There are no unconditional Schauder frames of exponentials in $L^p$ spaces, $p\ne 2$

Abstract

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega )$, $1\le p<2$, for any set $\Omega \subset {\mathbb{R}}^d$ of finite measure. For $p>2$, we proved that the same holds provided that $\Omega$ has nonempty interior, but the general case remained open. In this note we close this gap, i.e., we establish that exponentials cannot form an unconditional Schauder frame in $L^p(\Omega )$, $p\ne 2$, without any additional assumptions on the set $\Omega$ of finite measure other than measurability.