arXiv subjects
Publications and source records attributed to Oliver Brammen.
We show various uncertainty principles for the Fourier transform on harmonic manifolds of rank one. In particular, we derive a Heisenberg uncertainty principle, a Morgen theorem, an uncertainty principle for the Schr\"odinger equation and a version of H\"omanders theorem. Furthermore, we generalise the in the Euclidean case well-known Hausdorff-Young inequality for the Fourier transform, to harmonic manifolds of rank one.
We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial $L^1$ space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.
We show that Sullivan's method of constructing eigenfunctions of the Laplacian \cite{Sullivan1987RelatedAO} is applicable to non-compact simply connected harmonic manifolds of purely exponential volume growth. Thereby we extend the case of pinched negative curvature to non-symmetric Damek-Ricci spaces.
We study the dynamics of $L^p$-multipliers on non-compact simply connected harmonic manifolds of purely exponential volume growth. These are linear operators on the $L^p$-spaces which behave nicely on radial functions under Fourier transformation. In the process we complement the results of Kingshook Biswas and Rudra P. Sarkar by showing that if they are acting nicely on smooth function with compact support under Fourier transform for $p\leq 2$ these can not be chaotic. Furthermore, we use this to study the behaviour of the heat semi-group, the resolvent and the convolution algebra arising from convolution with radial functions. In the process, we obtain a Young inequality for the convolution on non-compact simply connected harmonic manifolds, an extension of the Kunz-Stein and study the domain of holomorphicity of the Fourier transform.
We solve the shifted wave equation \begin{align*} \frac{\partial^2}{\partial t^2}φ(x,t)=(Δ_x+ρ^2)φ(x,t) \end{align*} on a non compact simply connected harmonic manifold with mean curvature of the horospheres $2ρ>0$. We give an explicit representation of the solution as the inverse dual Abel transform of the spherical means of there initial conditions using the local injectivity of the Abel transform and symmetry properties of the spherical mean value operator. Furthermore we investigate the wave equation using the Fourier transform on harmonic manifolds of rank one. Additionally we show an analogous of the classical Paley-Wiener theorem and use it to show an asymptotic Huygens principle as well as asymptotic equidistribution of the energy of a solution of the shifted wave equation under assumptions on the $\mathbf{c}$-function.