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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 $L^{r}(R^{N};|x|^{c}dx):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{c}{r}}u\inL^{r}(R^{N})\}$.  This paper discusses the embedding question for $W_{\{a,b\}}^{(\infty,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{a}u\inL^{\infty}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$, which isnot the space obtained by the formal substitution $q=\infty$ in the previousdefinition 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 featureis that such embeddings exist only when $a\neq 0$ and, in particular, have noanalog in the unweighted setting.  It is also shown that the embeddings are always accounted for bymultiplicative rather than just additive norm inequalities. These inequalitiesare natural extensions of the Caffarelli-Kohn-Nirenberg inequalities which, intheir known form, are restricted to functions of $C_{0}^{\infty}(R^{N})$ and donot incorporate supremum norms.

Special embeddings of weighted Sobolev spaces with nontrivial power weights