Cesaro Summability with Respect to Two-Parameter Walsh Systems |
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Authors: | Péter Simon |
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Institution: | L. E?tv?s University, Budapest, Hungary, HU
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Abstract: | The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show
that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H
p
to L
p
for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T
α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T
α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the
usual two-parameter maximal operators of the (C, 1)-means are bounded from L
p
spaces to L
p
if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from
the two-dimensional case can be extended to higher dimension in a simple way.
(Received 20 April 2000; in revised form 25 September 2000) |
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Keywords: | 2000 Mathematics Subject Classifications: 42C10 43A75 42B08 42B30 |
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