Pointwise multipliers of Calderón-Lozanovskiĭ spaces (Q2841690)

If \((\Omega,\Sigma,\mu)\) is a \(\sigma\)-finite measure space, let the classes of \(\mu\)-measurable functions be denoted by \(L^0= L^0(\Omega)\). The Banach space \({\mathcal E}=(E,\|.\|_E)\) with norm \(\|.\|_E\) is a Banach ideal space on \(\Omega\) if \(E\) is a linear subspace of \(L^0(\Omega)\) with the ideal property, so that, if \(y\in E\), \(x\in L^0(\Omega)\), and \(|x(t)|\leq|y(t)|\) for almost all \(t\in \Omega\), then \(x\in E\) and \(\| x\|_E\leq\| y\|_E\). If \(w\in L^0(\Omega)\) is nonnegative, then the weighted space \(E(w)\) is defined by \(E(w)= \{x\in L^0(\Omega): xw\in E\}\) with norm \(\| x\|_{E(w)}=\| xw\|_E\). If \({\mathcal E}= (E,\|.\|_E)\), \({\mathcal F}= (F,\|.\|_E)\) are ideal Banach spaces on \(L^0(\Omega)\), then \(M(E,F)\) denotes the space of pointwise multipliers in the form NEWLINE\[NEWLINEM(E,F)= \{x\in L^0(\Omega): xy\in F\text{ for }y\in E\},NEWLINE\]NEWLINE and \(\| x\|_{M(E,F)}\) denotes \(\sup\{\| xy\|_F: y\in E\) and \(\| y\|_E\leq 1\}\).NEWLINENEWLINE A Young function (or Orlicz function) \(\varphi: [0,\infty)\to [0,\infty)\) is defined to be a nondecreasing convex function such that \(\varphi(0)= 0\), and if \(I_\varphi(x)\) is defined to be \(\|\varphi_{\circ}|x|\,\|_E\), then the space \(E_\varphi\) is defined to be NEWLINE\[NEWLINE\{x\in L^0(\Omega): I_\varphi(cx)<\infty\text{ for some number }c= c(x)> 0\}NEWLINE\]NEWLINE and has Luxemburg-Nakano norm \(\| x\|_{E_\varphi}= \text{inf}\{\lambda> 0: I_\varphi(x/\lambda)\geq 1\}\).NEWLINENEWLINE The main theorems of this paper involve conditions on the functions \(\varphi\), \(\varphi_1\), \(\varphi_2\), for confirming estimates of the form NEWLINE\[NEWLINE \| x\|_{M(E\varphi_1, E\varphi)}\leq C\| x\|_{E\varphi_2},\;E_{\varphi_2}\subseteq M(E_{\varphi_1}, E_\varphi);NEWLINE\]NEWLINE NEWLINE\[NEWLINE \| x\|_{E\varphi_2}\leq C_0\| x\|_{M(E\varphi_1,E\varphi_2)},\;M(E_{\varphi_1}, E_\varphi)\subseteq E_{\varphi_2}.NEWLINE\]NEWLINE In particular, some of the main conditions include NEWLINE\[NEWLINE\varphi^{-1}_1(u) \varphi^{-1}_2(u)\leq \varphi^{-1}\quad\text{or}\quad \varphi^{-1}(u)\leq \varphi^{-1}_1(u)\varphi^{-1}_2(u).NEWLINE\]