The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces

Let E be a rearrangement-invariant Banach function space. Let $\mathcal{P}\left(\Omega \right)$ denote the collection of all measurable variable exponents $p(·):\Omega \to (0,\infty )$ such that $0<\mathrm{ess}{\inf }_{w\in \Omega }p\left(w\right)\le \mathrm{ess}{\sup }_{w\in \Omega }p\left(w\right)<\infty$. In this paper, with the help of a new atomic decomposition of the variable Hardy–Lorentz–Karamata space $H_{p(·),q,b}^M$ via ${(s,p(·),E)}^M$-atoms, we characterize the dual space of $H_{p(·),q,b}^M$ for the two cases $0<q\le 1$ and $1<q\le \infty$, respectively. Using this, some new John–Nirenberg theorems associated with variable exponents are also established.