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This question arises when I am reading Klainerman&Machedon's paper "On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy". The author made a comment on page 3, which in effect is as follow …

Let $M$ be an orientable, compact smooth manifold with a metric $g$ and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2 d\mu<\infty\}$$ I know it is well known that (see Julie …

Assume $L$ acting on functional space over the cylinderal end is Fredholm, then restricted to this space it has finite dimensional kernel and cokernel. The kernel of functions $L$ acting on the functi …

I want to ask if a construction of discrete Gaussian free field has been done for a closed Riemannian manifold. Most of the literature I surveyed either need extra boundary condition and consider subm …

I take a very brief look at the paper and I did not see $\Psi DO$ on manifold with boundary being used heavily anywhere (no conormal distribution, multiple blow-ups, heavy handed symbol estimates, etc …

I think this is a good question. I am no longer working in this field. When I was working in this field, there is no general program and I do not know what are the important open problems. Instead of …

The "chart problem" you mentioned is not really a problem, because principal symbols can be defined via local coordinates and glue them together. After all we obtain pseudo-differential operators this …

The unit length hypothesis at here is not important, and very crude estimates are available using Sobolev embedding only. The main issue is that studying the spectrum on the manifold itself is not eno …

I took a reading course based on Sogge's book "Hangzhou Lectures on Eigenfunctions of the Laplacian" a long time ago. This may serve as a standard reference because most of the results mentioned in th …