Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On extensions of local Dirichlet forms   Let $\ce$ be a Dirichlet form on $L_2(X\,;\mu)$ where $(X,\mu)$ is locally
compact $\sigma$-compact measure space. Assume $\ce$ is inner regular, i.e.\
regular in restriction to functions of compact support, and local in the sense
that
  $\ce(\varphi,\psi)=0$ for all $\varphi, \psi\in D(\ce)$ with
$\varphi\,\psi=0$. We construct two Dirichlet forms $\ce_m$ and $\ce_M$ such
that $\ce_m\leq \ce\leq \ce_M$. These forms are potentially the smallest and
largest such Dirichlet forms. In particular $\ce_m\supseteq \ce_M$,
$(\ce_M)_m=\ce_m$ and $(\ce_m)_M=\ce_M$. We analyze the family of local, inner
regular, Dirichlet forms $\cf$ which extend $\ce$ and satisfy $\ce_m\leq
\cf\leq \ce_M$. We prove that the latter bounds are valid if and only if
$\cf_M=\ce_M$, or $\cf_m=\ce_m$, or $D(\ce_M)$ is an order ideal of $D(\cf)$.
Alternatively the $\cf$ are characterized by $D(\ce_M)\cap L_\infty(X)$ being
an algebraic ideal of $D(\cf)\cap L_\infty(X)$. As an application we show that
if $\ce$ and $\cf$ are strongly local then the Ariyoshi--Hino set-theoretic
distance is the same for each of the forms $\ce$, $\ce_M$ and $\cf$. If in
addition $\ce_m$ is strongly local then it also defines the same distance.
Finally we characterize the uniqueness condition $\ce_M=\ce_m$ by capacity
estimates.

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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an