Sequences of dilations and translations in function spaces

Let f∈L1[0,1] be a mean zero function and let fn, n=1,2,…, be the dyadic dilations and translations of f. We investigate conditions on f, under which the linear operator Tf defined by Tfhn=fn, n=1,2,…, where hn, n=1,2,…, are mean zero Haar functions, can be continuously extended to the closed linear span [hn] in a certain function space X. Among other results we prove that Tf is bounded in every symmetric space with nontrivial Boyd indices whenever f∈BMOd and f has “good” Haar spectral properties. In the special case of so-called Haar chaoses the above results can be essentially refined and sharpened. In particular, we find necessary and sufficient conditions, under which the operator Tf, generated by a Haar chaos f of order 1, is continuously invertible in Lp for all 1