Let $\mathbb I^d$ be the $d$-dimensional unit cube, and $f\in L^1(\mathbb I^d)$. Further let $x\in\mathbb I^d$ and assume that (some representative of) $f$ is differentiable at $x=(x_1,\dotsc, x_d)$ (or, if needed, in some neighborhood of $x$). Is it true that $$\lim_{n\to\infty}\sum_{m_1\leq n,\dotsc,m_d\leq n} \hat f_{m_1,\dotsc, m_k} e^{2\pi i (m_1x_1+\cdots m_d x_d)}=f(x) \qquad ?$$
For $d=1$, this is the classical Dirichlet(–Dini) criterion. For $d\geq 2$ it amounts to say that the $\ell^\infty$-Fourier partial sum $\sigma^\infty_n(f)$ converges to $f$ as $n\to\infty$ at every differentiability of $f$.
I would be interested in a reference for this statement, or (if false) in a counterexample.
In the multidimensional case the condition for a continuous function $f$ with the modulus of continuity $\omega(t)$ to have pointwise Pringsheim convergence it is enough that $\omega(t)\log^d\frac1t\to0$ then $t\to0+$. See referencies in the review B.I. Golubov, Multiple series and Fourier integrals.