On the orthogonal democratic systems in the $L^p$ spaces

Abstract

The concept of bidemocratic pair for a Banach space was introduced in \cite{KS:18}. We construct a family of orthonormal systems ${\mathfrak{F}}_l,$ $l\in (0,\infty )$ of functions defined on $[-1,1]$ such that the pair $({\mathfrak{F}}_l,{\mathfrak{F}}_l)$ is bidemocratic for $L^p[-1,1]$ and for $L^{p^{\prime }}[-1,1]$ if $l\in (0,\frac{p}{2(p-2)}]$, where $p>2$ and $p^{\prime }=\frac{p}{p-1}$. The system ${\mathfrak{F}}_l$ is not democratic for $L^{p^{\prime }}[-1,1]$ when $l\in (\frac{p}{2(p-2)},\frac{p}{p-2}).$ When $l>\frac{p}{2(p-2)}$ the pair $({\mathfrak{F}}_l,{\mathfrak{F}}_l)$ is not bidemocratic neither for $L^p[-1,1]$ nor for $L^{p^{\prime }}[-1,1]$.