On the Lebesgue summablility of truncated double Fourier series (Q519974)

Given \(1\leq p,q,r,s\leq\infty\) and \(M,N\) two positive integers, one considers the following two Banach spaces endowed with mixed norms: \(\ell^{p,q}\left(M,N\right) ,\) the space of all matrices \(A=\left(a_{mn}\right)_{m,n} \in\mathbb{C}^{M\times N}\) normed by \[ \left\| A\right\|_{p,q}=\left\| \left(\left\| (a_{m1})_{m}\right\|_{p},\ldots,\left\| (a_{mn})_{m}\right\|_{p}\right) \right\|_{q}, \] and \(L^{r,s}([0,1]\times[0,1]),\) the space of all Lebesgue measurable functions \(f:[0,1]\times[0,1]\rightarrow\mathbb{C}\) such that \(\left\| f\right\|_{r,s}=\left\| \left\| f(x,y)\right\|_{L_{x}^{r}[0,1]}\right\|_{L_{y}^{s}}<\infty.\) The paper under review offers valuable estimates for the norm of the operator \(T_{M,N}:\ell ^{p,q}\left(M,N\right) \rightarrow L^{r,s}([0,1]\times[0,1]),\) that associates to a matrix \(A=\left(a_{mn}\right)_{m,n}\) the double trigonometric sum \(S_{M,N}(x,y)=\sum_{n=1}^{N}\sum_{m=1}^{M}a_{mn}e^{2\pi i\left((m-1)x+(n-1)y\right)}\). The investigation covers the whole range of parameters \(p\), \(q\), \(r\), \(s\). For example, it is shown that \(\left\| S_{M,N} \right\|_{r,s}\leq M^{1/2-1/p}N^{1/2-1/q}\left\| A\right\|_{p,q}\) if \(p,q\geq2\) and \(1\leq r,s\leq2.\) A discussion on the optimality of these estimates and the asymptotic behavior of \(\left\| T_{M,N}\right\| \) as \(M,N\rightarrow\infty\) are also included.