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 $L\_p$, $p \neq 2$, can be semi-normalized in $L\_q$ for $q \neq p$, thus extending a classical theorem of Kadets and Pełczyński from 1962.