Equality of some classical Lorentz spaces
If \(w\) is a nonnegative `weight function' on \(\mathbb R^+= (0,\infty)\) and \(1\leq p<\infty\), then \(L^p(w)\) denotes \(\{[f]:\| [f]\|_{p,w}=\| f\|_{p,w}< \infty\}\) where \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure \(0\) and \(\| f\|_{p,w}= (\int^\infty_0|f(t)|^p w(t)\,dt)^{1/p}\), and \(L^p(w)^{\text{dec}}\) denotes \(\{f\in L^p(w)\): \(f\) is nonincreasing on \(\mathbb R^+\}\). The Hardy operator \(P\) and its weighted form \(P_\lambda\), and the adjoint \(Q\) of the Hardy operator and its weighted form \(Q_\mu\) are defined by \(P= P_1\), \(P_\lambda(f)(x)= x^{\lambda- 1}\int^x_0 t^{-\lambda} f(t)\,dt\); \(Q= Q_1\), \(Q_\mu(f)(x)= x^{\mu-1}\int^\infty_x t^{-\mu}f(t)\,dt\). The product formula and composition identities satisfied by the operators have the form \(\int^\infty_0 g(t) P_\lambda(f)(t)\,dt= \int^\infty_0 f(x) Q_{1-\lambda}(g)(x)\,dx\); \((\mu- \lambda)Q_\mu P_\lambda(f)= Q_\mu(f)+ P_\lambda(f)\) \((\lambda< \mu)\). If \(M\) denotes the Hardy-Littlewood maximal function, defined by \(M(f)(x)= \sup\{(x- t)^{-1}|\int^x_t f(\xi)\,d\xi|\), \(0\leq t< x\}\), and if \(f^*\) denotes the nonincreasing rearrangement of the function \(f\), so that \(P(|f|)(x)\leq M(|f|)(x)\) and for some constant \(C\), \((1/C)P(f^*)(s)\leq M(f^*(s)\leq CP(f^*)(s)\), \(s> 0\), let \(\Lambda^p(w)\), \(M_p\), \(B_p\) denote, respectively, \(\{f: f^*\in L^p(w)\}\), \(\{w:P\) is bounded in \(L^p(w)\}\), \(\{w: M\) is bounded in \(\Lambda^p(w)\}\). In the main results of this paper, the authors show that (i) if \(w\in B_p\), then there is a weight \(\widehat w\in M_p\) such that \(L^p(w)= L^p(\widehat w)\); (ii) if \(Q: L^p(w)^{\text{dec}}\to L^p(w)^{\text{dec}}\) is bounded, then there is a weight function \(\widehat w\) such that \(\Lambda^p(w)= \Lambda^p(\widehat w)\); (iii) if \(M: \Lambda^p(w)\to \Lambda^p(w)\) and \(Q: L^p(w)^{\text{dec}}\to L^p(w)^{\text{dec}}\) are bounded operators, then there is a weight function \(\overline\omega\) in \(B_p\) such that \(Q: L^p(\overline\omega)^{\text{dec}}\to L^p(\overline\omega)^{\text{dec}}\), and \(\Lambda^p(w)= \Lambda^p(\overline\omega)\).
scientific article; zbMATH DE number 5070752