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• # Artykuł - szczegóły

## Colloquium Mathematicum

2002 | 94 | 2 | 263-280

## Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

EN

### Abstrakty

EN
Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis ${Y_{δ,j}: δ ∈ K̂₀, 1 ≤ j ≤ d_{δ}}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ}(iλ + ϱ)$ be the Kostant polynomials. We establish the following version of Hardy's theorem for the Helgason Fourier transform: Let f be a function on G/K which satisfies $|f(ka_{r})| ≤ Ch_{t}(r)$. Further assume that for every δ and j the functions
$F_{δ,j}(λ) = Q_{δ}(iλ +ϱ)^{-1} ∫_{K/M} f̃(λ,b)Y_{δ,j}(b)db$
satisfy the estimates $|F_{δ,j}(λ)| ≤ C_{δ,j}e^{-tλ²}$ for λ ∈ ℝ. Then f is a constant multiple of the heat kernel $h_{t}$.

263-280

wydano
2002

### Twórcy

autor
• Stat-Math Division, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore-560 059, India