A multiplier in double Fourier series (Q1180813)

Let \(f(x)=f(x_ 1,x_ 2)\) be a function defined on the two-dimensional torus and let \[ S_ \Gamma f(x)\sim\sum_{j\in\Gamma\cap Z^ 2}\hat f_ j e^{2\pi ijx}, \] where \(\Gamma\) is a broken line with vertices on the graph of a strictly convex curve \(x_ 2=x_ 2(x_ 1)\), \(x_ 1\geq 0\), \(x_ 2\geq 0\). The author studies boundedness of the operator \(S_ \Gamma\) in \(L^ p\). The case of a finite number of components of \(\Gamma\) is known to follow from the results of M. Riesz for all \(p\in(1,\infty)\), while the case when the number of integer points of \(Z^ 2\) on every component of \(\Gamma\) is bounded is connected with some R. Cooke's results for \(4/3\leq p\leq 4\). The author proves that \[ \sup_{f\in L^ p, \| f\|\leq 1}\| S_ \Gamma f\|<\infty \tag{1} \] for \(4/3\leq p<4\) whatever \(\Gamma\) is. He also gives an example of a polygon \(\Gamma\) for which (1) is certainly false in the cases \(1<p<5/4\) and \(p>5\).