Pointwise convergence of double Fourier series

Question

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:

1. 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 $f_D$?
2. Is the sequence of rectangular (or square) partial Fourier sums uniformly bounded?

Answer 1

It is known that the square partial sums of any $L^2$ (or, for that matter, $L^p$ for $1 < p \leq \infty$) function on $\mathbb{T}^2$ converges almost everywhere. This can be deduced from (the one dimensional) Carleson's theorem. This gives an affirmative answer to the (square version) of your first question. See*: C. Fefferman, On the convergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 (1971), 744–745

It is known that $2$-d rectangular summation can fail to converge pointwise almost everywhere even for $L^\infty$ functions. However, it is unclear to me if the known counterexample can be modified to give a function in the form your considering above. See*: C. Fefferman, On the divergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 1971 191–195.

*Confusingly, there is an error with mathscinet and the reviews for these two papers have been transposed. Update: It appears the mathscinet has now corrected the issue.