A Hardy type inequality for W(0, 1) functions.

In this paper, we consider functions $${u\in W^{m,1}(0,1)}$$ where m ≥ 2 and u(0) = Du(0) = · · · = D u(0) = 0. Although it is not true in general that $${\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}$$ for $${j\in \{0,1,\ldots,m-1\}}$$, we prove that $${\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}$$ if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality, where the constant is optimal. [ABSTRACT FROM AUTHOR]