Permutations of the Haar system
نویسنده
چکیده
Let us briefly describe the setting in which we are working. D denotes the set of all dyadic intervals contained in the unit interval. π : D → D denotes a permutation of the dyadic intervals. The operator induced by π is determined by the equation TπhI = hπ(I) where hI denotes the L∞-normalised Haar function supported on the dyadic intervall I. The main result of this paper treats general permutations on BMO and on Lipschitz spaces. The condition on π which controlles the boundedness of Tπ is given in termes of the Carleson constant of collections of dyadic intervals. The proof of the general result given below is quite complicated. We start therefore by considering first a special class of permutation operators. To study these operators on Lp E.M. Semyonov introduced the parameter,
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