Using a method developed by Pérez and Wheeden and the representation of smooth functions by integral operators whose kernels are gradients of the Green functions, we obtain weighted Sobolev's inequalities for bounded domains which improve and unify several kinds of inequalities. From these results we establish Green functions and the existence, uniqueness and regularity results for a class of singular elliptic equations. Theorem 3.1. Let 1 < p ≤ q < ∞ and and let v, w be nonnegative measurable functions on Ω. Suppose that for some γ, 0 < γ ≤ α, (3.2) sup Q: (Q)≤4d(Ω)
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