PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 9, September 2007, Pages 2795–2802 S 0002-9939(07)08839-9 Article electronically published on February 9, 2007 SOME WEIGHTED GAGLIARDO-NIRENBERG INEQUALITIES AND APPLICATIONS JAVIER DUOANDIKOETXEA AND LUIS VEGA (Communicated by Andreas Seeger) Abstract. We obtain conditions on the measure μ so that the L 2 (μ)-norm of a function is controlled by the L 2 -norms of the function and its gradient. Applications to eigenvalues of the Schr¨odinger operator and to other inequalites are also given. 1. Introduction We will be interested in inequalities of the type (1.1)  R d |f (x)| 2 dμ(x)  1/2 ≤ C ‖f ‖ θ 2 ‖∇f ‖ 1−θ 2 . We would like to give sufficient conditions on μ so that (1.1) holds and to study the best constant in the inequality if possible. Inequalities of this type with an unweighted L q -norm on the left-hand side are known as Gagliardo-Nirenberg in- equalities. In this paper we will focus attention on the case θ =1/2. Other values of θ and other norms different from L 2 -norms are also of interest but will not be considered here. We consider the Sobolev space H 1 (R d ) of complex-valued functions f ∈ L 2 (R d ) such that |∇f |∈ L 2 (R d ). The basic theorem for d ≥ 2 and dμ(x)= w(x) dx is the following. Theorem 1.1. Let d ≥ 2. Let f be a function in H 1 (R d ) and w(x) ≥ 0. Then (1.2)  R d w(x)|f (x)| 2 dx ≤ 2K(w)‖f ‖ 2 ‖∇f ‖ 2 , where K(w)= inf a∈R d sup x∈R d |x|  1 0 w(tx + a)t d−1 dt. Equality holds when w(x) is a multiple of |x − b| −1 for some b ∈ R d . In that case, f (x) must be a multiple of e −c|x−b| with c> 0, and K(|x − b| −1 )=(d − 1) −1 . Received by the editors February 22, 2006 and, in revised form, May 12, 2006. 2000 Mathematics Subject Classification. Primary 26D10. The first author was supported by the grant MTM2005-08430 of MEC (Spain) and FEDER. The second author was supported by the grant MTM 2004-03029 of MEC (Spain) and FEDER. Both authors were supported by the European Project HPRN-CT-2001-00273-HARP. c 2007 American Mathematical Society 2795 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use