Sobolev embeddings in infinite dimensions

In this paper, we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems. Our underlying spaces are l(r) for r is an element of [1,infinity), equipped with corresponding probability measures. For the weighted Sobolev space W-b(1, p) (l(r), gamma) with a weight alpha is an element of l(r) of the Gaussian measure gamma(a) and a gradient weight b is an element of l(infinity), we characterize the relation between the weights (a and b) and the continuous (resp. compact) log-Sobolev embedding for p is an element of [1, infinity) (resp. p is an element of (1, infinity)). Several counterexamples are also constructed, which are of independent interest.