Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 2 (1999), 323-344.

The failure of the Hardy inequality and interpolation of intersections

Natan Krugljak, Lech Maligranda, and Lars-Erik Persson

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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 (NLp(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 (NLp(w0), NLp (w1)), which turns out to be essentially different from the K-functional for (Lp(w0), Lp(w1)), even for the case when NLp(wi) is dense in Lp(wi) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.

Note

The second author was partly supported by a grant M-AA/MA 06857-306 of the Swedish Natural Science Research Council (NFR).

Article information

Source
Ark. Mat., Volume 37, Number 2 (1999), 323-344.

Dates
Received: 13 February 1998
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898638

Digital Object Identifier
doi:10.1007/BF02412218

Mathematical Reviews number (MathSciNet)
MR1714765

Zentralblatt MATH identifier
1021.46024

Rights
1999 © Institut Mittag-Leffler

Citation

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


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