Fourier coefficients of functions in power-weighted $L_2$-spaces and conditionality constants of bases in Banach spaces

Abstract

We prove that, given $2<p<\infty$, the Fourier coefficients of functions in $L_2(\mathbb{T}, \lvert t \rvert^{1-2/p}\, dt)$ belong to $\ell_p$, and that, given $1<p<2$, the Fourier series of sequences in $\ell_p$ belong $L_2(\mathbb{T}, \lvert t \rvert^{2/p-1}\, dt)$. Then, we apply these results to the study of conditional Schauder bases and conditional almost greedy bases in Banach spaces. Specifically, we prove that, for every $1<p<\infty$ and every $0\le \alpha<1$, there is a Schauder basis of $\ell_p$ whose conditionality constants grow as $(m^\alpha)_{m=1}^\infty$, and there is an almost greedy basis of $\ell_p$ whose conditionality constants grow as $((\log m)^\alpha)_{m=2}^\infty$.