Unconditional and quasi-greedy bases in $L_p$ with applications to Jacobi polynomials Fourier series

Abstract

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in $L_p$ does not converge unless $p=2$. As a by-product of our work on quasi-greedy bases in $L_p(\mu )$, we show that no normalized unconditional basis in $L_p$, $p\ne 2$, can be semi-normalized in $L_q$ for $q\ne p$, thus extending a classical theorem of Kadets and Pełczy{ń}ski from 1968.