Wyniki wyszukiwania

1

Heat kernels and Riesz transforms on nilpotent Lie groups

100%

ter Elst A. F. M., Robinson D. W., Sikora A.

Colloquium Mathematicum

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1997

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tom 74

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nr 2

191-218

EN

We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp -spaces with p ∈ (1, ∞). Finally, for second-order operators with real coefficients we derive matching Gaussian lower bounds and deduce Harnack inequalities valid for all times.

2

Asymptotics of sums of subcoercive operators

100%

Dungey N., ter Elst A. F. M., Robinson D. W.

Colloquium Mathematicum

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1999

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tom 82

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nr 2

231-260

EN

We examine the asymptotic, or large-time, behaviour of the semigroup kernel associated with a finite sum of homogeneous subcoercive operators acting on a connected Lie group of polynomial growth. If the group is nilpotent we prove that the kernel is bounded by a convolution of two Gaussians whose orders correspond to the highest and lowest orders of the homogeneous subcoercive components of the generator. Moreover we establish precise asymptotic estimates on the difference of the kernel and the kernel corresponding to the lowest order homogeneous component. We also prove boundedness of a range of Riesz transforms with the range again determined by the highest and lowest orders. Finally we analyze similar properties on general groups of polynomial growth and establish positive results for local direct products of compact and nilpotent groups.

3

On positive Rockland operators

100%

Auscher P., ter Elst A. F. M., Robinson D. W.

Colloquium Mathematicum

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1994

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tom 67

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nr 2

197-216

EN

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.

Ograniczanie wyników

3 Colloquium Mathematicum

3 Robinson D. W.

3 ter Elst A. F. M.

1 Auscher P.

1 Dungey N.

1 Sikora A.

1 1999

1 1998

1 1994

Heat kernels and Riesz transforms on nilpotent Lie groups

Asymptotics of sums of subcoercive operators

On positive Rockland operators