On some properties of generalized Marcinkiewicz spaces (Q2717558)

Let \((X_0,X_1)\) be a couple of interpolation of Banach spaces, and, for \(t>0\), and \(x\in X_0+X_1\): NEWLINE\[NEWLINEK(t,x)= \inf\{\|x_0\|_{X_0}+t \|x_1\|_{X_1}, \;x=x_0+x_1\} .NEWLINE\]NEWLINE For any quasiconcave function \(\varphi: (0,\infty)\to (0,\infty)\), the generalized Marcinkiewicz space \(M_\varphi=M_\varphi(X_0,X_1)\) is the set of \(x\in X_0+X_1\) for which NEWLINE\[NEWLINE\|x\|_{M_\varphi}=\sup_{0<t<+\infty} K(t,x)/\varphi(t)<+\infty .NEWLINE\]NEWLINE For \(X_0\cap X_1\subseteq X\subseteq X_0+X_1\), \(X^\circ\) is the closure in \(X\) of \(X_0\cap X_1\). The author shows:NEWLINENEWLINENEWLINETheorem 2: Let \(\varphi,\psi\) be two quasiconcave functions. If there exists a sequence \(t_n\rightarrow_{n\to+\infty} +\infty\) (or \(t_n\rightarrow_{n\to+\infty} 0\)) such that \(\varphi_n(t_n)\sim \psi(t_n)\), then there exists a Banach couple \((X_0,X_1)\) for which \(M_\varphi(X_0,X_1)= M_\psi(X_0,X_1)\). If such a sequence exists neither on \((0,1)\), when \(X_0^\circ\not\subseteq X_1^\circ\), nor on \((1,+\infty)\), when \(X_1^\circ\not\subseteq X_0^\circ\), then \(M_\varphi(X_0,X_1)\not= M_\psi(X_0,X_1)\) for any nontrivial Banach couple. NEWLINENEWLINENEWLINETheorem 3: Let \(\varphi\) be a quasiconcave function which is not equivalent to any of the functions \(1, t, \min(1,t), \max(1,t)\). Then, a Banach space \(X\) is equal to the space \(M_\varphi(X_0,X_1)\) for some nontrivial Banach couple \((X_0,X_1)\) if and only if it contains a subspace isomorphic to \(\ell_\infty\).NEWLINENEWLINENEWLINEThe above excluded functions correspond respectively to \(X_0\), \(X_1\), \(X_0\cap X_1\) and \(X_0+X_1\).NEWLINENEWLINENEWLINETheorem 4: If \(X_0\) and \(X_1\not\subseteq M_\varphi(X_0,X_1)\), then no embedding \(M_\psi(X_0,X_1)\subset_{\not=} M_\varphi(X_0,X_1)\) can be dense. NEWLINENEWLINENEWLINEThe main tool is the notion of \textit{\(K\)-envelope} of \(X\) (for \(X_0\cap X_1\subseteq X\subseteq X_0+X_1\)): NEWLINE\[NEWLINE\mu(t,X)=\sup_{\|x\|_X\leq 1} K(t,x) ,\quad t>0 .NEWLINE\]