Wave equation and multiplier estimates on Damek-Ricci spaces

TL;DR

This paper establishes pointwise and gradient estimates for wave operators on Damek-Ricci spaces, generalizing prior results and deriving Sobolev estimates for wave equation solutions.

Contribution

It extends wave and multiplier estimates from ax+b-groups to Damek-Ricci spaces and provides new proofs and Sobolev estimates for wave equations.

Findings

Pointwise estimates for convolution kernels of wave operators.

Gradient estimates for these convolution kernels.

Sobolev estimates for solutions to wave equations.

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 associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.