Embeddings of weighted sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities

Abstract

We characterize all the real numbers a, b, c and 1 ≤ p, q, r < ∞ such that the weighted Sobolev space

$$W_{\{ a,b\} }^{\{ q,q\} }({R^N}\backslash \{ 0\} ): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{a/q}} \in {L^q}({R^{N),}}{\left| x \right|^{b/p}}\nabla u \in {({L^p}({R^N}))^N}\} $$

is continuously embedded into

$${L^r}({R^N};{\left| x \right|^c}dx): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{c/r}}u \in {L^r}({R^N})\} $$

with norm ‖·‖ _c,r . It turns out that, except when N ≥ 2 and a = c = b − p = −N, such an embedding is equivalent to the multiplicative inequality

$${\left\| u \right\|_{c,r}} \le C\left\| {\nabla u} \right\|_{b,p}^\theta \left\| u \right\|_{a,q}^{1 - \theta }$$

for some suitable θ ∈ [0, 1], which is often but not always unique. If a, b, c > −N, then C ^∞₀ (ℝ^N) ⊂ W ^(q,p)_{a,b} (ℝ^N{0}) ∩ L ^r(ℝ^N; |x|^c dx) and such inequalities for u ∈ C ^∞₀ (ℝ^N) are the well-known Caffarelli-Kohn-Nirenberg inequalities; but their generalization to W ^(q,p)_{a,b} (ℝ^N{0}) cannot be proved by a denseness argument. Without the assumption a, b, c > −N, the inequalities are essentially new, even when u ∈ C ^∞₀ (ℝ^N{0}), although a few special cases are known, most notably the Hardy-type inequalities when p = q.

In a different direction, the embedding theorem easily yields a generalization when the weights |x|^a, |x|^b and |x|^c are replaced with more general weights w _a ,w _b and w _c , respectively, having multiple power-like singularities at finite distance and at infinity.