Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces

Boundedness of dyadic maximal operators on variable Lebesgue spaces

“…Next, we verify that V b is bounded on L pðÁÞ if 1\p À p þ \1 and b [ 0, without any additional condition. This was stated in [12] without proof.…”

“…Instead of the log-Hölder continuity condition, in [4,12], we introduced the slightly more general condition kðIÞ p À ðIÞÀp þ ðIÞ C ð1Þ for all dyadic intervals I & ½0; 1Þ. By a dyadic interval, we mean one of the form ½k2 Àn ; ðk þ 1Þ2 Àn Þ for some k; n 2 N, 0 k\2 n .…”

“…Usually, we suppose that pðÁÞ satisfies the Hölder continuity condition. In [4,5,12] as well as in this paper, we suppose a slightly more general condition. It is known (see [4,5]) that the usual dyadic (or martingale) maximal operator is bounded on L pðÁÞ with 1\p À p þ \1.…”

“…In [12], we investigated two more dyadic maximal operators denoted by U b;s and V b , where b and s are positive parameters. We stated there that U b;s is bounded on L pðÁÞ if 1\p À p þ \1 and b; s [ 0.…”

“…In [12], we used the boundedness of the above dyadic maximal operators to prove that the maximal Cesàro (or ðC; aÞ) and Riesz operators of the Walsh-Fourier series are bounded from the variable Hardy space H pðÁÞ to L pðÁÞ if 1=ða þ 1Þ\p À \1.…”