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On Gaussian kernel estimates on groups

Nick Dungey — 2004

Colloquium Mathematicae

We give new and simple sufficient conditions for Gaussian upper bounds for a convolution semigroup on a unimodular locally compact group. These conditions involve certain semigroup estimates in L²(G). We describe an application for estimates of heat kernels of complex subelliptic operators on unimodular Lie groups.

A Littlewood-Paley-Stein estimate on graphs and groups

Nick Dungey — 2008

Studia Mathematica

We establish the boundedness in L q spaces, 1 < q ≤ 2, of a “vertical” Littlewood-Paley-Stein operator associated with a reversible random walk on a graph. This result extends to certain non-reversible random walks, including centered random walks on any finitely generated discrete group.

A Gaussian bound for convolutions of functions on locally compact groups

Nick Dungey — 2006

Studia Mathematica

We give new and general sufficient conditions for a Gaussian upper bound on the convolutions K m + n K m + n - 1 K m + 1 of a suitable sequence K₁, K₂, K₃, ... of complex-valued functions on a unimodular, compactly generated locally compact group. As applications, we obtain Gaussian bounds for convolutions of suitable probability densities, and for convolutions of small perturbations of densities.

Heat kernel estimates for a class of higher order operators on Lie groups

Nick Dungey — 2005

Studia Mathematica

Let G be a Lie group of polynomial volume growth. Consider a differential operator H of order 2m on G which is a sum of even powers of a generating list A , . . . , A d ' of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list A , . . . , A d ' which is sufficient to ensure that the semigroup kernel of H satisfies global Gaussian estimates for all times. For G not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global Gaussian estimates....

On an integral of fractional power operators

Nick Dungey — 2009

Colloquium Mathematicae

For a bounded and sectorial linear operator V in a Banach space, with spectrum in the open unit disc, we study the operator V ̃ = 0 d α V α . We show, for example, that Ṽ is sectorial, and asymptotically of type 0. If V has single-point spectrum 0, then Ṽ is of type 0 with a single-point spectrum, and the operator I-Ṽ satisfies the Ritt resolvent condition. These results generalize an example of Lyubich, who studied the case where V is a classical Volterra operator.

Heat kernel and semigroup estimates for sublaplacians with drift on Lie groups.

Nick Dungey — 2005

Publicacions Matemàtiques

Let G be a Lie group. The main new result of this paper is an estimate in L2 (G) for the Davies perturbation of the semigroup generated by a centered sublaplacian H on G. When G is amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth.

High order regularity for subelliptic operators on Lie groups of polynomial growth.

Nick Dungey — 2005

Revista Matemática Iberoamericana

Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup St = e-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n'. The regularity...

A Class of Contractions in Hilbert Space and Applications

Nick Dungey — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ( T ) n = 1 , 2 , . . . by the continuous semigroup ( e - t ( I - T ) ) t 0 . Moreover, we give a stronger quadratic form inequality which ensures that s u p n T - T n + 1 : n = 1 , 2 , . . . < . The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

Some Gradient Estimates on Covering Manifolds

Nick Dungey — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

Asymptotics of sums of subcoercive operators

Nick DungeyA. ter ElstDerek Robinson — 1999

Colloquium Mathematicae

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...

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