Equivalent characterizations of martingale Hardy–Lorentz spaces with variable exponents

Abstract We prove that under the log-Hölder continuity condition of variable exponent $$p(\cdot )$$ p ( · ) , a new type maximal operators, $$U_{\gamma ,s}$$ U γ , s is bounded from martingale Hardy–Lorentz space $$H_{p(\cdot ),q}$$ H q to $$L_{p(\cdot L whenever $$0<p_-\le p_+ <\infty $$ 0 < - ≤ + ∞ $$0<q \le \infty $$0<\gamma ,s<\infty and $$1/p_- - 1/p_+ < \gamma +s$$ 1 / . Moreover, operator generates equivalent quasi-norms on spaces