Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

(1,p)-Sobolev spaces based on strongly local Dirichlet forms (2310.11652v2)

Published 18 Oct 2023 in math.PR and math.FA

Abstract: In the framework of quasi-regular strongly local Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ on $L2(X;\mathfrak{m})$ admitting minimal $\mathscr{E}$-dominant measure $\mu$, we construct a natural $p$-energy functional $(\mathscr{E}{\,p},D(\mathscr{E}{\,p}))$ on $Lp(X;\mathfrak{m})$ and $(1,p)$-Sobolev space $(H{1,p}(X),|\cdot|_{H{1,p}})$ for $p\in]1,+\infty[$. In this paper, we establish the Clarkson type inequality for $(H{1,p}(X),|\cdot|_{H{1,p}})$. As a consequence, $(H{1,p}(X),|\cdot|_{H{1,p}})$ is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of $(H{1,p}(X),|\cdot|_{H{1,p}})$, we prove that (generalized) normal contraction operates on $(\mathscr{E}{\,p},D(\mathscr{E}{\,p}))$, which has been shown in the case of various concrete settings, but has not been proved for such general framework. Moreover, we prove that $(1,p)$-capacity ${\rm Cap}_{1,p}(A)<\infty$ for open set $A$ admits an equilibrium potential $e_A\in D(\mathscr{E}{\,p})$ with $0\leq e_A\leq 1$ $\mathfrak{m}$-a.e. and $e_A=1$ $\mathfrak{m})$-a.e.~on $A$.

Citations (4)

Summary

We haven't generated a summary for this paper yet.