Abstract
We characterize all the real numbers a, b, c and 1 ≤ p, q, r < ∞ such that the weighted Sobolev space
is continuously embedded into
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
for some suitable θ ∈ [0, 1], which is often but not always unique. If a, b, c > −N, then C ∞0 (ℝN) ⊂ W (q,p){a,b} (ℝN{0}) ∩ L r(ℝN; |x|c dx) and such inequalities for u ∈ C ∞0 (ℝN) are the well-known Caffarelli-Kohn-Nirenberg inequalities; but their generalization to W (q,p){a,b} (ℝN{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 (ℝN{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.
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Rabier, P.J. Embeddings of weighted sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities. JAMA 118, 251–296 (2012). https://doi.org/10.1007/s11854-012-0035-1
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DOI: https://doi.org/10.1007/s11854-012-0035-1