We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces whenp≥1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains inℂN. Our theory applies in a wide range of settings.

No abstract

Let (𝒳, d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. Assuming that μ satisfies certain estimates from below and there exists a suitable Calderón reproducing formula in L 2(𝒳), the authors establish a Lusin‐area characterization for the atomic Hardy spaces H p at(𝒳) of Coifman and Weiss for p ∈ (p 0, 1], where p 0 = n /(n + ε 1) depends on the “dimension” n of 𝒳 and the “regularity” ε 1 of the Calderón reproducing formula. Using this characterization, the authors further obtain a Littlewood–Paley g *λ ‐function characterization for Hp (𝒳) when λ > n + 2n /p and the boundedness of Calderón–Zygmund operators on Hp (𝒳). The results apply, for instance, to Ahlfors n ‐regular metric measure spaces, Lie groups of polynomial volume growth and boundaries of some unbounded model domains of polynomial type in ℂN . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Abstract. The notion of an adapted coordinate system for a given realanalytic function, introduced by V. I. Arnol'd, plays an important role, for instance, in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A. N. Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components. Varchenko's proof is based on a two-dimensional resolution of singularities result.In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko's conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by

No abstract