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This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. Then there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.

2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked on MO here Estimates on the Green function of an elliptic second order differential operator.

This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. Then there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.

2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked on MO here Estimates on the Green function of an elliptic second order differential operator.

This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. The there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.

2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked here Estimates on the Green function of an elliptic second order differential operator.

This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. Then there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.

2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked on MO here Estimates on the Green function of an elliptic second order differential operator.

This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. The there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.
2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked here Estimates on the Green function of an elliptic second order differential operator.

This is not an answer but a long comment. There is the following result which does not cover your example. Let $M$ be a compact smooth manifold. Let $P$ be a linear differential second order elliptic operator with smooth coefficients on functions on $M$. The there exists a Green function $G(x,y)$ on $M\times M$ such that

1. $G$ is smooth outside of the diagonal.

2. $G$ is bounded from below.

More precise statement and indication of the proof can be found in the Appendix to this paper http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf

Closely related question was also asked here Estimates on the Green function of an elliptic second order differential operator.