The exact $L_1$-norm supremum of Walsh--Kaczmarz--Fejér kernels

We study the $L_1(G)$-norms of Fejér kernels for the Walsh–Kaczmarz system. In the Walsh–Paley case the identity $$\mathrm{\parallel }{K}_{2^n}^w{\mathrm{\parallel }}_1=1(n\in \mathbb{N})$$ follows directly from the binary structure. Toledo’s sharp result gives the global supremum $$\underset{n\in \mathbb{P}}{\sup }\mathrm{\parallel }{K}_n^w{\mathrm{\parallel }}_1=\frac{17}{15}\mathrm{.}$$ The Walsh–Kaczmarz kernels behave differently. We first prove that $$\underset{n\in \mathbb{N}}{\sup }\mathrm{\parallel }{K}_{2^n}^{\kappa }{\mathrm{\parallel }}_1=\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{K}_{2^n}^{\kappa }{\mathrm{\parallel }}_1=\frac{4}{3},$$ and we prove that the sequence $(\mathrm{\parallel }{K}_{2^n}^{\kappa }{\mathrm{\parallel }}_1:n\in \mathbb{N})$ is non-decreasing and is strictly increasing from $n=2$ on. Then we use this result, an exact splitting formula from Skvortsov’s decomposition, and the sharp Walsh–Paley estimates. This gives the exact global supremum $$\underset{n\in \mathbb{P}}{\sup }\mathrm{\parallel }{K}_n^{\kappa }{\mathrm{\parallel }}_1=\frac{71}{50}\mathrm{.}$$