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Sobolev Inequalities on Homogeneous Spaces

Chapter

Abstract

We consider a homogeneous space X = (X, d, m) of dimension ν ≥ 1 and a local regular Dirichlet form a in L 2(X, m). We prove that if a Poincaré inequality of exponent 1 ≤ p < ν holds on every pseudo-ball B(x, R) of X, then Sobolev and Nash inequalities of any exponent q ∈ [p, ν), as well as Poincaré inequalities of any exponent q ∈ [p, +∞), also hold on B(x, R).

AMS Class

46E35 31C25 35J70 

Key words

Sobolev spaces Dirichlet forms Homogeneous spaces 

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References

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly

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