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Some improved Caffarelli-Kohn-Nirenberg inequalities

Authors:	B Abdellaoui E Colorado I Peral

Institution:	(1) Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Abstract:	For 1 < p < N and $-\infty < \gamma < \frac{N-p}{p}$ we obtain the following improved Hardy-Sobolev Inequalities $\int\limits_\Omega \vert\nabla \phi\vert^p\vert x\vert^{-p\gamma}dx -\left(\frac{N-p(\gamma +1)}{p}\right)^p \int\limits_\Omega \frac{\vert\phi\vert^p}{\vert x\vert^{p(\gamma + 1)}}dx$ $\ge C(p,q,r,\gamma,\vert\Omega \vert)\left(\int\limits_\Omega \vert\nabla\phi\vert^q\vert x\vert^{-r\gamma}dx\right)^{\frac{p}{q}},$ where 1 < q < p and $q\le r < \infty$ if $\gamma \le 0$ , $1\le r < p + \rho(N,p,q,\gamma)$ if $\gamma > 0$ , for some positive constant $\rho(N,p,q,\gamma)$ . Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in 16]. Received: 2 February 2004, Accepted: 12 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35P05, 35R05, 46E30, 46E35 Partially supported by Project BFM2001-0183

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