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We find lower bounds for linear and Alexandrov's cowidths of Sobolev's classes on Compact Riemannian homogeneous manifolds M d . Using these results we give an explicit solution of the problem of optimal reconstruction of functions from Sobolev's classes

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We study weighted inequalities for vector valued extensions of the conditioned square function operator and of the maximal operators of matrix type in the case of regular martingales. As applications we obtain weighted inequalities for vectorvalued extensions of the Hardy᎐Littlewood maximal operator and of singular integral transforms on martingales.