Original Question: $\displaystyle v=\sum_{n\neq 0}v_ne^{in\theta}, w=\sum_{n\neq 0}w_ne^{in\theta}$ are $C^{3/2+\epsilon}$ functions on $S^1$ ($v_n$ and $w_n$ are Fourier coefficients of $v$ and $w$), prove $\displaystyle \sum_{n\geq 2}v_nw_n n^3$ is absolutely convergent.
This is a proposition of some paper, but it didn't give the definition of $C^{3/2+\epsilon}$ function on $S^1$, I searched the Internet and found nothing.
Q1: What is the definition of $C^{3/2+\epsilon}$ function on $S^1$?
In the proof, it asserts the fact that the Fourier coefficients of a $C^{k+\epsilon}$ function on $S^1$ decay at least as fast as $1/n^{k+\epsilon}$. (See Katznelson An introduction to Harmonic analysis, p. 24-25.) I thought the book just talked about when $k$ is integer.
Q2: How to get above fact?
The proof said if $v$ and $w$ are $C^{3/2+\epsilon}$ smooth on $S^1$, then $\{v_nn^{3/2}\}$ and $\{w_nn^{3/2}\}$ are in $\ell^2$, by Cauchy-Schwarz inequality we can get the convergence.
Q3:If $v$ and $w$ are $C^{3/2+\epsilon}$ smooth on $S^1$, why $\{v_nn^{3/2}\}$ and $\{w_nn^{3/2}\}$ are in $\ell^2$?
Any help will be appreciated!