Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e(it root L)psi(root L/lambda) for arbitrary time t and arbitrary lambda > 0, where psi is a smooth bump function supported in [-2, 2] if lambda < 1 and supported in [1, 2] if lambda >= 1. This generalizes previous results in Muller and Thiele (Studia Math. 179: 117-148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37-61, 2003) and Vallarino (J. Lie Theory 17:163-189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L
Muller, D., & Vallarino, M. (2010). Wave equation and multiplier estimates on Damek-Ricci spaces. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 16(2), 204-232 [10.1007/s00041-009-9088-7].
Wave equation and multiplier estimates on Damek-Ricci spaces
Vallarino, M
2010
Abstract
Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e(it root L)psi(root L/lambda) for arbitrary time t and arbitrary lambda > 0, where psi is a smooth bump function supported in [-2, 2] if lambda < 1 and supported in [1, 2] if lambda >= 1. This generalizes previous results in Muller and Thiele (Studia Math. 179: 117-148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37-61, 2003) and Vallarino (J. Lie Theory 17:163-189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L
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