Sobolev Spaces on Metric Measure Spaces

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Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology (2012, background): “; see Shanmugalingam [37] and Haj lasz [19].”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “A weak upper gradient is an analog of |∇ f | when f is a measurable function on the metric space; |∇ f | satisfies a variant of the fundamental theorem of calculus along most rectifiable curves in X , and it has the property that if f is constant on a Borel set E ⊂ X , then |∇ f | = 0 almost everywh”
Gradient estimates for heat kernels and harmonic functions (2017, methodology): “The proof uses a telescopic argument, which is by now classical; see for instance [60] and the monographs [19, 63].”