Regularity and pointwise convergence of solutions of the Schr\"odinger operator with radial initial data on Damek-Ricci spaces

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition f of the Schr\"odinger equation given by {equation*}{cases} i∂ u/∂ t =Δ u\:,\: (x,t) ∈ R^n × R \\ u(0,·)=f\:, on R^n \:, {cases}{equation*} in terms of the index α such that f belongs to the inhomogeneous Sobolev space H^α(Rⁿ) , so that the solution of the Schr\"odinger operator u converges pointwise to f, limt → 0+ u(x,t)=f(x), almost everywhere. In this article, we consider the Carleson's problem for the Schr\"odinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint α ≥ 1/4, which agrees with the classical Euclidean case.