Equivalent characterizations of martingale Hardy–Lorentz spaces with variable exponents

Abstract

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