DiPerna-Majda measures and uniform integrability (Q2713589)

The topic of uniform integrability is discussed from the point of view of relations to Young measures and DiPerna-Majda measures. A formula for the Rosenthal modulus of uniform integrability of \(\{|u_k|^p\}\) is obtained, namely NEWLINE\[NEWLINE \eta (\{|u_k|^p\})=\sup _{(\sigma ,\widehat \nu)\in \mathcal U}\int _{\bar \Omega } \int _{\beta _{\mathcal R}{\mathbb R}^m\setminus {\mathbb R}^m}\widehat \nu (d\lambda)\sigma (dx), NEWLINE\]NEWLINE where \(\mathcal U\) is the set of all DiPerna-Majda measures \((\sigma ,\widehat \eta)\) that are generated by the same subsequence of \(\{u_k\}\) and computed with respect to a given ring \(\mathcal R\) of bounded continuous functions on \({\mathbb R}^m\). As an application, the following inequality related to Fatou's lemma is established: NEWLINE\[NEWLINE \int _{\Omega }\liminf _{k\to \infty }|u_k(x)|^p dx \leq \inf _{\nu \in \mathcal V}\int _{\Omega }\int {{\mathbb R}^m}|\lambda |^p\nu _x(d\lambda) dx \leq \liminf _{k\to \infty }\int _{\Omega }|u_k(x)|^p dx , NEWLINE\]NEWLINE where \(\mathcal V\) is the set of all Young measures generated by some subsequence of \(\{u_k\}\).