Integral Geometric Properties of Non-compact Harmonic Spaces (original) (raw)
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Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E) supports a scale-invariant L 2-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p): L p-estimate for the gradient of the associated heat semigroup; (ii) (RH p): L p-reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p): L p-boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.
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Transactions of the American Mathematical Society, 1993
We shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce sharp results for the pointwise almost everywhere convergence of Fourier integrals of radial functions in the Lorentz spaces LP'q(W). Also we shall briefly apply the above approach to the study of the harmonic analysis of radial functions on noneuclidean hyperbolic spaces. In 1869 Hermann Hankel proved what is by now known as Hankel's inversion formula for Fourier-Bessel expansions: Revisiting [Hankel] we shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce sharp results for the pointwise almost everywhere convergence of Fourier integrals of radial functions in the Lorentz spaces Lp'q(R"). In particular we shall prove that the partial sums of the Fourier integrals of radial functions in L^m'1(R") + L"-i ''(R") converge almost everywhere, while for the other Lorentz spaces either the partial sums cannot be defined, or they may diverge at every point. Finally we shall briefly study the equiconvergence between Fourier-Jacobi and cosine expansions. This is related to the harmonic analysis of radial functions on noneuclidean hyperbolic spaces. It is noteworthy that in this case pointwise convergence holds only for V^ spaces with p in a nonsymmetric range around 2.
The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth
The Journal of Geometric Analysis, 2019
Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by h > 0 the mean curvature of horospheres in X, and set ρ = h/2. Fixing a basepoint o ∈ X, for ξ ∈ ∂X, denote by B ξ the Busemann function at ξ such that B ξ (o) = 0. then for λ ∈ C the function e (iλ−ρ)B ξ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue −(λ 2 + ρ 2). For a function f on X, we define the Fourier transform of f bỹ f (λ, ξ) := X f (x)e (−iλ−ρ)B ξ (x) dvol(x) for all λ ∈ C, ξ ∈ ∂X for which the integral converges. We prove a Fourier inversion formula f (x) = C 0 ∞ 0 ∂Xf (λ, ξ)e (iλ−ρ)B ξ (x) dλo(ξ)|c(λ)| −2 dλ for f ∈ C ∞ c (X), where c is a certain function on R − {0}, λo is the visibility measure on ∂X with respect to the basepoint o ∈ X and C 0 > 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon. Contents 1. Introduction 2. Basics about harmonic manifolds 3. Radial and horospherical parts of the Laplacian 4. Analysis of radial functions 4.1. Chebli-Trimeche hypergroups 4.2. The density function of a harmonic manifold 4.3. The spherical Fourier transform 5. Fourier inversion and Plancherel theorem 6. An integral formula for the c-function 7. The convolution algebra of radial functions 8. The Kunze-Stein phenomenon References