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The failure of the Hardy inequality and interpolation of intersections
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
1999 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 37, no 2, 323-244 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
1999. Vol. 37, no 2, 323-244 p.
Research subject
URN: urn:nbn:se:ltu:diva-10505DOI: 10.1007/BF02412218Local ID: 950edd90-7f8c-11db-8824-000ea68e967bOAI: diva2:983450
Godkänd; 1999; 20061129 (evan)Available from: 2016-09-29 Created: 2016-09-29Bibliographically approved

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Kruglyak, NatanMaligranda, LechPersson, Lars-Erik
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