Dual spaces for variable martingale Lorentz–Hardy spaces

Let $$H_{p(\cdot ),q}$$ be the variable Lorentz–Hardy martingale spaces. In this paper, we give a new atomic decomposition for these spaces via simple $$L_r$$ -atoms $$(1<r \le \infty )$$ . Using decomposition, consider dual of Lorentz-Hardy case $$0<p(\cdot )\le 1$$ , $$0<q\le and )<2$$ $$1<q<\infty $$ respectively, prove that they are equivalent to BMO with exponent. Furthermore, also obtain several John-Nirenberg theorems based on results.