Let \(\left\{k\in \mathbb{Z}^{n}:1 \leq \prod_{j=1}^{n}|k_{j}|^{\gamma_{j}}\leq R^{\gamma_{1}+\ldots+\gamma_{n}}\right\},\) where γ1,…,γn > 0, be a “hyperbolic crosses” dilated homothetically as R → +∞. In our study, their Lebesgue constants are, as expected, always of power growth Rp, p > 0, maybe up to a logarithmic factor. What turned out to be surprising is that contrary to the expected \(p={n-1\over 2}\) in any case, p may become, for an appropriate choice of γ1,…,γn, an arbitrary number larger than that fraction. In many cases, the estimates of the Lebesgue constants are sharp in the sense that those from above and from below differ from one another only by coefficients.
