Rademacher functions in Cesaro type spaces

The Rademacher sums are investigated in the Cesaro spaces Ces(p) (1 <= p <= infinity) and in the weighted Korenblyum-Krein-Levin spaces K(p,w) on [0,1]. They span l(2) space in Ces(p) for any 1 <= p < infinity and in K(p,w) if and only if the weight w is larger than t log(2)(p/2)(2/t) on (0,1). Moreover, the span of the Rademachers is not complemented in Ces(p) for any 1 <= p < infinity or in K(1,w) for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l(2), this span is a complemented subspace in K(p,w).