Estimates of ‖e ^itf‖ _A(Γ), when Γ ⊂ ℝ _n is a curve andf is a real-valued functionis a curve andf is a real-valued function

Abstract

Γ is a simple curve in ℝ _n with an equationx=γ(u),u∈[0. 1], γ∈C ^k, where for somek, 2≦k−1≦n, γ′(u), γ″(u),…,γ _k−1(u) are linearly independent for everyu. It is then proved that if t ∈ ℝ and f is a real-valued function in C ^k(Γ),t ^−1/k ‖e ^itf‖ _A(Γ) is bounded as |t|→∞. An example shows that the estimate cannot be improved in general, whenn=2,k=3. The result is interpreted in terms of properties of the space of pseudomeasures on Γ.