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The Grushin spaces, as one of the most important models in theCarnot-Carath\'eodory space, are a class of locally compact and geodesic metricspaces which admit a dilation. Function spaces on Grushin spaces and somerelated geometric problems are always the research hotspots in this field.Firstly, we investigate two classes of Besov type spaces based on the Grushinsemigroup and the fractional Grushin semigroup, respectively, and prove someimportant properties of these two Besov type spaces. Moreover, we also revealthe relationship between them. Secondly, we establish the isoperimetricinequality for the fractional perimeter, which is defined by theGrushin-Laplace operator on Grushin spaces. Finally, we combine the semigrouptheory with a nonlocal calculus for the Grushin-Laplace operator to obtain theSobolev type inequality. As a corollary, we also obtain the embedding theoremfor Besov type spaces.

Geometric topics related to Besov type spaces on the Grushin setting