The Bishop-Phelps-Bollob\'{a}s theorem for operators on $L_1(\mu)$

In this paper we show that the Bishop-Phelps-Bollob\'as theorem holds for $\mathcal{L}(L_1(\mu), L_1(\nu))$ for all measures $\mu$ and $\nu$ and also holds for $\mathcal{L}(L_1(\mu),L_\infty(\nu))$ for every arbitrary measure $\mu$ and every localizable measure $\nu$. Finally, we show that the Bishop-Phelps-Bollob\'as theorem holds for two classes of bounded linear operators from a real $L_1(\mu)$ into a real $C(K)$ if $\mu$ is a finite measure and $K$ is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators.