On positive Rockland operators

Volume 67 / 1994

Pascal Auscher, A. ter Elst, Derek Robinson Colloquium Mathematicum 67 (1994), 197-216 DOI: 10.4064/cm-67-2-197-216

Abstract

Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on $L_p(G;dg)$. Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on $L_2$ we prove that it is closed on each of the $L_p$-spaces, p ∈ 〈 1,∞〉, and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the $L_p$-spaces, p ∈ [1,∞]. Further extensions of these results to nonhomogeneous operators and general representations are also given.

Authors

  • Pascal Auscher
  • A. ter Elst
  • Derek Robinson

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