L Version of Hardy’s Theorem on Semisimple Lie Groups
نویسندگان
چکیده
We prove an analogue of the Lp version of Hardy’s theorem on semisimple Lie groups. The theorem says that on a semisimple Lie group, a function and its Fourier transform cannot decay very rapidly on an average.
منابع مشابه
An Analogue of Hardy’s Theorem for Very Rapidly Decreasing Functions on Semi-simple Lie Groups
A celebrated theorem of L. Schwartz asserts that a function f on R is ‘rapidly decreasing’ (or in the ‘Schwartz class’) iff its Fourier transform is ‘rapidly decreasing’. Since this theorem is of fundamental importance in harmonic analysis, there is a whole body of literature devoted to generalizing this result to other Lie groups. (For example, see [18].) In sharp contrast to Schwartz’s theore...
متن کاملACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE
A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...
متن کاملAn Lp-Lq version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups
We consider a real semisimple Lie group G with finite center and K a maximal compact subgroup of G. We prove an Lp −Lq version of Hardy’s theorem for the spherical Fourier transform on G. More precisely, let a, b be positive real numbers, 1 ≤ p, q ≤ ∞, and f a K-bi-invariant measurable function on G such that h−1 a f ∈ Lp(G) and eb‖λ‖ (f )∈ Lq(a∗ +) (ha is the heat kernel on G). We establish th...
متن کاملHardy’s Uncertainty Principle on Semisimple Groups
A theorem of Hardy states that, if f is a function on R such that |f(x)| ≤ C e−α|x|2 for all x in R and |f̂(ξ)| ≤ C e−β|ξ|2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. Sitaram and Sundari generalised this theorem to semisimple groups with one conjugacy class of Cartan subgroups and to the K-invariant case for general semisimple groups. We extend the theorem to all semisimple gr...
متن کاملHardy’s Theorem and Rotations
We prove an extension of Hardy’s classical characterization of real Gaussians of the form e−παx 2 , α > 0 to the case of complex Gaussians in which α is a complex number with positive real part. Such functions represent rotations in the complex plane of real Gaussians. A condition on the rate of decay of analytic extensions of a function f and its Fourier transform f̂ along some pair of lines in...
متن کامل