On positive Rockland operators

Pascal Auscher; A. ter Elst; Derek Robinson

Colloquium Mathematicae (1994)

  • Volume: 67, Issue: 2, page 197-216
  • ISSN: 0010-1354

Abstract

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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 ; d g ) . 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.

How to cite

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Auscher, Pascal, ter Elst, A., and Robinson, Derek. "On positive Rockland operators." Colloquium Mathematicae 67.2 (1994): 197-216. <http://eudml.org/doc/210273>.

@article{Auscher1994,
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.},
author = {Auscher, Pascal, ter Elst, A., Robinson, Derek},
journal = {Colloquium Mathematicae},
keywords = {fundamental solution; homogeneous Lie group; homogeneous operator; positive Rockland operator; homogeneous differential operator},
language = {eng},
number = {2},
pages = {197-216},
title = {On positive Rockland operators},
url = {http://eudml.org/doc/210273},
volume = {67},
year = {1994},
}

TY - JOUR
AU - Auscher, Pascal
AU - ter Elst, A.
AU - Robinson, Derek
TI - On positive Rockland operators
JO - Colloquium Mathematicae
PY - 1994
VL - 67
IS - 2
SP - 197
EP - 216
AB - 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.
LA - eng
KW - fundamental solution; homogeneous Lie group; homogeneous operator; positive Rockland operator; homogeneous differential operator
UR - http://eudml.org/doc/210273
ER -

References

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