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Hajłasz-Sobolev type spaces and $$p$$-energy on the Sierpinski gasket. (English) Zbl 1062.43006
The authors study Hajłasz-Sobolev type spaces that depend on quasi-distances. They consider a metric space $$(F,d)$$ and a Borel measure $$\mu$$ on the metric space $$(F,d)$$ and introduce the Hajłasz-Sobolev space $$M^p(\mu)$$ ($$1\leq p\leq\infty$$) that depends on the quasi-distance defined by $$q(x,y)=d(x,y)^\sigma$$ ($$0<\sigma<\infty$$). The case $$\sigma=1$$ and the case $$\sigma>1$$ were analyzed in earlier papers when $$F$$ is a fractal in the Euclidean settings. In this paper the authors generalize the earlier results to the non-Euclidean setting. Afterwards they give an example. They show that $$M^p(\mu)$$ is non-trivial for any $$1<p<\infty$$ and $$\sigma>1$$. The paper is very well written and the subject is very actual and interesting.

##### MSC:
 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 46B20 Geometry and structure of normed linear spaces 28A80 Fractals
##### Keywords:
Hajlasz-Sobolev type space; p-energy
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