Admissible function spaces for weighted sobolev inequalities

Let k, N ∈ N with 1 ≤ k ≤ N and let Ω = Ω1 × Ω2 be an open set in Rk × RN−k. For p ∈ (1, ∞) and q ∈ (0, ∞), we consider the following weighted Sobolev type inequality: f Ω |g1(y)

g2(z)

u(y, z)|q dydz ≤ C ( Ω |∇u(y, z)|p dydz ) pq , ∀u ∈ Cc1(Ω), (0.1) for some C > 0. Depending on the values of N, k, p, q we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for (g1, g2) so that (0.1) holds. Furthermore, we give a sufficient condition on g1, g2 so that the best constant in (0.1) is attained in the Beppo-Levi space D01,p(Ω)-the completion of Cc1(Ω) with respect to ∇uLp(Ω)