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

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript upper J Baseline left-parenthesis upper X Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mi>J</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {P}_J(X_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of multivariate analytic polynomials. This class consists of all polynomials whose monomial coefficients vanish outside the set of multi-indices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and it is equipped with the supremum norm on the unit sphere of the finite-dimensional Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript n Baseline equals left-parenthesis double-struck upper C Superscript n Baseline comma double-vertical-bar dot double-vertical-bar right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mo> ⋅ </mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_n = (\mathbb {C}^n, \|\cdot \|)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.