Rademacher functions in Morrey spaces

TL;DR

This paper explores the structure and properties of Rademacher functions within Morrey spaces, revealing conditions for their span, complementability, and geometric subspace structures depending on parameters p and weight w.

Contribution

It characterizes when Rademacher functions span, are complemented, and describes the geometric subspace structure in Morrey spaces for various parameters.

Findings

Rademacher functions span l_2 in M(p,w) if w is sufficiently small.

R_p is complemented in M(p,w) iff it is isomorphic to l_2 for 1 < p < ∞.

Rademacher subspace is not complemented in M(1,w) for any quasi-concave w.

Abstract

The Rademacher functions are investigated in the Morrey spaces M(p,w) on [0,1] for 1 \le p <\infty and weight w being a quasi-concave function. They span l_2 space in M(p,w) if and only if the weight w is smaller than the function log_2^{-1/2}(2/t) on (0,1). Moreover, if 1 < p < \infty the Rademacher sunspace R_p is complemented in M(p,w) if and only if it is isomorphic to l_2. 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 R_p the following alternative holds: either X is isomorphic to l_2 or X contains a subspace which is isomorphic to c_0 and is complemented in R_p.