For every p>1 and r>2, the weighted L_p norm of the pathwise r-variation of a martingale is bounded by C_p sqrt(r/(r-2)) times the A_p characteristic of the weight to a power, times the weighted L_p norm of the martingale.
Sharp Weighted $L^2$ inequalities for square functions
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abstract
Using Bellman function approach, we present new proofs of weighted $L^2$ inequalities for square functions, with the optimal dependence on the $A_2$ characteristics of the weight and further explicit constants. We study the estimates both in the analytic and probabilistic context, and, as application, obtain related estimates for the classical Lusin and Littlewood-Paley square functions.
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math.PR 1years
2019 1verdicts
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Weighted L\'epingle inequality
For every p>1 and r>2, the weighted L_p norm of the pathwise r-variation of a martingale is bounded by C_p sqrt(r/(r-2)) times the A_p characteristic of the weight to a power, times the weighted L_p norm of the martingale.