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.

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.

We give new and general sufficient conditions for a Gaussian upper bound on the convolutions ${K}_{m+n}\ast {K}_{m+n-1}\ast \cdots \ast {K}_{m+1}$ 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.

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 $A\u2081,...,{A}_{{d}^{\text{'}}}$ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list $A\u2081,...,{A}_{{d}^{\text{'}}}$ 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....

For a bounded and sectorial linear operator V in a Banach space, with spectrum in the open unit disc, we study the operator $V\u0303={\int}_{0}^{\infty}d\alpha {V}^{\alpha}$. 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.

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.

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 S_{t} = 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...

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 ${\left(T\u207f\right)}_{n=1,2,...}$ by the continuous semigroup ${\left({e}^{-t(I-T)}\right)}_{t\ge 0}$. Moreover, we give a stronger quadratic form inequality which ensures that $supn\parallel T\u207f-{T}^{n+1}\parallel :n=1,2,...<\infty $. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

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.

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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