Maximal Averages on the Affine Group $G_n$ and applications

Let \(G_n=\mathbb R^n\rtimes\mathbb R_+\) be equipped with the left Haar measure \( dμ(x,y)=\frac{dx\,dy}{y^{n+1}}. \) We study maximal averages associated with three basic motions on \(G_n\): horizontal translations, vertical dilations, and fixed hyperbolic geodesics in the upper half-space model. The translation maximal operator is the Euclidean Hardy--Littlewood maximal operator on each horizontal slice. The Haar-compatible dilation maximal operator is of weak type \((1,1)\) and bounded on \(L^p(G_n)\) for \(1