Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

Abstract. For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p> 2. 1.

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

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Abstract. Let M be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, R n \ B(0, R) for some R> 0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from L p (M) → L p (M; T ∗ M) for 1 < p < n and unbounded for p ≥ n if there is more than one end. It follows from known results that in such a case the Riesz transform on M is bounded for 1 < p ≤ 2 and unbounded for p> n; the result is new for 2 < p ≤ n. We also give some heat kernel estimates on such manifolds. We then consider the implications of boundedness of the Riesz transform in L p for some p> 2 for a more general class of manifolds. Assume that M is a n-dimensional complete manifold satisfying the Nash inequality and with an O(r n) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on L p for some p> 2 implies a Hodge-de Rham interpretation of the L p cohomology in degree 1, and that the map from L 2 to L p cohomology in this degree is injective. 1.

Abstract. Let X be an RD-space, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of localized Hardy spaces H1 ρ(X) associated with ρ, which includes several maximal function characterizations of H1 ρ (X), the relations between H1 ρ (X) and the classical Hardy space H1 (X) via constructing a kernel function related to ρ, the atomic decomposition characterization of H1 ρ(X), and (X) via a finite atomic the boundedness of certain localized singular integrals on H1 ρ decomposition characterization of some dense subspace of H1 ρ (X). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on Rn, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality. 1

Let K denote the kernel of the continuous semigroup S generated by H = ( 1) m=2 d 0 X i=1 A m i where A 1 ; : : : ; A d 0 are a generating set of right-invariant elds acting on L 2 (G) with G a solvable Lie group of polynomial growth and m an even positive integer. If G is connected, simply connected, and has an abelian nilshadow we establish that jK t (g)j c Z Na dh G (m) b;t (gh 1 ) G (2) b;t (h) for all g 2 G and all t 1, where N a is a subgroup of the abelian nilradical, G (m) denotes an m-th order Gaussian over G and G (2) the second-order Gaussian over N a . The group N a is determined by the choice of the generating set and in general is non-zero. Analogous estimates are derived for various derivatives of the kernel. Further, through the use of homogenization theory, we establish asymptotic estimates for S and K. These estimates imply that the above kernel bounds give the correct asymptotic behaviour of K, e.g., if m 4 and N a 6= f0g then K decreases...

On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p, pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d’exposants p. One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same

Let (M, g) be a complete Riemannian manifold with infinite volume, we denote by ∆ = ∆ g its Laplace operator, it has an unique self-adjoint extension on L 2 (M, dvolg) which is also denoted by ∆. The Green formula and the spectral theorem show that for any ϕ ∈ C ∞ 0 (M):

Abstract. Let G be the Lie group R 2 ⋉ R + endowed with the Riemannian symmetric space structure. Let X0, X1, X2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian ∆ = −(X 2 0 +X 2 1 +X 2 2). In this paper we consider the first order Riesz transforms Ri = Xi ∆ −1/2 and Si = ∆ −1/2 Xi, for i = 0, 1, 2. We prove that the operators Ri, but not the Si, are bounded from the Hardy space H 1 to L 1. We also show that the second-order Riesz transforms Tij = Xi ∆ −1 Xj are bounded from H 1 to L 1, while the Riesz transforms Sij = ∆ −1 XiXj and Rij = XiXj ∆ −1, for i, j = 0, 1, 2, are not. 1.