Some properties of the relative rearrangement
Let \(\Omega \subset R^ N\) be a bounded Lebesgue measurable set of measure \(| \Omega |\). The decreasing rearrangement of a real measurable function u on \(\Omega\) is defined by \[ u_*(s)=\inf \{\theta \in R,| \{x,u(x)>\theta \}| \leq s\},\quad s\in {\bar \Omega}^*=[0,| \Omega |]. \] Given \(t\in R\) we put \(P(t)=\{x\in \Omega,\quad u(x)=t\}.\) If \(| P(t)| >0,\) then P(t) is called the flat region of value t of the function u. \textit{J. Mossino} and \textit{R. Temam} [Duke Math. J. 48, 475-495 (1981; Zbl 0476.35031)] introduced the notion of relative rearrangement: For \(v\in L^ p(\Omega),\quad 1\leq p\leq \infty,\) define a function w on \({\bar \Omega}{}^*\) by \[ w(s)=\int_{u>u_*(s)}v(x)dx+\int^{s-| \{u>u_*(s)\}|}_{0}(v|_{P(u_*(s))})_*(\sigma)d\sigma, \] where \(v|_ E\) is the restriction of v to the set E. The function \(v_{*u}=dw/ds\) satisfies \(\| v_{*u}\|_{L^ p(\Omega^*)}\leq \| v\|_{L^ p(\Omega)}\) and is called the relative rearrangement of v with respect to u. The author proves various properties of the relative rearrangement, among others the following generalization of the well-known Hardy-Littlewood inequality: If u is a real measurable function on \(\Omega\), \(v_ 1\in L^ p(\Omega),\quad v_ 2\in L^ q(\Omega^*),\quad 1/p+1/q=1,\) then \[ \int_{\Omega^*}v_{1*u}v_ 2 d\sigma \leq \int_{\Omega^*}v_{1*}v_{2*} d\sigma. \]
scientific article; zbMATH DE number 4124024
scientific article