The failure of the Hardy inequality and interpolation of intersections

Natan Krugljak Department of Mathematics, Yaroslavl State University Lech Maligranda Department of Mathematics, Luleå University of Technology Lars-Erik Persson Department of Mathematics, Luleå University of Technology

TBD mathscidoc:1701.332919

Arkiv for Matematik, 37, (2), 323-344, 1998.2
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}(w_{0}), N∩L_{p}(w_{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 ($N$∩$L$_{p}(w_{0}),$N$∩$L$_{p}(w_{1})), which turns out to be essentially different from the$K$-functional for ($L$_{p}(w_{0}), L_{p}(w_{1})), 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.
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@inproceedings{natan1998the,
  title={The failure of the Hardy inequality and interpolation of intersections},
  author={Natan Krugljak, Lech Maligranda, and Lars-Erik Persson},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556411408728},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={2},
  pages={323-344},
  year={1998},
}
Natan Krugljak, Lech Maligranda, and Lars-Erik Persson. The failure of the Hardy inequality and interpolation of intersections. 1998. Vol. 37. In Arkiv for Matematik. pp.323-344. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203556411408728.
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