A scale of linear spaces related to the \(L_ p\) scale
Let (\(\Omega\),\(\Sigma\),m) be a \(\sigma\)-finite measure space, and let \(f^*\) represent the non-increasing rearrangement of \(| f|\). If \(0<p<\infty\), then the spaces \(Y_ p(\Omega)\), \(M_ p(\Omega)\) are defined in the introduction of this paper by \[ Y_ p(\Omega)=\{f:\quad \| f\|_{Y_ p}<\infty,\quad where\quad \| f\|_{Y_ p}=\| f^*\chi_{(0,1)}\|_ p+\| f^*\chi_{(1,\infty)}\|_ 2\}; \] \[ M_ p(\Omega)=\{f:\quad \| f\|_{M_ p}<\infty,\quad where\quad \| f\|_{M_ p}=\| f^*\chi_{(0,1)}\|_ 2+\| f^*\chi_{(1,\infty)}\|_ p\}, \] and it is observed that if \(\Omega =(0,\infty)\), then \(L_ p(0,\infty)\cap L_ 2(0,\infty)=M_ p(\Omega)\) if \(0<p\leq 2\) \((=\) \(Y_ p(\Omega)\) if \(2\leq p<\infty)\); \(L_ p(0,\infty)+L_ 2(0,\infty)=M_ p(\Omega)\) if \(2\leq p<\infty\) \((=\) \(Y_ p(\Omega)\) if \(0<p\leq 2).\) The results stated in the paper confirm isomorphism or otherwise between \(M_ p(0,\infty)\) and specified subspaces of \(L_ p\) or \(\ell_ p\)- spaces. In particular, some of the conclusions of the paper extend results involving \(L_ p\)-scales of \(L_ 1(0,\infty)\cap L_ 2(0,\infty)\), previously derived by the author in Proc. Am. Math. Soc. 103, No.4, 1185-1188 (1988; Zbl 0661.46015).
scientific article; zbMATH DE number 4134817
scientific article