  We introduce a new notion of G-expectation-weighted Sobolev spaces, or in
short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian
motion are in fact path dependent PDEs in the corresponding Sobolev spaces
under G-norms. For the linear case of G corresponding the classical Wiener
probability space with Wiener measure P, we have established a 1-1
correspondence between BSDE and such new type of quasilinear PDE in the
corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1
correspondence between a fully nonlinear PDE in the corresponding G-Sobolev
space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness
of such type of fully nonlinear path-dependence PDE in G-Sobolev space have
been obtained via a recent results of BSDE driven by G-Brownian motion.

Probability