Essential divergence in measure of multiple orthogonal Fourier series (Q2510985)

Let \(\{\varphi_{m,n}\}\), \((m,n= 1,2,\dots)\) be a uniformly bounded orthonormal system on \(l= [1,2]^2\) and denote by \(L_{m,p}\) the \((m,p)\)-th Lebesgue function of the system. If there exist \(\varepsilon> 0\) and an increasing sequence \(\{r_n\}\), \((n= 1,2,\dots)\), of positive integers such that \(L_{r_n,r_m}(x,y)> \ln^{1+\varepsilon}r_n\) a.e. in \(l\), then there exists \(f\in L(l)\) such that its Fourier series with respect to the system \(\{\varphi_{m,n}\}\) diverges in measure on every square \(Q\), \(Q\subset l\) and \(\mu(Q)> 0\).