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Oscillating multipliers on Lie groups and Riemannian manifolds. (English) Zbl 0835.22008

Let \(G\) be a Lie group of polynomial volume growth or a Riemannian manifold of nonnegative Ricci curvature. The author first defines the local dimension \(d\) and the dimension at infinity \(D\) of \(G\) by making use of the ball \(B_r (x)\) of radius \(r > 0\) with center \(x \in G\), and its volume \(|B_r (x) |\). Let \(L\) be the Laplace-Beltrami operator on \(G\). Using the spectral resolution \(L = \int^\infty_0 \lambda d E_\lambda\), we can define for a bounded measurable function \(m\) on \(G\) the operator \(m(L) = \int^\infty_0 m (\lambda) dE_\lambda\), which is bounded on \(L^2 (G)\). The author investigates the boundedness of the operator \(m(L)\) on \(L^p (G)\), in terms of \(d\), \(D\) and \(p\) for oscillating multipliers \(m = m_{\alpha, \beta}\) defined by \[ m_{\alpha, \beta} (\lambda) = \psi \bigl( |\lambda |\bigr) |\lambda |^{-\beta/2} \exp \bigl( i |\lambda |^{\alpha/2} \bigr) \] \((\alpha, \beta > 0)\), where \(\psi (\lambda)\) is a \(C^\infty\) function with value 0 for \(|\lambda |< 1\) and 1 for \(|\lambda |> 2\).

MSC:

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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