Gaussian Poincare Inequalities on the Half-Space with Singular Weights

We prove Rellich-Kondrachov-type theorems and weighted Poincare inequalities on the half-space R-+(N+1) = {z = (x, y) : x is an element of R-N, y > 0} endowed with the weighted Gaussian measure mu := y (c) e(-a|z|2)dz where c + 1 > 0 and a > 0. We prove that for some positive constant C > 0, one has ||u-u(-)||(2)(L mu)(R+(N+1)) <= C||del u||(N+1)(L mu 2(R+)),for all u is an element of H-mu(1)(R-+(N+1)), where (u) over bar = 1/mu(R-+(N+1))integral(N+1)(R+) u d mu(z). Besides this, we also consider the local case of bounded domains of R-+(N+1) where the measure mu is y (c) dz.