Proves necessity of martingale cotype q for vector q-variational inequalities of averaging operators and characterizes UMD and cotype q via Hilbert transform variational inequalities.
Xu, Littlewood-Paley theory for functions with values in unifo rmly convex spaces , J
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Vector-valued $q$-variational inequalities for averaging operators and Hilbert transform
Proves necessity of martingale cotype q for vector q-variational inequalities of averaging operators and characterizes UMD and cotype q via Hilbert transform variational inequalities.