Rademacher functions in Morrey spaces

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The Rademacher sums are investigated in the Morrey spaces M_p,w

M_{p, w}

on [0,1]

[0, 1]

for 1≤p<∞

1 \leq p < \infty

and weight w being a quasi-concave function. They span l₂

l_{2}

space in M_p,w

M_{p, w}

if and only if the weight w is smaller than

\log_{2}^{- 1 / 2} \frac{2}{t}

on (0,1)

(0, 1)

. Moreover, if 1<p<∞

1 < p < \infty

the Rademacher subspace R_p,w

R_{p, w}

is complemented in M_p,w

M_{p, w}

if and only if it is isomorphic to l₂

l_{2}

. However, the Rademacher subspace R_1,w

R_{1, w}

is not complemented in M_1,w

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

M_{p, w}

is described. It turns out that for any infinite-dimensional subspace X of R_p,w

R_{p, w}

the following alternative holds: either X is isomorphic to l₂

l_{2}

or X contains a subspace which is isomorphic to c₀

c_{0}

and is complemented in R_p,w

R_{p, w}

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