A priori bounds and existence of smooth solutions to a $L_p$ Aleksandrov problem for Codazzi tensor with log-convex measure

In the present paper, we prove the existence of smooth solutions to a $L_p$ Aleksandrov problem for Codazzi tensor with a log-convex measure in compact Riemannian manifolds $(M,g)$ with positive constant sectional curvature under suitable conditions. Our proof is based on the solvability of a Monge-Ampère equation on $(M,g)$ via the method of continuity whose crucial factor is the a priori bounds of smooth solutions to the Monge-Ampère equation mentioned above. It is worth mentioning that our result can be seen as an extension of the classical $L_p$ Aleksandrov problem in Euclidian space to the frame of Riemannian manifolds with weighted measures and that our result can also be seen as some attempts to get some new results on geometric analysis for Codazzi tensor.