Abstract. We prove that in presence of L 2 Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces. Contents

Abstract. Let Γ be a graph endowed with a reversible Markov kernel p, and P the associated operator, defined by Pf(x) = ∑ y p(x, y)f(y). Denote by ∇ the discrete gradient. We give necessary and/or sufficient conditions on Γ in order to compare ‖∇f‖ p and ∥ ∥ (I − P)

Abstract. We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The method relies upon a good-λ method that does not use any size or smoothness estimates for the kernels. 1.

On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on R n and related estimates

Abstract. Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces L p (R n; X) of X-valued functions on R n. We characterize Kato’s square root estimates ‖ √ Lu‖p � ‖∇u‖p and the H ∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative L p space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X = C, we get a new approach to the L p theory of square roots of elliptic operators, as well as an L p version of Carleson’s inequality.

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz transforms.

We correct an inaccuracy in the proof of a result in [Aus1]. 2000 MSC: 42B20, 46E35 Key words: Calderón-Zygmund decomposition; Sobolev spaces. We recall the lemma. Lemma 0.1. Let n ≥ 1, 1 ≤ p ≤ ∞ and f ∈ D ′ (R n) be such that ‖∇f‖p < ∞. Let α> 0. Then, one can find a collection of cubes (Qi), functions g and bi such that (0.1) f = g + ∑ and the following properties hold:

Abstract. We study the interpolation property of Sobolev spaces of order 1 denoted by W 1 p,V, arising from Schrödinger operators with positive potential. We show that for 1 ≤ p1 < p < p2 < q0 with p> s0, W 1 p,V is a real interpolation space between W 1 p1,V and W 1 p2,V on some classes of manifolds and Lie groups. The constants s0, q0 depend on our hypotheses.

Noname manuscript No. (will be inserted by the editor) Riesz transforms associated to Schrödinger operators with negative potentials

The aim of this paper is to propose weak assumptions to prove maximal L q regularity for Cauchy problem: du (t) − Lu(t) = f(t). dt Mainly we only require “off-diagonal ” estimates on the real semigroup (e tL)t>0 to obtain maximal L q regularity. The main idea is to use a one kind of Hardy space H 1 adapted to this problem and then use interpolation results. These techniques