Factorization in Lorentz spaces, with an application to centralizers (Q333875)

A classical result by Lozanovsky provides a factorization for any function belonging to \(L^1\) as a product of two functions, one of them belonging to a function space \(X\) and the other one to its Köthe dual in a way that the product of the norms is (almost) equal to the norm of \(f\) in \(L^1\). In the same direction of this fundamental result, we can find in the mathematical literature several relevant theorems providing factorizations for functions belonging to other function spaces and giving some control on their norms. However, very often -- beyond the case of the \(L^p\)-spaces -- it is not easy to explicitly compute such factorization.NEWLINENEWLINEThe present paper provides this factorization for the case of Lorentz spaces. The main theorem gives the followingNEWLINENEWLINETheorem. Consider extended real numbers \(0<p_0,p_1,q_0,q_1,p,q\) and the Lorentz spaces \(L(p_0,q_0),\) \(L(p_1,q_1)\) and \(L(p,q)\). Suppose that \( (p_0,q_0)^{-1} + (p_1,q_1)^{-1}=(p,q)^{-1}\). Then there is a constant \(M\) depending only on \(p\) and \(q\) such that {\parindent=0.7cm \begin{itemize}\item[(a)] If \(f_i \in L(p_i,q_i)\), \(i=0,1\), then \(f_0 f_1 \in L(p,q)\) and NEWLINE\[NEWLINE \|f_0 f_1 \|_{p,q} \leq M \|f_0\|_{p_0,q_0} \, \|f_1\|_{p_1,q_1}. NEWLINE\]NEWLINE \item[(b)] If \(f \in L(p,q)\), then there are \(f_i \in L((p_i,q_i)\), \(i=0,1\), such that \(f=f_0 f_1\) and NEWLINE\[NEWLINE \|f_0\|_{p_0,q_0} \, \|f_1\|_{p_1,q_1} \leq M \|f \|_{p,q}. NEWLINE\]NEWLINE \item[(c)] If \( q< \infty\) and \(f \geq 0\), one can take \(f_i= f^{q q_i^{-1}} \, r_f^{qp^{-1} q_i^{-1}- p_i^{-1}}\) for \(i=0,1\) in (b), where \(r_f\) is the rank function for \(f\). For \(q= \infty\) and \(f \geq 0\), one may take \(f_i=f^{p p_i^{-1}}\) in (b). NEWLINENEWLINE\end{itemize}} Given a function \(f\), the rank function \(r_f\) is given by NEWLINE\[NEWLINE r_f(t) = m \{s: |f(s)| > |f(t)| \, \text{or}\, |f(s)|= |f(t)| \, \text{and}\, s \leq t \}. NEWLINE\]NEWLINE Some applications to centralizers are also given.