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1
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Heat kernel estimates for a class of higher order operators on Lie groups

100%
Dungey N.
Studia Mathematica
|
2005
|
tom 169
|
nr 1
71-80
EN
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 $A₁, ..., A_{d'}$ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list $A₁, ..., A_{d'}$ 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. Our results extend previously known results for nilpotent groups.
2
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Some Gradient Estimates on Covering Manifolds

100%
Dungey N.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
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tom 52
|
nr 4
437-443
EN
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.
3
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A Littlewood-Paley-Stein estimate on graphs and groups

100%
Dungey N.
Studia Mathematica
|
2008
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tom 189
|
nr 2
113-129
EN
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.
4
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A Class of Contractions in Hilbert Space and Applications

100%
Dungey N.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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tom 55
|
nr 4
347-355
EN
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 $(Tⁿ)_{n=1,2,...}$ by the continuous semigroup $(e^{-t(I-T)})_{t≥0}$. Moreover, we give a stronger quadratic form inequality which ensures that $sup {n∥Tⁿ - T^{n+1}∥: n = 1,2,...} < ∞$. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.
5
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On an integral of fractional power operators

100%
Dungey N.
Colloquium Mathematicum
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2009
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tom 117
|
nr 2
157-164
EN
For a bounded and sectorial linear operator V in a Banach space, with spectrum in the open unit disc, we study the operator $Ṽ = ∫_{0}^{∞} dα V^{α}$. 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.
6
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On Gaussian kernel estimates on groups

100%
Dungey N.
Colloquium Mathematicum
|
2004
|
tom 100
|
nr 1
77-90
EN
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.
7
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A Gaussian bound for convolutions of functions on locally compact groups

100%
Dungey N.
Studia Mathematica
|
2006
|
tom 176
|
nr 3
201-213
EN
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.
8
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Asymptotics of sums of subcoercive operators

51%
Dungey N., ter Elst A. F. M., Robinson D. W.
Colloquium Mathematicum
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1999
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tom 82
|
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.
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