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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
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