Suppose $f : \mathbb{T}^d \to \mathbb{R}$. I would like to know under what circumstances the Fourier series of $f$ is absolutely convergent, i.e., $\sum_{k \in \mathbb{Z}^d} |\hat f(k)| < \infty.$
I would like this to be based upon $f$ belonging to some Holder-space $\mathcal{C}^{k,\alpha}(\mathbb{T}^d)$ or some space of continuous functions $\mathcal{C}^{k}(\mathbb{T}^d)$, where $k=0,1,2, ...$ and $\alpha \in (0,1].$
From here, I know that $f \in \mathcal{C}^{0,\alpha}(\mathbb{T})$ with $\alpha>1/2$ ensures absolute convergence -- for $d=1.$
I came across an old thesis (not publicly available) that states if $f \in \mathcal{C}^{\ell}(\mathbb{T}^d)$ and $\ell \geq d/2$ then the Fourier series converges absolutely. But there are some problems with that: firstly, what does $\mathcal{C}^{d/2}$ mean when $d=1,3,5,...$? Does that mean $\mathcal{C}^{1,1/2}$ for $d=3$, for example? Secondly, with $d=1$, the $\geq$ rather than $>$ is different to the result cited above, but maybe the stronger result is valid?
I am interested to know for arbitrary $d$, what the requirement is. Ideally I'd like a reference to a paper stating such a result or a textbook. Thanks!
