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Scaling and variants of Hardy’s inequality. (English) Zbl 1403.35140
The authors consider Hardy’s inequality and its important generalization by Caffarelli, Kohn, and Nirenberg We recall that these inequalities are used in connection with establishing wellposedness and illposedness for certain singular parabolic equations. The Hardy and Caffarelli, Kohn, and Nirenberg inequalities for functions on \(\mathbb R^+\) can be proved by scaling techniques. In this paper, the authors look at these issues from a reverse perspective. They start with two suitable equations that possess scaling properties and from them they derive an associated Hardy-type inequality. They apply their result to prove some non-wellposedness results when the constant in the singular potential term is large enough.
35K65 Degenerate parabolic equations
26D10 Inequalities involving derivatives and differential and integral operators
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI
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