was initially given by Malliavin and then extended by Stroock, Shigekawa, Kusuoka and Sugita in a series of papers [@Malliavin; @stroock; @Shige; @Kusuoka; @Sugita]. In addition, Sugita proved in [@Sugita] that a r.v. $F$ is Malliavin differentiable if it is *ray absolutely continuous*[^6] and if it is stochastically Gâteaux differentiable. Using the main ideas of [@Sugita] we provide incidentally a new formulation of the characterization of the Malliavin-Sobolev type spaces $\D^{1,p}$ in Theorem \[th:newcarD\]. Since we did not find explicitly this characterization in the literature, we believe that this result is new and maybe interesting by itself. The main point is that this formulation is especially handy when dealing with stochastic equations like BSDEs. With this result at hand, we obtain new conditions (see Assumptions (D), ($H_1$) and ($H_2$) at the beginning of Section \[section:BSDEs\]) for $Y,Z$ to be Malliavin differentiable, see Theorem \[thm:yzd12\]. Our assumptions refine those of [@pardouxpeng; @EPQ] in the Markovian case, and our approach is directly applicable to quadratic growth BSDEs since we do not rely on any approximation procedure. We refer the reader to Section \[section:applidiscus\] for some examples and a discussion on the differences between our approach and the one of [@pardouxpeng; @EPQ].

The 