Restricted summability of the multi-dimensional Cesàro means of Walsh–Kaczmarz–Fourier series

The properties of the maximal operator of the (C, α)-means (α = (α1, . . ., α_d)) of the multi-dimensional Walsh–Kaczmarz–Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from dyadic Hardy space H_p ^γ to Lebesgue space Lp for p0 < p (p0 = max{1/(1 + α_k): k = 1, . . ., d}) and is of weak type (1, 1). As a corollary, we get a theorem of Simon on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = p0, we show that the maximal operator σ_L ^κ,α, is not bounded from the dyadic Hardy space H_p ^γ ₀ to the Lebesgue space Lp0.