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Unconditionality for m-homogeneous polynomials on $ℓⁿ_{∞}$

100%
Defant A., Sevilla-Peris P.
Studia Mathematica
|
2016
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tom 232
|
nr 1
45-55
EN
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 𝔻ⁿ. We prove that the quotient of $sup_{m} \sqrt[m]{sup_{m} χ(m,n)}$ 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 of our results in terms of tensor products, and as an application a solution for a problem on Bohr radii.
2
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Invertibility in tensor products of Q-algebras

100%
Dineen S., Sevilla-Peris P.
Studia Mathematica
|
2002
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tom 153
|
nr 3
269-284
EN
We consider, using various tensor norms, the completed tensor product of two unital lmc algebras one of which is commutative. Our main result shows that when the tensor product of two Q-algebras is an lmc algebra, then it is a Q-algebra if and only if pointwise invertibility implies invertibility (as in the Gelfand theory). This is always the case for Fréchet algebras.
3
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Multilinear Hölder-type inequalities on Lorentz sequence spaces

81%
Carando D., Dimant V., Sevilla-Peris P.
Studia Mathematica
|
2009
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tom 195
|
nr 2
127-146
EN
We establish Hölder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.
4
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The Dirichlet-Bohr radius

71%
Carando D., Defant A., Garcí D., Maestre M., Sevilla-Peris P.
Acta Arithmetica
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2015
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tom 171
|
nr 1
23-37
EN
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 $∑_{n ≤ x} |a_n| r^{Ω(n)} ≤ sup_{t∈ ℝ} |∑_{n ≤ x} a_n n^{-it}|$. We prove that the asymptotically correct order of L(x) is $(log x)^{1/4} x^{-1/8}$. 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 radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.
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