Doob’s and Burkholder-Davis-Gundy inequalities with variable exponent

Let p ( ⋅ ) p(· ) be a measurable function defined on a probability space Ω Ω and p − ≔ inf x ∈ Ω p ( x ) p_- inf x∈ Ωp(x) , p + ≔ sup x ∈ Ω p ( x ) p_+ x∈ Ωp(x) . Under a probabilistic version of the log-Hölder continuity of 1 / p ( ⋅ ) 1/p(· ) , Doob’s inequality is proved if 1 > p − ≤ p + ≤ ∞ 1>p_- ≤ p_+ ≤ ∞ . Dual Doob’s inequality, the Davis decomposition and the generalization of the Burkholder-Davis-Gundy inequality is also verified for 1 ≤ p − ≤ p + > ∞ 1 ≤ p_- ≤ p_+>∞ .