be relaxed to hold only in $L^{2+\varepsilon}$ for some $\varepsilon>0$. Besides, as noted in [@EPQ Remark at the bottom of p. 59], if $K$ is bounded then the proof can be modified so that the extra integrability condition on $\xi$ (i.e. $\E[|\xi|^4]<+\infty$) can be dropped. However, even in this case, one can check that in the Markovian framework, Conditions (EPQ1)-(EPQ2) are strictly stronger than Conditions (PP1)-(PP2).

Since these two seminal papers, the most notable extension was concerned with the study of the Malliavin differentiability of $(Y,Z)$ in a quadratic setting, that is to say when the generator $f$ has quadratic growth in the $z$ variable, a problem addressed in [@ank; @DosReis_Imkeller; @dosreis; @irr]. Notice nonetheless that the proofs in these references are strongly influenced by the ones in the Lipschitz setting of [@pardouxpeng; @EPQ], as they all start by approximating the quadratic generators by Lipschitz ones, to which they apply the results of [@pardouxpeng; @EPQ]. The applications of the Malliavin differentiability of BSDEs also received a lot of attention in the literature. Hence, it was used in the context of numerical schemes for BSDEs in, among others, [@briand_labart; @hns], or to study the existence and regularity of densities for the 