On everywhere divergent Fourier series of continuous functions of two variables (Q1594006)

The author sketches the construction of a continuous function \(G(x,y)\), periodic in each variable, such that for its modulus of continuity \(\omega(G,\delta)\) we have \[ \omega(G, \delta)= O\{(\log(1/\delta))^{- 1}\}\quad\text{as}\quad \delta\to +0, \] and the sequence of the rectangular partial sums \(s_{mn}(G, x,y)\) of the Fourier series of \(G\) diverges everywhere as \(m,n\to\infty\) and \(1/2\leq m/n\leq 2\).