Rademacher functions in Morrey spaces
This is an interesting paper. In the well-known Morrey spaces on \([0,1]\), \(M_{p,w},\) where \( 1\leq p<\infty,\) and \(w\) is a non-decreasing positive weight, consisting of all Lebesgue measurable functions on \([0,1]\) which satisfy the condition \[ ||f||_{M_{p,w}}:=\sup_{I}w(|I|)\left( \frac{1}{|I|}\int_{I}|f(t)|^{p}dt \right)^{1/p}<\infty, \] the authors study the subspace \(\mathcal{R}_{p,w}\) generated by the Rademacher functions \(r_{n}\), \(n\in \mathbb{N}\), proving four theorems. The first one gives the estimation of the sum \(\sum_{k=1}^{\infty}a_{k}r_{k}\) in the norm in \(M_{p,w}.\) Theorem 1. With a constant depending only on \(p\), we have \[ \left\|\sum_{k=1}^{\infty}a_{k}r_{k}\right\|_{M_{p,w}}\approx ||(a_{k})_{k=1}^{\infty} ||_{\ell_{2}}+\sup_{m\in \mathbb{N}}\left(w(2^{-m})\sum_{k=1}^{m}|a_{k}| \right). \] The second theorem gives a necessary and sufficient condition in order that \(\mathcal{R}_{p,w}\) is a complemented subspace of \(M_{p,w}\). More specifically, this condition is \[ \sup w(t)\log_{2}^{1/2}(2/t)<\infty \] when \(t\in (0,1]\). The third theorem states that the subspace \(\mathcal{R}_{p,w}\) is not complemented in the Morrey space \(M_{1,w}\). Finally, the fourth theorem gives some information about the structure of \(\mathcal{R}_{p,w}\). Namely, if \(1\leq p<\infty\) and \(\lim_{t\to 0^{+}} w(t)=0\), then every infinite-dimensional subspace of \(\mathcal{R}_{p,w}\) is either isomorphic to \(\ell_{2}\) or it contains a subspace which is isomorphic to \(c_{0}\) and is complemented in \(\mathcal{R}_{p,w}.\)
scientific article; zbMATH DE number 6618078
scientific article