On the Dirichlet kernels with respect to the Walsh-Kaczmarz system

Abstract: In this paper we give a form of the Dirichlet kernels $D_n^\kappa $ with respect to the Walsh-Kaczmarz system. Define the operator of Sunouchi $U^\kappa f:=(\sum_{n=1}^\infty {\vert S_{n}^\kappa f-\sigma_{n}^\kappa f\vert^2 \over n})^{1\over 2}$ $(f\in L^1)$ with respect to the Walsh-Kaczmarz system. We prove that the operator $U^\kappa $ is not of type $(H^1,L^1).$

Keywords: Walsh-Kaczmarz system, Dirichlet kernel

Classification (MSC91): 42C10

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