J. Pseudo-Differ. Oper. Appl. (2010) 1:207–231 DOI 10.1007/s11868-010-0003-4 Invertibility for a class of degenerate elliptic operators Julio Delgado · Alex M. Zamudio Received: 22 October 2009 / Revised: 3 February 2010 / Accepted: 7 February 2010 / Published online: 10 March 2010 © Birkhäuser / Springer Basel AG 2010 Abstract In this work we study fundamental solutions for a class of highly degen- erate elliptic operators of type ∑ n i =1 D 2 x i + ∑ n i =1 x 2k i ∑ m i =1 D 2 t i on R n+m . Introducing a suitable weight and metric the authors prove that the fundamental solutions are in a Hörmander’s class of pseudo-differential operators. Keywords Degenerate elliptic operators · Nonhomogeneous calculus · Microlocal analysis Mathematics Subject Classification (2000) Primary 35J70; Secondary 35A27 · 47G30 1 Introduction In [1] Beals et al. have found exact fundamental solutions for operators on R n+m of type ∑ n i =1 D 2 x i +|x| 2k ∑ m i =1 D 2 t i , where k = 1, 2, 3,... ; n + m > 2. In [19] Xu and Zhu have studied the operators D 2 x 1 +ˆ x 2k 1 D 2 x 2 + c on R 2 , where c > 0, k is a positive integer, ˆ x is in the class C ∞ (R), ˆ x = x for |x|≤ 2, ˆ x is increasing, ˆ x = ωs gnx if |x|≥ 4, and ω = Sup ˆ x > 1. They have proved that the fundamental solution is in the class of nonhomogeneous pseudo-differential operators. In this paper we prove that Julio Delgado was partially supported by Universidad del Valle, Vicerrectoria Inv. Grant #7795. J. Delgado (B ) Universidad del Valle, Calle 13, 100-00 Cali, Colombia e-mail: delgado@math.jussieu.fr A. M. Zamudio Menadostraat 22, 3532 SM Utrecht, The Netherlands e-mail: A.Zamudio@students.uu.nl