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Heat kernels on metric spaces and a characterisation of constant functions. (English) Zbl 1062.60076
The main result states that the only \(u\in L^ {2}(X, m)\) for which \[ \int _ {X} \int _ {X} \frac{(u(x)-u(y))^ {2}}{\rho (x, y)^ {d+d_ {w}}} \,dm(x) \,dm(y) <+\infty \] are the constants. Here \((X, \rho)\) is a locally compact metric space and \(m\) is a Borel measure on \(X\), for which the volume of the balls of radius \(r\) behaves as \(r^ {d}\). The proof uses the Dirichlet forms theory, hence the existence of a strongly continuous semigroup of contractions on \(L^ {2}(X, m)\), whose transition density satisfies some two-sided Gaussian estimate, is assumed. Examples are included, showing the relation with recent results of J. Bourgain, H. Brézis, Y. Li, P. Mironescu, L. Nirenberg; see H. Brézis [Russ. Math. Surv. 57, No. 4, 693–708 (2002); translation from Usp. Mat. Nauk 57, No. 4, 59–74 (2002; Zbl 1072.46020)] for a detailed account of these results.

MSC:
60J35 Transition functions, generators and resolvents
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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