The exact $L_1$-norm supremum of Walsh--Kaczmarz--Fej\'er kernels

We study the $L_1(G)$-norms of Fejér kernels for the Walsh--Kaczmarz system. In the Walsh--Paley case the identity \[
\|K^w_{2^n}\|_1=1 \qquad (n\in\mathbb N) \] follows directly from the binary structure. Toledo's sharp result gives the global supremum \[
\sup_{n\in\mathbb P}\|K^w_n\|_1=\frac{17}{15}. \] The Walsh--Kaczmarz kernels behave differently. We first prove that \[
\sup_{n\in\mathbb N}\|K^\kappa_{2^n}\|_1=\lim_{n\to\infty}\|K^\kappa_{2^n}\|_1=\frac43, \] and we prove that the sequence $(\|K^\kappa_{2^n}\|_1:n\in\mathbb N)$ 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 \[
\sup_{n\in\mathbb P}\|K^\kappa_n\|_1=\frac{71}{50}. \]