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

Let E be a rearrangement-invariant Banach function space. Let P(Ω) denote the collection of all measurable variable exponents p(·):Ω→(0,∞) such that 0<essinfw∈Ωp(w)≤esssupw∈Ωp(w)<∞. In this paper, with the help of a new atomic decomposition of the variable Hardy–Lorentz–Karamata space Hp(·),q,bM via (s,p(·),E)M-atoms, we characterize the dual space of Hp(·),q,bM for the two cases 0<q≤1 and 1<q≤∞, respectively. Using this, some new John–Nirenberg theorems associated with variable exponents are also established.