Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H p to the space L p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σα,κ,* is bounded from the martingale Hardy space H 1/(1+α) to the space weak- L 1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H p such that the maximal operator σα,κ,* f does not belong to the space L p .
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 2, pp. 158–166, February, 2010.
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Goginava, U., Nagy, K. On the maximal operator of (C, α)-means of Walsh–Kaczmarz–Fourier series. Ukr Math J 62, 175–185 (2010). https://doi.org/10.1007/s11253-010-0342-6
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DOI: https://doi.org/10.1007/s11253-010-0342-6