Heat and Harmonic Extensions for Function Spaces of Hardy–Sobolev–Besov Type on Symmetric Spaces and Lie Groups
✍ Scribed by Leszek Skrzypczak
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 192 KB
- Volume
- 96
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Function spaces of Hardy Sobolev Besov type on symmetric spaces of noncompact type and unimodular Lie groups are investigated. The spaces were originally defined by uniform localization. In the paper we give a characterization of the space F s p, q (X ) and B s p, q (X ) in terms of heat and Poisson semigroups, for 1 p, q and s # R. The main tool we use, is an atomic decomposition of function spaces on manifolds.
1999 Academic Press 1. PRELIMINARIES Let (X, g) be an n-dimensional connected Riemannian manifold with the Riemannian metric tensor g. Let r inj denote the injectivity radius of X. The manifold X is called a manifold of bounded geometry if the following two conditions are satisfied: (a) r inj >0, (b) |{ k R| C k , k=0, 1, 2, ..., (i.e., every covariant derivative of the Riemannian curvature tensor is bounded).
Examples of manifolds of bounded geometry include all compact manifolds and all homogeneous spaces, i.e., manifolds with a transitive group of isometries (symmetric spaces, Lie groups with left (right) Riemannian structure).
Let [0(x j , r)] j be a covering of X by geodesic balls. The maximal number of the balls with non-empty intersection in this covering is called the multiplicity of the covering. A covering with finite multiplicity is called uniformly locally finite. For the manifold X of bounded geometry there exists a number 0<r 0 <r inj such that if r # (0, r 0 ) then there exists a countable uniformly locally finite covering of X by balls of radius r, cf.