I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations $L = -divA\nabla$ over the domain ${\mathbb R}^{n+1}_+$, where $\lambda^{-1}\omega(x)\xi^2 \leq A(x)\xi\xi \leq \lambda\omega(x)\xi^2$ and $\omega \in A_2$

Wheeden and Chanillo had some kind of that results but only for bounded domain (they called Green function) on their paper "Existence and estimates of Greens functions for degenerate elliptic equations." However, they did not have the estimates for the size or the smoothness of the Green function.


Update: It looks like the problem is still untouched and we may not have fundamental solution for unbounded domain.

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    $\begingroup$ Hi, welcome to MO. Some of the equations and symbols here are, at least for me, un-decipherable. Can you explain what are $\xi, \lambda, A_2$ and $\omega$? $\endgroup$
    – Amir Sagiv
    Apr 10, 2017 at 6:13

1 Answer 1


Here is an interesting reference, https://arxiv.org/abs/1612.05583. This is not about Green's function, but about sobolev space estimates. Maybe this could be useful/interesting for you.

This paper looks at the equation you are interested in.


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