Rademacher functions in weighted Cesàro spaces (Q2847825)

For \(p\in[1,\infty]\) and \(\omega\) a weight (i.e., a measurable function on \([0,1]\) with \(0<\omega(x)<\infty\) a.e.), the Cesàro space \(\mathrm{Ces}(p,\omega)\) is defined as the space of all measurable functions \(f:I\rightarrow\mathbb{R}\) such that \(\left\| f\right\| _{\mathrm{Ces}(p,\omega )}=\left\| H_{\omega}(f)\right\| _{L^{p}}<\infty,\) where \(H_{\omega }(f)=\frac{1}{\omega(x)}\int_{0}^{x}\left| f(t)\right| dt,\) for \(x\in[0,1].\) Let \((r_{n}(t))_{n}\) be the sequence of Rademacher functions. As is well known, a Rademacher series \(\sum_{n=1}^{\infty} c_{n}r_{n}(t)\) converges a.e. if and only if the sequence of its coefficients \(c_{n}\) is square summable. This paper describes (in terms of \(p\) and \(\omega)\) the behavior of the space \(\{\sum_{n=1}^{\infty}c_{n}r_{n}:(c_{n})_{n}\in\ell^{2}\}\cap \mathrm{Ces}(p,\omega)\) in \(\mathrm{Ces}(p,\omega)\).