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 fractional and introduce $$M_{\gamma ,s,\alpha }$$ M γ , s α , a new for martingales. prove that under log-Hölder continuity condition of variable exponents $$p(\cdot )$$ p ( · ) $$q(\cdot q is bounded from Lebesgue space $$L_{q(\cdot )}$$ L to $$L_{p(\cdot Hardy $$H_{q(\cdot H whenever $$0 \le \alpha <1$$ 0 ≤ < 1 $$0<q_-\le q_+ 1/\alpha $$ - + / $$0<\gamma ,s<\infty ∞ $$1/p(\cdot )= 1/q(\cdot )- = $$1/p_- - 1/p_+ < \gamma +s$$ . Moreover, $$\alpha =0$$ ,s,0}$$ generates equivalent quasi-norms on spaces $$H_{p(\cdot