The failure of the Hardy inequality and interpolation of intersections

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 alpha not equal 0 but which fail for the "critical" point alpha=0. This means that we cannot interpolate these inequalities between the noncritical points alpha=1 and alpha=-1 and conclude that it is also true at the critical point alpha=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N boolean AND L-p(omega(0)), N boolean AND L-p(omega(1))), 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 boolean AND L-p(omega(0)), N boolean AND L-p(omega(1))), which turns out to be essentially different from the K-functional for (L-p(omega(0)), L-p(omega(1))), even for the case when N boolean AND L-p(omega(i)) is dense in L-p(omega(i)) (i=0,1) This essential difference is the reason why the "naive" interpolation above gives an incorrect result.