Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Expansion of the fundamental solution of a second-order elliptic
  operator with analytic coefficients   Let $L$ be a second-order elliptic operator with analytic coefficients
defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a
fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$
such that $Lu=\delta_0$.
  As a consequence of our construction, we obtain an expansion of the
fundamental solution in homogeneous terms (homogeneous polynomials divided by a
power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the
dimension $n$ is even) which improves the classical result of F. John (1950).
The control we have on the "complexity" of each homogeneous term is optimal and
in particular, when $L$ is the Laplace-Beltrami operator of an analytic
Riemannian manifold, we recover the construction of the fundamental solution
due to K. Kodaira (1949).
  The main ingredients of the proof are a harmonic decomposition for singular
functions and the reduction of the convergence of our construction to a
nontrivial estimate on weighted paths on a graph with vertices indexed by
$\mathbb Z^2$.

25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics