A generalization of affine fractional $ L^p $ Sobolev inequalities

Wentao Guo; Yuke Li; Xiaolin Zeng; Wentao Guo; Yuke Li; Xiaolin Zeng

doi:10.3934/math.2026925

AIMS Mathematics

2026, Volume 11, Issue 7: 22959-22982. doi: 10.3934/math.2026925

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A generalization of affine fractional $ L^p $ Sobolev inequalities

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

Received: 02 April 2026 Revised: 25 June 2026 Accepted: 22 July 2026 Published: 29 July 2026
MSC : 26D10, 52A20

General affine fractional $ L^p $ Sobolev inequalities were established by Monika Ludwig and Julián Haddad in their foundational work for fractional parameters $ s\in(0, 1) $ and $ 1 < p < n/s $. In this paper, we extend the admissible range of the fractional parameter to $ s\in(0, n)\setminus\mathbb{N} $ and $ 1 < p < n/s $. The affine fractional $ L^p $ Sobolev inequalities are obtained for both the classical fractional case $ s\in(0, 1) $ and the higher-order fractional case. In the course of the proof, we define the higher-order fractional $ L^p $ Sobolev polar projection body and use the method of symmetric decreasing rearrangement.
- fractional Sobolev inequality,
- fractional $ L^p $ polar projection body,
- symmetric decreasing rearrangement,
- Pólya–Szegö inequality
Citation: Wentao Guo, Yuke Li, Xiaolin Zeng. A generalization of affine fractional $ L^p $ Sobolev inequalities[J]. AIMS Mathematics, 2026, 11(7): 22959-22982. doi: 10.3934/math.2026925

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Abstract

General affine fractional $ L^p $ Sobolev inequalities were established by Monika Ludwig and Julián Haddad in their foundational work for fractional parameters $ s\in(0, 1) $ and $ 1 < p < n/s $. In this paper, we extend the admissible range of the fractional parameter to $ s\in(0, n)\setminus\mathbb{N} $ and $ 1 < p < n/s $. The affine fractional $ L^p $ Sobolev inequalities are obtained for both the classical fractional case $ s\in(0, 1) $ and the higher-order fractional case. In the course of the proof, we define the higher-order fractional $ L^p $ Sobolev polar projection body and use the method of symmetric decreasing rearrangement.

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