Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh–Fourier Series
Tóm tắt
In the paper, we are going to prove that the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series is bounded from
$$L\log L\left( (0,1]\times (0,1]\right) $$
to
$$L_{1,\infty }((0,1]\times (0,1])$$
. As a consequence, it can be obtained that
$$L\log L\left( (0,1]\times (0,1]\right) $$
is maximal Orlicz space, where the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series for the functions from this space converge in measure.
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