For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$: \begin{equation*} D_J(x,y) = J(x) - J(y) - \langle \xi,x-y\rangle. \end{equation*} One quickly realizes that the mapping $x\mapsto D_J(x,y)$ shares many properties with $J$ (e.g. it is also convex and in $J$ is (weakly) lower semicontinuous, the same holds here). However, the situation is different for $y\mapsto D_J(x,y)$.

To make is simpler, consider, that $J$ is Gateaux-differentiable. Then one has \begin{equation*} D_J(x,y) = J(x) - J(y) - \langle \nabla J(y),x-y\rangle. \end{equation*}

My question in general is: Is there a good reference which deals with properties of the mapping $y\mapsto D_J(x,y)$ or even $(x,y)\mapsto D_J(x,y)$?

Or, more focused: What is known about the (weak) lower semiconituiuty of this mappings?

(You may, or may not assume that $X$ is reflexive.)