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 $L_2(\mathbb{T},|t{|}^{1-2/p}dt)$ belong to ${\ell }_p$, and that, given $1<p<2$, the Fourier series of sequences in ${\ell }_p$ belong $L_2(\mathbb{T},|t{|}^{2/p-1}dt)$. 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 $((\log m{)}^{\alpha }{)}_{m=2}^{\infty }$.