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 interval of p's.MSC numbers 2000: 58J35, 42B20
In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John-Nirenberg inequality holds on these spaces and they interpolate with L p spaces by the complex interpolation method. We then give applications on L pboundedness of singular integrals whose kernels do not satisfy the Hörmander condition.
Abstract. Let X bea space of homogeneous type. The aims of this paper are as follows.i) Assuming that T is a bounded linear operator on L 2 (X ), we give a su cient condition on the kernel of T so that T is of weak type (1 1), hence bounded on L p (X ) for 1 < p 2 our condition is weaker than the usual H ormander integral condition.ii) Assuming that T is a bounded linear operator on L 2 ( ) where is a measurable subset of X, we give a su cient condition on the kernel of T so that T is of weak type (1 1), hence bounded on L p ( ) for 1 < p 2. iii) We establish su cient conditions for the maximal truncated operator T , which is de ned by T u(x) = sup ">0 jT " u(x)j, to be L p bounded, 1 < p < 1. Applications include weak (1 1) estimates of certain Riesz transforms, and L p boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.
Let L L be the infinitesimal generator of an analytic semigroup on L 2 ( R n ) L^2({\mathbb R}^n) with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space H L 1 H_L^1 by means of an area integral function associated with the operator L L . By using a variant of the maximal function associated with the semigroup { e − t L } t ≥ 0 \{e^{-tL}\}_{t\geq 0} , a space BMO L \textrm {BMO}_L of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if L L has a bounded holomorphic functional calculus on L 2 ( R n ) L^2({\mathbb R}^n) , then the dual space of H L 1 H_L^1 is BMO L ∗ \textrm {BMO}_{L^{\ast }} where L ∗ L^{\ast } is the adjoint operator of L L . We then obtain a characterization of the space BMO L \textrm {BMO}_L in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces K L {\mathcal K}_L of BMO L _{ L} when L L is a second-order elliptic operator of divergence form and when L L is a Schrödinger operator, and study the inclusion between the classical BMO space and BMO L \textrm {BMO}_L spaces associated with operators.
Abstract. It has been asked (see R. Strichartz, Analysis of the Laplacian. . . , J. Funct. Anal. 52 (1983), 48-79) whether one could extend to a reasonable class of non-compact Riemannian manifolds the L p boundedness of the Riesz transforms that holds in R n . Several partial answers have been given since. In the present paper, we give positive results for 1 ≤ p ≤ 2 under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for p > 2 under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of R n .
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