In this short article, we give a necessary and sufficient condition for the form (1.3), like the one in Hamza’s theorem, to be closable and prove that every regular and strongly local Dirichlet form is the closure of a form of Hamza’s type, which can be represented in terms of effective intervals introduced in [7].
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[6] L. Li, W. Sun and J. Ying: Effective intervals and regular Dirichlet subspaces, to appear in Stochastic Process. Appl. https://doi.org/10.1016/j.spa.2020.05.003
[7] L. Li and J. Ying: On symmetric linear diffusions, Trans. Amer. Math. Soc. 371 (2019), 5841–5874.
[8] H.L. Royden and P. Fitzpatrick: Real Analysis, 4th ed., Prentice Hall, Boston, 2010.
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