The failure of the Hardy inequality and interpolation of intersections

Abstract

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=−1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L_p(w₀), N∩L_p(w₁)), where N is the linear space which consists of all functions with the integral equal to 0. We calculate the K-functional for the couple (N∩L_p(w₀), N∩L_p (w₁)), which turns out to be essentially different from the K-functional for (L_p(w₀), L_p(w₁)), even for the case when N∩L_p(w_i) is dense in L_p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.