Special embeddings of weighted Sobolev spaces with nontrivial power weights

Abstract

In prior work, the author has characterized the real numbers $a,b,c$ and $1\leq p,q,r<\infty $ such that the weighted Sobolev space $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{a}{q}}u\in L^{q}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$ is continuously embedded into $L^{r}(R^{N};|x|^{c}dx) :=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{c}{r}}u\in L^{r}(R^{N})\}$. This paper discusses the embedding question for $W_{\{a,b\}}^{(\infty, p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{a}u\in L^{\infty}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$, which is not the space obtained by the formal substitution $q=\infty$ in the previous definition of $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}),$ unless $a=0$. The corresponding embedding theorem identifies all the real numbers $a,b,c$ and $1\leq p,r<\infty $ such that $W_{\{a,b\}}^{(\infty, p)}(R^{N} \backslash \{0})$ is continuously embedded in $L^{r}(R^{N};|x|^{c}dx)$. A notable feature is that such embeddings exist only when $a\neq 0$ and, in particular, have no analog in the unweighted setting. It is also shown that the embeddings are always accounted for by multiplicative rather than just additive norm inequalities. These inequalities are natural extensions of the Caffarelli-Kohn-Nirenberg inequalities which, in their known form, are restricted to functions of $C_{0}^{\infty}(R^{N})$ and do not incorporate supremum norms.