SEMINARIO DE ANÁLISIS COMPLEJO (COMPLEX ANALYSIS SEMINAR)
Counterexample of normability in Hardy and Bergman
spaces with 0 < p < 1

Iván Jiménez

U.A.M. - ICAI, Universidad Pontificia de Comillas

Wednesday, September 18, 2024 at 15:30
Room 520, Module 17, Department of Mathematics,
Universidad Autónoma de Madrid (Autonomous University of Madrid)
Abstract:

It is well-known from the literature on Hardy spaces that the Hardy spaces Hp,
0 < p < 1, are not normable. The same is also true for the Bergman spaces Ap.
However, none of the standard sources offer proofs of this fact. In 1953, Livingston
published an article demonstrating this fact, using a convexity argument based on a
theorem by Kolmogorov in order to prove the non-normability of Hp when 0 < p < 1.
In the case of the Bergman spaces, there is no proof of this fact in the literature, as
far as we know.
In this talk, we will present a direct proof (based on a counterexample) that the
usual expression for the norm when 1 ≤ p <∞ is not a norm in the Hardy spaces Hp,
0 < p < 1. In addition, we will show some counterexamples for the triangle inequality
in Ap, 0 < p < 1. This is a joint work with Dragan Vukoti´c.