It is known that the maximal operator σκ,∗(f):=supn∈P|σκn(f)|σκ,∗(f):=supn∈P|σnκ(f)| is bounded from the dyadic Hardy space HpHp into the space LpLp for p>2/3p>2/3 [6]. Moreover, Goginava and Nagy showed that σκ,∗σκ,∗ is not bounded from the Hardy space H2/3H2/3 to the space L2/3L2/3 [9]. The main aim of this paper is to investigate the case 0