Doob’s and Burkholder-Davis-Gundy inequalities with variable exponent
Ferenc Weisz
Source abstract
Let p ( ⋅ ) p(\cdot ) be a measurable function defined on a probability space Ω \Omega and p − ≔ inf x ∈ Ω p ( x ) p_- \colonequals \inf _{x\in \Omega }p(x) , p + ≔ sup x ∈ Ω p ( x ) p_+\colonequals \sup _{x\in \Omega }p(x) . Under a probabilistic version of the log-Hölder continuity of 1 / p ( ⋅ ) 1/p(\cdot ) , Doob’s inequality is proved if 1 > p − ≤ p + ≤ ∞ 1>p_- \leq p_+ \leq \infty . Dual Doob’s inequality, the Davis decomposition and the generalization of the Burkholder-Davis-Gundy inequality is also verified for 1 ≤ p − ≤ p + > ∞ 1 \leq p_- \leq p_+>\infty .
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