Fejér summability of multi-parameter bounded Ciesielski systems

Abstract

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejer means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space Hp([0, 1)d1×…×[0, 1)dl to Lp ([0, 1)d) if 1/2<p<∞ and mj≥0, ‖kj‖≤mj+1. By an interpolation theorem, we get that the maximal operator is also of weak type ( $$(H_1^{\# _i } ,L_1 )$$ ) (i=1,…,l), where the Hardy space $$H_1^{\# _i } $$ is defined by a hybrid maximal function and $$H_1^{\# _i } \supset L(\log L)^{l - 1} $$ . As a consequence, we obtain that the Fejer means of the Ciesielski-Fourier series of a function f converge to f a.e. if f ∈ $$H_1^{\# _i } $$ and converge in a cone if feL1.