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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∩Lp(w1)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L p(w0),N∩L p (w1)), which turns out to be essentially different from theK-functional for (L p(w0), Lp(w1)), even for the case whenN∩L p(wi) is dense inL p(wi) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.Godkänd; 1999; 20061129 (evan

The failure of the Hardy inequality and interpolation of intersections