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

We prove that, given $2<p<\mathrm{\infty }$, the Fourier coefficients of functions in $L_2(\mathbb{T},\mid t{\mid }^{1-2\mathrm{/}p}dt)$ belong to ${\mathrm{\ell }}_p$, and that, given $1<p<2$, the Fourier series of sequences in ${\mathrm{\ell }}_p$ belong $L_2(\mathbb{T},\mid t{\mid }^{2\mathrm{/}p-1}dt)$. 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 $1<p<\mathrm{\infty }$ and every $0\le \alpha <1$, there is a Schauder basis of ${\mathrm{\ell }}_p$ whose conditionality constants grow as $(m^{\alpha }{)}_{m=1}^{\mathrm{\infty }}$, and there is an almost greedy basis of ${\mathrm{\ell }}_p$ whose conditionality constants grow as $((\log m)^\alpha)_{m=2}^\infty$.