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 |
Keywords:
homogeneous space; local regular Dirichlet form; Poincaré inequality; Sobolev and Nash inequalitiesReferences:
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