Pointwise convergence of Fourier series in two dimensions

Question

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$ f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.} $$ Suppose now that $f\in L^2(\mathbb{T}^2)$. Using Carleson's Theorem, which would be the easiest way to prove $$ f(x,y)=\lim_{N\rightarrow\infty}\sum_{k,l=-N}^N\hat{f}(k,l)e^{2\pi i(kx+ly)}\;\text{ a.e.}? $$ Is it also true that $$ f(x,y)=\lim_{M,N\rightarrow\infty}\sum_{|k|\leq M,\,|l|\leq N}\hat{f}(k,l)e^{2\pi i(kx+ly)}\;\text{ a.e.}? $$ (by $\lim_{M,N\rightarrow\infty}$ I mean the limit of a double sequence: given $\{a_{m,n}\}\subseteq\mathbb{C}$, we say that $\lim_{m,n\rightarrow\infty}a_{m,n}=L$ if for all $\epsilon>0$ there exists an $N_{\epsilon}\in\mathbb{N}$ such that $|L-a_{m,n}|<\epsilon$ for every $m,n\geq N_{\epsilon}$).

Answer 1

Both of your questions were answered in short papers by Charles Fefferman:

• C. Fefferman. On the convergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 (1971), 744-745.

• C. Fefferman. On the divergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 (1971), 191-195.

In the second one he shows that the answer to your second question is no. The counterexample is very simple: consider $f(x,y)=e^{i\lambda xy}$ for large $\lambda$ and suppose that the linearizing functions associated to the supremum are $N(x,y)=\lambda x$ and $M(x,y)=\lambda y$ (you can safely ignore that these are not integers). Then one can show an estimate from below by $C\log(\lambda)$. Now let $\lambda\to\infty$, contradiction. Details in the paper.

In the first one he answers your first question. It is a 2-page proof and uses only Carleson's theorem in one dimension. The key here is that there is still only a single scaling parameter $N$. Maybe it is reasonable to believe that the bound for the Carleson maximal operator associated to the rectangles $[-N,N]\times [-N,N]=N[-1,1]^2$ can be deduced from the appropriate bounds with $N [-1,1]\times (-\infty,\infty)$ and $N(-\infty,\infty)\times [-1,1]$ which each follow from Carleson's one-dimensional theorem by simply fixing the second (or first) variable. Again, see the paper for a way to make this idea precise.

Alternatively, it is also possible to essentially repeat the one-dimensional proof of Carleson's theorem in the two-dimensional (and higher-dimensional) setting. This is way more involved and there are some technical complications, but it has some benefits (e.g. it allows you to obtain more precise information for $f$ very close to $L^1$). To this end, the original reference is

• P. Sjölin. On the convergence almost everywhere of certain singular integrals and multiple Fourier series. Ark. Mat. 9 (1971).

This work was essentially redone using modern time-frequency analysis tools in

• M. Pramanik, E. Terwilleger. A weak $L^2$ estimate for a maximal dyadic sum operator on $\mathbb{R}^n$. Illinois J. Math. Volume 47, Number 3 (2003), 775-813.