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On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions
Adam Mickiewicz Univ, Fac Math & Comp Sci, Ul Umultowska 87, PL-61879 Poznan, Poland..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2016 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 45, no 1, 65-81 p.Article in journal (Refereed) Published
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Text
Abstract [en]

The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r(1)(x), r(2)(x)), where r(1)(x) = root x(1)(2) + x(2)(2) and r(2)(x) = root x(3)(2) + x(4)(2) from the subspace (H) over dot((2,2))(1,3) (R-4) of the Sobolev space (H) over dot(1,3) (R-4), radially symmetric in variables (x(1), x(2)) and in variables (x(3), x(4)): integral(R4) u(2)/r(1)(x)r(2)(x) dx <= C integral(R4) vertical bar del u vertical bar(2)dx, Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the radial estimate in Strauss (Comm. Math. Phys. 55, 149-162 1977). Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity, e.g. 1/r(1)(x)r(2)(x) >= 1/2 vertical bar x vertical bar(2).

Place, publisher, year, edition, pages
2016. Vol. 45, no 1, 65-81 p.
Keyword [en]
CKN inequalities, Sobolev embeddings, Hardy inequalities, Block-radial symmetry, Concentration compactness
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-299860DOI: 10.1007/s11118-016-9535-4ISI: 000378557400003OAI: oai:DiVA.org:uu-299860DiVA: diva2:950250
Funder
Wenner-Gren Foundations
Available from: 2016-07-28 Created: 2016-07-28 Last updated: 2016-07-28Bibliographically approved

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