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Sobolev inequalities on homogeneous spaces. (English) Zbl 0833.46020
Summary: We consider a homogeneous space $$X= (X, d, m)$$ of dimension $$\nu\geq 1$$ and a local regular Dirichlet form $$a$$ in $$L^2 (X, m)$$. We prove that if a Poincaré inequality of exponent $$1\leq p<\nu$$ holds on every pseudo-ball $$B(x,R)$$ of $$X$$, then Sobolev and Nash inequalities of any exponent $$q\in [p, \nu)$$, as well as Poincaré inequalities of any exponent $$q\in [p, +\infty)$$, also hold on $$B(x, R)$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 31C25 Dirichlet forms 35J70 Degenerate elliptic equations
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##### References:
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