Special embeddings of weighted Sobolev spaces with nontrivial power weights

Patrick J. Rabier

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Special embeddings of weighted Sobolev spaces with nontrivial power weights

Autores: Patrick J. Rabier
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 3, 2013, págs. 409-426
Idioma: inglés
DOI: 10.1007/s13348-012-0068-x
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Resumen
- In prior work, the author has characterized the real numbers a,b,c and 1\le p,q,r<\infty such that the weighted Sobolev space W_{\{a,b\}}^{1,(q,p)}(\mathbb R ^{N}\backslash \{0\}):=\{u\in L_{loc}^{1}(\mathbb R ^{N}\backslash \{0\}):|x|^{\frac{a}{q}}u\in L^{q}(\mathbb R ^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(\mathbb R ^{N}))^{N}\} is continuously embedded into L^{r}(\mathbb R ^{N};|x|^{c}dx):=\{u \text{ measurable} :|x|^{\frac{c}{r}}u\in L^{r}(\mathbb R ^{N})\}. This paper discusses the embedding question for W_{\{a,b\}}^{1,(\infty ,p)}(\mathbb R ^{N}\backslash \{0\}):=\{u\in L_{loc}^{1}(\mathbb R ^{N}\backslash \{0\}):|x|^{a}u\in L^{\infty }(\mathbb R ^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(\mathbb R ^{N}))^{N}\}, which is not the space obtained by the formal substitution q=\infty in the previous definition of W_{\{a,b\}}^{1,(q,p)}(\mathbb R ^{N}\backslash \{0\}), unless a=0. The corresponding embedding theorem identifies all the real numbers a,b,c and 1\le p,r<\infty such that W_{\{a,b\}}^{1,(\infty ,p)}(\mathbb R ^{N}\backslash \{0\}) is continuously embedded in L^{r}(\mathbb R ^{N};|x|^{c}dx). A notable feature is that such embeddings exist only when a\ne 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 }(\mathbb R ^{N}) and do not incorporate supremum norms.