Rademacher functions in Morrey spaces

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The Rademacher functions are investigated in the Morrey spaces M(p,w) on [0,1] for 1 ≤ p <∞ and weight w being a quasi-concave function. They span l₂ space in M(p,w) if and only if the weight w is smaller than the function log₂⁻¹/²(2/t) on (0,1). Moreover, if 1 < p < ∞ the Rademacher sunspace Rₚ is complemented in M(p,w) if and only if it is isomorphic to l₂. However, the Rademacher subspace is not complemented in M(1,w) for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces M(p,w) is described. It turns out that for any infinite-dimensional subspace X of Rₚ the following alternative holds: either X is isomorphic to l₂ or X contains a subspace which is isomorphic to c₀ and is complemented in Rₚ.