On the convergence of double Fourier series of functions in \(L_p[0,1]^2\), where \(p=(p_1,p_2)\) (Q1857406)

Let \(1<p_1,p_2<+\infty\) and \({p}=(p_1,p_2)\). A function \(f\) on \(\left[0,1\right]^2\) belongs to \(L_{p}\left[0,1\right]^2\) if \[ \left\|f\right\|_{L_{p}}=\left(\int_0^1\left(\int_0^1 \left|f(x,y)\right|^{p_1}dx\right)^{p_2/p_1}dy\right)^{1/p_2}<+\infty. \] Denote by \((\phi_m)_{m\geq 1}\) either the trigonometric system or the Walsh system. For any function \(f\in L_{p}\left[0,1\right]^2\), let \(a_{m_1,m_2}, m_1,m_2\geq 0\) be the Fourier coefficients of \(f\). Assume that, for all \(m_i\leq n_i\), \[ a_{m_1,m_2}\geq a_{n_1,n_2}. \tag{1} \] The main theorems of the paper are the following: Let \({p}=(p_1,p_2)\) with \(1<p_1,p_2<+\infty\) and \(1/p_1+1/p_2<1\) and \(f\in L_{p}\left[0,1\right]^2\). If condition (1) is fulfilled, then the series \[ \sum_{m_1\geq 0} \sum_{m_2\geq 0} a_{m_1,m_2}\phi_{m_1}(x)\phi_{m_2}(y)\tag{2} \] converges in Pringsheim's sense to \(f\) everywhere in \(\left[0,1\right]^2\). Moreover, if \(1/p_1+1/p_2<3/2\), then the series (2) converges almost everywhere to \(f\) in \(\left[0,1\right]^2\). Furthermore, these results are the best possible. More precisely, if \(\frac 1{p_1}+\frac 1{p_2}>1\), then there exists a function \(f\in L_{p}\left[0,1\right]^2\) such that the square partial sums of its trigonometric Fourier series diverge at the points of a dense subset of \(\left[0,1\right]^2\) and whose Fourier coefficients satisfy (1). If \(\frac 1{p_1}+\frac 1{p_2}>3/2\), then there exists a function \(f\in L_{p}\left[0,1\right]^2\) such that each sequence of the rectangular partial sums of its Fourier series diverges almost everywhere on \(\left[0,1\right]^2\) and whose Fourier coefficients satisfy (1).