Triangular projection on Sₚ,~0<p<1, as p approaches 1

This is a continuation of our recent paper. We continue studying properties of the triangular projection Pₙ on the space of n× n matrices. We establish sharp estimates for the p-norms of Pₙ as an operator on the Schatten--von Neumann class Sₚ for $0<p<1$. Our estimates are uniform in n and p as soon as p is separated away from 0. The main result of the paper shows that for p∈(0,1), the p-norms of Pₙ on Sₚ behave as n→∞ and p→1 as n1/p-1min\(1-p)⁻¹,log n\.