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

Abstract

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