Estimates for Lebesgue constants in dimension two (Q1813403)

Fourier series of integrable functions fail to converge in the mean. The divergence is measured by the \(L^ 1\)-norms of the Dirichlet kernel; they are called Lebesgue constants and they provide positive summability results for certain classes of functions [cf. \textit{D. I. Cartwright} and \textit{P. M. Soardi}, J. Approximation Theory 38, 344-353 (1983; Zbl 0516.42020)]. An important result of K. Babenko (see the above paper for the reference) shows that for the \(N\)-dimensional torus there are constants \(A\) and \(B\) such that \(AR^{(N-1)/2}\leq\| D_ R\|_ 1\leq BR^{(N-1)/2}\), where \(D_ R\) is the Dirichlet kernel associated to a ball of radius \(R\). Several extensions of Babenko's results have been proved. The paper under review is one of them: roughly speaking, it shows that in the 2- dimensional case the disk can be substituted by the interior of a piecewise regular curve with curvature different from zero in at least one point. The proof is direct and the result is best possible since, e.g., the Lebesgue constants associated to a polyhedron have a logarithmic growth.