Fundamental solution and Fourier series in eigenfunctions of degenerate elliptic operator

We study the degenerate elliptic differential operator of the second order in the divergence form. The operator is assumed to be symmetric. The weight function which is describing the degeneration of the coefficients (or singularity) assumed to be in the Muckenhoupt class.We prove the uniform estimates for the fundamental solution of this operator and obtain the conditions which guarantee the absolute and uniform convergence of Fourier series in eigenfunctions. These results might be applied to the ground of Fourier method. © 2006 Elsevier Inc. All rights reserved