)) when 1 < α < ∞.

The Grothendieck-Pietsch domination principle for nonlinear summing integral operators

Karl Lermer (1998)

Studia Mathematica

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We transform the concept of p-summing operators, 1≤ p < ∞, to the more general setting of nonlinear Banach space operators. For 1-summing operators on B(Σ,X)-spaces having weak integral representations we generalize the Grothendieck-Pietsch domination principle. This is applied for the characterization of 1-summing Hammerstein operators on C(S,X)-spaces. For p-summing Hammerstein operators we derive the existence of control measures and p-summing extensions to B(Σ,X)-spaces. ...

Four characterizations of scalar-type operators with spectrum in a half-line

Peter Vieten (1997)

Studia Mathematica

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$C^{0}$ -scalar-type spectrality criterions for operators A whose resolvent set contains the negative reals are provided. The criterions are given in terms of growth conditions on the resolvent of A and the semigroup generated by A. These criterions characterize scalar-type operators on the Banach space X if and only if X has no subspace isomorphic to the space of complex null-sequences.

The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

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Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_{Ω} x y d μ \leq ϕ^{- 1} (ʃ_{Ω} ϕ \circ x d μ) ψ^{- 1} (ʃ_{Ω} ψ \circ x d μ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then...