On Damek-Ricci spaces, radial initial data in H^β with β>a/4 give almost everywhere pointwise convergence for dispersive equations with asymptotically concave phase of degree a; the threshold is sharp up to the endpoint.
Dispersive estimates and generalized Boussinesq equation on hyperbolic spaces with rough initial data
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abstract
We consider the generalized Boussinesq (GBq) equation on the real hyperbolic space $\mathbb{H}^{n}$ ($n\geq2$) in a rough framework based on Lorentz spaces. First, we establish dispersive estimates for the GBq-prototype group, which is associated with a core term of the linear part of the GBq equation, through a manifold-intrinsic Fourier analysis and estimates for oscillatory integrals in $\mathbb{H}^{n}$. Then, we obtain dispersive estimates for the GBq-prototype and Boussinesq groups on Lorentz spaces in the context of $\mathbb{H}^{n}$. Employing those estimates, we obtain local and global well-posedness results and scattering properties in such framework. Moreover, we prove the polynomial stability of mild solutions and leverage this to improve the scattering decay.
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Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
On Damek-Ricci spaces, radial initial data in H^β with β>a/4 give almost everywhere pointwise convergence for dispersive equations with asymptotically concave phase of degree a; the threshold is sharp up to the endpoint.