New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

Abstract

We generalize the usual Doob maximal operator as well as the fractional maximal operator and introduce M_{gamma ,s,alpha }, a new fractional maximal operator for martingales. We prove that under the log-Hölder continuity condition of the variable exponents p(cdot ) and q(cdot ), the maximal operator M_{gamma ,s,alpha } is bounded from the variable Lebesgue space L_{q(cdot )} to L_{p(cdot )} and from the variable Hardy space H_{q(cdot )} to L_{p(cdot )}, whenever 0 le alpha <1, 0<q_-le q_+ le 1/alpha , 0<gamma ,s<infty , 1/p(cdot )= 1/q(cdot )- alpha and 1/p_- - 1/p_+ < gamma +s. Moreover, for alpha =0, the operator M_{gamma ,s,0} generates equivalent quasi-norms on the Hardy spaces H_{p(cdot )}.