Higher order Hardy inequalities

This note deals with the inequality (∫ b a |u(x)|w0(x)dx )1/q ≤ C (∫ b a |u(x)|wk(x)dx )1/p , (1) more precisely, with conditions on the parameters p > 1, q > 0 and on the weight functions w0, wk (measurable and positive almost everywhere) which ensure that (1) holds for all functions u from a certain class K with a constant C > 0 independent of u. Here −∞ ≤ a < b ≤ ∞ and k ∈ N and we will consider classes K of functions u = u(x) defined on (a, b) whose derivatives of order k− 1 are absolutely continuous and which satisfy the “boundary conditions” u(a) = 0 for i ∈ M0 , u(b) = 0 for j ∈ M1 (2) where M0,M1 are subsets of the set M = {0, 1, . . . , k− 1}; we will suppose that the number of conditions in (2) is exactly k. This class will be denoted by AC(k−1)(a, b;M0,M1). (3) The conditions (2) are reasonable since they allow to exclude functions like polynomials of order ≤ k−1 for which the right hand side in (1) is zero while the left hand side is positive. 147