The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

Abstract

Let, ( ) be the best constant that fulfills the following inequality: for every -homogeneous polynomial ( ) = ∑ | |= in complex variables,For every degree , and a wide range of values of , ∈ [1, ∞] (including any in the case ∈ [1, 2], and any and for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as (the number of variables) tends to infinity.Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., -homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of , . As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality. K E Y W O R D SHardy-Littlewood inequalities, multivariable von Neumann's inequality, unconditionality in spaces of polynomials, unimodular polynomials M S C ( 2 0 1 0 ) 11C08, 15A60, 15A69, 46G25, 47A30, 47H60 Problem 1.3. Let , ( ) and , ( ) be the smallest constants that fulfill the following inequalities: for every -homogeneous polynomial in complex variables,

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

“…., (ℓ p 0 , ℓ p 1 ) got the injective θ-property? A positive answer would have interesting consequences (see [16] again). Unfortunately, we show that this is false provided p 0 is small enough and p 1 is big enough.…”

“…Nevertheless, nothing seems known about the interpolation of m tensor products, m ≥ 3 except when all but one of the spaces are equal to ℓ ∞ . In particular, in [16], the authors ask the following question: let m ≥ 3, 1 ≤ p 0 < p 1 ≤ 2, θ ∈ (0, 1). Is it true that…”

“…Results are known for the tensor products of two spaces (see [17] and [23]) but almost nothing is known about the interpolation of the tensor products of m spaces with m ≥ 3. We then give a negative answer to a question asked in [16], showing for instance that the interpolating space of ⊗ m,ε ℓ 1 and ⊗ m,ε ℓ 2 , m ≥ 3, is not the expected ⊗ m,ε ℓ p . In Section 3, we will prove that any m-linear form ℓ 1 × · · · × ℓ 1 × Z → C with Z a cotype 2 space is multiple (1, 1)-summing.…”

Mixed Bohr radius in several variables

“…When X = ℓ p and Y = ℓ q we will denote χ M (P ( m X n ), P ( m Y n )), the (p, q)-mixed unconditionally constant for the monomial basis of P ( m C n ), as χ M (P ( m ℓ n p ), P ( m ℓ n q )). It should be mentioned that, for any fixed m ∈ N, the asymptotic growth of χ M (P ( m ℓ n p ), P ( m ℓ n q )) as n → ∞ was studied in [GMM16].…”

“…The previous lemma shows that understanding K (B ℓ n p , B ℓ n q ) translates into seeing how the constant χ M (P ( m ℓ n p ), P ( m ℓ n q )) 1/m behaves. It should be mentioned that, for any fixed m ∈ N, the asymptotic growth of χ M (P ( m ℓ n p ), P ( m ℓ n q )) as n → ∞ was studied in [GMM16]. These results unfortunately are not useful because, as can be seen in Lemma 2.2, one needs to comprehend how…”