Lower bounds of Copson type for the transposes of lower triangular matrices

Let A=(an,k)n,k⩾0 be a non-negative matrix. Denote by Lp,q(A) the supremum of those L satisfying the following inequality:(∑n=0∞(∑k=0∞an,kxk)q)1/q⩾L(∑k=0∞xkp)1/p(X∈ℓp,X⩾0). In this paper, we focus on the evaluation of Lp,p(At) for a lower triangular matrix A, where 0