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Volume 38, Issue 2
Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$

Shutao Zhang & Yazhou Han

Anal. Theory Appl., 38 (2022), pp. 178-203.

Published online: 2022-07

[An open-access article; the PDF is free to any online user.]

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  • Abstract

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.

  • Keywords

Hardy-Littlewood-Sobolev inequality, reversed Hardy-Littlewood-Sobolev inequality, rearrangement free method.

  • AMS Subject Headings

39B62, 26A33, 26D10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-38-178, author = {Shutao and Zhang and and 24076 and and Shutao Zhang and Yazhou and Han and and 24077 and and Yazhou Han}, title = {Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$}, journal = {Analysis in Theory and Applications}, year = {2022}, volume = {38}, number = {2}, pages = {178--203}, abstract = {

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2021-0025}, url = {http://global-sci.org/intro/article_detail/ata/20798.html} }
TY - JOUR T1 - Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$ AU - Zhang , Shutao AU - Han , Yazhou JO - Analysis in Theory and Applications VL - 2 SP - 178 EP - 203 PY - 2022 DA - 2022/07 SN - 38 DO - http://doi.org/10.4208/ata.OA-2021-0025 UR - https://global-sci.org/intro/article_detail/ata/20798.html KW - Hardy-Littlewood-Sobolev inequality, reversed Hardy-Littlewood-Sobolev inequality, rearrangement free method. AB -

For conformal Hardy-Littlewood-Sobolev(HLS) inequalities [22] and reversed conformal HLS inequalities [8] on $\mathbb{S}^n,$ a new proof is given for the attainability of their sharp constants. Classical methods used in [22] and [8] depends on rearrangement inequalities. Here, we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method. The merit of the method is that it does not rely on rearrangement inequalities.

Shutao Zhang & Yazhou Han. (2022). Rearrangement Free Method for Hardy-Littlewood-Sobolev Inequalities on $\mathbb{S}^n$. Analysis in Theory and Applications. 38 (2). 178-203. doi:10.4208/ata.OA-2021-0025
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