On holomorphic functions attaining their weighted norms

TL;DR

This paper investigates the properties of holomorphic functions that attain their weighted norms, exploring their density in various function spaces and introducing new geometric methods for analysis.

Contribution

It establishes the denseness of weighted norm-attaining holomorphic functions in uniformly convex spaces and introduces a novel geometric technique for the proof.

Findings

Existence of polynomials on ℓ_p that attain weighted but not supremum norm

Equivalence of weighted and supremum norms for fixed-degree polynomials

Denseness of weighted norm-attaining functions in uniformly convex spaces

Abstract

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their supremum norm and viceversa. Nevertheless, we also prove that in the context of polynomials of fixed degree both norms are in fact equivalent. This leads us to the main problem of the paper, namely, whether the holomorphic functions attaining their weighted norm are dense. Although we exhibit an example where this does not hold, as the main theorem of our paper, we prove the denseness provided the domain space is uniformly convex. In fact, we provide a Bollob\'as type theorem in this setting. For the proof of such a result we develop a new geometric technique.