Integral Geometric Properties of Non-compact Harmonic Spaces (original) (raw)
Gradient estimates for heat kernels and harmonic functions
Journal of Functional Analysis, 2019
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.
Some new function spaces and their applications to Harmonic analysis
Journal of Functional Analysis, 1985
... SOME NEW FUNCTION SPACES 331 The proof proceeds by induction on k and a reduction via Theorem 5 from k to Al ... PROPOSITION 7. Suppose a is bounded; then u(x, t) = H,(aH,(a H,(a)) satisfies the condition (9.2) of Theorem 5. We start by observing that H,=(IP,)H, where H ...
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.
An introduction to some modern aspects of harmonic analysis
1996
The following notes were prepared as the text of a set of lectures given at the Workshop in Geometry and Analysis, held at the Australian National University in January 1995, and aimed at a mixed audience of postgraduate students and academics. can be extended to a bounded linear map LP -t Lq, where l + l = 1. We shall study this p q
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
Operators and harmonic analysis on the sphere
Transactions of the American Mathematical Society, 1966
The main result of this paper concerns operators which commute with all rotations on certain spaces of functions on Sk, the /¿-dimensional sphere (k^2), namely C, L1, L". The proofs use harmonic analysis of various spaces of functions and measures on Sk, which involves the ultraspherical polynomials. Notation. Sk admits a group of rotations, namely the special orthogonal group SOfc + 1. The result of the action of the rotation a on the point x will be denoted by xa. The "rotation" operator Ra acting on functions (f) and measures (/*) is defined by RJ(x) = /(*«) for all x e Sk, Rap(E) = n(Ea) for all /¿-measurable subsets E of Sk. P% is the ultraspherical polynomial of index A and degree n (normalized bŷ nO)=l)-Considering Sk imbedded as the unit sphere in Rk + 1, let x-y be the ordinary inner product of the vectors which correspond to the points x and y (-1 ^x-y¿ 1). Sk has a unique rotation invariant Borel measure, say m, such that m(Sk)=l, and the use of this measure is implied by notations such as "dx"; further let Lp(Sk)=Lp(Sk, ni). Statement of results. With each ¡j, e M(Sk), the space of finite regular Borel measures on Sk, and each /eL1^'), the following continuous functions are associated : fin(x) = jgk Pf-»'2(x y)My) " = 0, 1, 2,...