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Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities

Andreas DefantMarius Junge — 1997

Studia Mathematica

We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator T : L q ( μ ) L p ( ν ) , each n ∈ ℕ and functions f 1 , . . . , f n L q ( μ ) , ( ʃ ( k = 1 n | T f k | r ) p / r d ν ) 1 / p c T ( ʃ ( k = 1 n | f k | r ) q / r d μ ) 1 / q . This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

Composition of (E,2)-summing operators

Andreas DefantMieczysław Mastyło — 2003

Studia Mathematica

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.

Factorization and extension of positive homogeneous polynomials

Andreas DefantMieczysław Mastyło — 2014

Studia Mathematica

We study the following problem: Given a homogeneous polynomial from a sublattice of a Banach lattice to a Banach lattice, under which additional hypotheses does this polynomial factorize through L p -spaces involving multiplication operators? We prove that under some lattice convexity and concavity hypotheses, for polynomials certain vector-valued norm inequalities and weighted norm inequalities are equivalent. We combine these results and prove a factorization theorem for positive homogeneous polynomials...

Weyl numbers versus Z-Weyl numbers

Bernd CarlAndreas DefantDoris Planer — 2014

Studia Mathematica

Given an infinite-dimensional Banach space Z (substituting the Hilbert space ℓ₂), the s-number sequence of Z-Weyl numbers is generated by the approximation numbers according to the pattern of the classical Weyl numbers. We compare Weyl numbers with Z-Weyl numbers-a problem originally posed by A. Pietsch. We recover a result of Hinrichs and the first author showing that the Weyl numbers are in a sense minimal. This emphasizes the outstanding role of Weyl numbers within the theory of eigenvalue distribution...

Unconditionality for m-homogeneous polynomials on

Andreas DefantPablo Sevilla-Peris — 2016

Studia Mathematica

Let χ(m,n) be the unconditional basis constant of the monomial basis z α , α ∈ ℕ₀ⁿ with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit polydisc ⁿ. We prove that the quotient of s u p m s u p m χ ( m , n ) m and √(n/log n) tends to 1 as n → ∞. This reflects a quite precise dependence of χ(m,n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a reformulation...

The Dirichlet-Bohr radius

Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n x a n n - s we have n x | a n | r Ω ( n ) s u p t | n x a n n - i t | . We prove that the asymptotically correct order of L(x) is ( l o g x ) 1 / 4 x - 1 / 8 . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr...

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