The UMD conjecture is reduced to L^p boundedness of a pair of simple dyadic shift operators via equivalence with the Hilbert transform.
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Martingale square functions with matrix weights satisfy L_p bounds characterized by matrix A_p conditions, with sharpness for 1<p≤2 and optimal exponents achieved in the scalar case for all 1<p<∞.
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A twosided linear estimate and a dyadic reduction of the UMD Conjecture
The UMD conjecture is reduced to L^p boundedness of a pair of simple dyadic shift operators via equivalence with the Hilbert transform.
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Sharp weighted norm estimates for martingale square functions
Martingale square functions with matrix weights satisfy L_p bounds characterized by matrix A_p conditions, with sharpness for 1<p≤2 and optimal exponents achieved in the scalar case for all 1<p<∞.