Abstract

In this paper, we consider functions u is an element of W-m,W-1 (0, 1) where m >= 2 and u(0) = Du(0) = ... = D(m-1)u(0) = 0. Although it is not true in general that D-j u(x)/x(m-j) is an element of L-1(0, 1) for j is an element of {0, 1,..., m - 1}, we prove that D-j u(x)/x(m-j-k) is an element of W-k,W-1 (0, 1) if k >= 1 and 1 = j + k = m, with j, k integers. Furthermore, we have the following Hardy type inequality, [GRAPHICS] where the constant is optimal.