Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials

TL;DR

The paper develops a unifying framework to relate projection, unconditional basis, and Gordon-Lewis constants for finite-dimensional polynomial spaces ${\mathcal P}_J(X_n)$ on Banach lattices, and leverages Lozanovskii factorization, convexity/concavity, and coefficient-functionals to derive sharp asymptotics. It establishes a tight connection between these constants and Bohr radii, enabling asymptotic results for general index sets and, in particular, Lorentz spaces $\ell_{r,s}^n$, including tetrahedral index sets. The authors achieve comprehensive bounds for the projection and unconditional basis constants, provide probabilistic lower bounds, and translate these into precise Bohr-radius asymptotics in high dimensions. The work significantly advances the understanding of local Banach space theory, Bohr phenomena in several variables, and the role of lattice structure in polynomial function spaces, with broad implications for Lorentz-type spaces and related Banach lattices.

Abstract

The primary aim of this work is to develop methods that provide new insights into the relationships between fundamental constants in Banach space theory--specifically, the projection constant, the unconditional basis constant and the Gordon-Lewis constant--for the Banach space $\mathcal{P}_J(X_n)$ of multivariate analytic polynomials. This class consists of all polynomials whose monomial coefficients vanish outside the set of multi-indices $J$, and it is equipped with the supremum norm on the unit sphere of the finite-dimensional Banach space $X_n = (\mathbb{C}^n, \|\cdot\|)$. We establish a~general framework for proving quantitative results on the asymptotic optimal behavior of these constants, which depend on both the dimension of the space and the degree of the polynomials. Using the tools developed, we derive asymptotic estimates of the Bohr radius for general Banach sequence lattices. Additionally, we apply our results to the asymptotic study of local constants and the Bohr radius within finite-dimensional Lorentz sequence spaces, which requires a~refined analysis of the combinatorial structure of the associated index sets. As a consequence, we obtain optimal results across a broad range of parameters.

Theorems & Definitions (96)

• Remark 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• proof : Proof of Theorem \ref{['gl_versus_proj']}