The problem addressed is the exact determination of the norms of the classical Hilbert, Copson and averaging operators on weighted `p spaces and the corresponding Lorentz sequence spaces d(w, p), with the power weighting sequence wn = n−α or the variant defined by w1 + · · ·+wn = n1−α. Exact values are found in each case except for the averaging operator with wn = n−α, for which estimates deriv...

In this article we prove exactness of the homotopy sequence overconvergent $p$-adic fundamental groups for a smooth and projective morphism in characteristic $p$. We do so by first proving corresponding result rigid analytic varieties $0$, following dos Santos algebraic case. $p$, then proceed series reductions to case liftable family curves, where can apply result.