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

作者：Rabier Patrick J^*

来源：Journal d Analyse Mathematique, 2012, 118: 251-296.

DOI：10.1007/s11854-012-0035-1

We characterize all the real numbers a, b, c and 1 <= p, q, r < infinity such that the weighted Sobolev space
W-{a,b}((q,p))(R-N\{0}) := {u is an element of L-loc(1)(R-N\{0}) : vertical bar x vertical bar(a/q)u is an element of L-q(R-N), vertical bar x vertical bar(b/p)del u is an element of (L-p(R-N))(N)} is continuously embedded into
L-r(R-N; vertical bar x vertical bar(c)dx) := {u is an element of L-loc(1)(R-N\{0}) : vertical bar x vertical bar(c/r)u is an element of L-r(R-N)}
with norm parallel to.parallel to(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
parallel to u parallel to(c,r) <= C parallel to del u parallel to(theta)(b,p)parallel to u parallel to(1-theta)(a,q)
for some suitable theta is an element of [0, 1], which is often but not always unique. If a, b, c > -N, then C-0(infinity)(R-N) subset of W-{a,b}((q,p))(R-N\{0}) boolean AND L-r(R-N; vertical bar x vertical bar(c)dx) and such inequalities for u is an element of C-0(infinity)(R-N) are the well-known Caffarelli-Kohn-Nirenberg inequalities; but their generalization to W-{a,b}((q,p))(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 is an element of C-0(infinity)(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 vertical bar x vertical bar(a), vertical bar x v

• 出版日期2012-10