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
• Texto completo no disponible (Saber más ...)
• 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.

• Referencias bibliográficas
  • Bradley, S.C.: Hardy inequalities with mixed norms. Can. Math. Bull. 21, 405–408 (1978)
  • Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
  • Catrina, F., Costa, D.G.: Sharp weighted-norm inequalities for functions with compact support in {R}^{N}\backslash \{0\}. J. Diff. Equ. 246,...
  • Marcus, M., Mizel, V.J.: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45, 294–320 (1972)
  • Maz’ja, V.G.: On certain integral inequalities for functions of many variables. J. Math. Sci. 1, 205–234 (1973)
  • Maz’ja, V.G.: Sobolev spaces. Springer, Berlin (1985)
  • Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations, Grund. Math. Wiss. vol. 342. Springer, Heidelberg...
  • Muckenhoupt, B.: Hardy’s inequality with weights. Stud. Math. 44, 31–38 (1972)
  • Rabier, P.J.: Embeddings of weighted Sobolev spaces and generalized Caffarelli–Kohn–Nirenberg inequalities. J. Anal. Math. Arxiv, preprint...
  • Rabier, P.J.: Boundedness properties of Sobolev functions in spaces with power weights (2012) (in preparation)
  • Ziemer, W.P.: Weakly differentiable functions. Springer, Berlin (1989)