I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function.

Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of finite perimeter with piecewise $C^1$ boundary. Let $\mathbf{1}_D : [-\pi, +\pi]^2 \to \{ 0, 1 \}$ be the indicator function of the set $D$, and $f_D : \mathbb{R}^2 \to \{ 0, 1 \}$ be the periodic extension of $\mathbf{1}_D$.

My questions are:

- Considering rectangular (or square) partial Fourier sums, does there exist a theorem that tells us that the double Fourier series expansion of $f_D$ converges pointwise everywhere and that the pointwise limit function is almost everywhere equal to $f_D$?
- Is the sequence of rectangular (or square) partial Fourier sums uniformly bounded?