Arkiv för Matematik
- Ark. Mat.
- Volume 37, Number 2 (1999), 323-344.
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 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∩Lp(w0), N∩Lp(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∩Lp(w0), N∩Lp (w1)), which turns out to be essentially different from the K-functional for (Lp(w0), Lp(w1)), even for the case when N∩Lp(wi) is dense in Lp(wi) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.
The second author was partly supported by a grant M-AA/MA 06857-306 of the Swedish Natural Science Research Council (NFR).
Ark. Mat., Volume 37, Number 2 (1999), 323-344.
Received: 13 February 1998
First available in Project Euclid: 31 January 2017
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1999 © Institut Mittag-Leffler
Krugljak, Natan; Maligranda, Lech; Persson, Lars-Erik. The failure of the Hardy inequality and interpolation of intersections. Ark. Mat. 37 (1999), no. 2, 323--344. doi:10.1007/BF02412218. https://projecteuclid.org/euclid.afm/1485898638