### Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for Lp-weighted Hardy inequalities

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Title: | Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for Lp-weighted Hardy inequalities |

Authors: | Ruzhansky, M Suragan, D Yessirkegenov, N |

Item Type: | Journal Article |

Abstract: | In this paper we give an extension of the classical Caffarelli-Kohn- Nirenberg inequalities: we show that for 1 <p,q< ∞ ,0 <r< ∞ with p + q ≥ r , δ ∈ [0 , 1] ∩ [ r − q r , p r ] with δr p + (1 − δ ) r q =1and a , b , c ∈ R with c = δ ( a − 1) + b (1 − δ ), and for all functions f ∈ C ∞ 0 ( R n \{ 0 } )wehave ‖| x | c f ‖ L r ( R n ) ≤ ∣ ∣ ∣ ∣ p n − p (1 − a ) ∣ ∣ ∣ ∣ δ ‖| x | a ∇ f ‖ δ L p ( R n ) ∥ ∥ ∥ | x | b f ∥ ∥ ∥ 1 − δ L q ( R n ) for n = p (1 − a ), where the constant ∣ ∣ ∣ p n − p (1 − a ) ∣ ∣ ∣ δ is sharp for p = q with a − b =1or p = q with p (1 − a )+ bq = 0. In the critical case n = p (1 − a )we have ‖| x | c f ‖ L r ( R n ) ≤ p δ ‖| x | a log | x |∇ f ‖ δ L p ( R n ) ∥ ∥ ∥ | x | b f ∥ ∥ ∥ 1 − δ L q ( R n ) . Moreover, we also obtain anisotropic ve rsions of these inequalities which can be conveniently formulated in the language of Folland and Stein’s homoge- neous groups. Consequently, we obtain remainder estimates for L p -weighted Hardy inequalities on homogeneous groups, which are also new in the Eu- clidean setting of R n . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p -weighted Hardy inequalities in- volving a distance and stability estimat e. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also inves- tigated. We also establish sharp Hardy type inequalities in L p ,1 <p< ∞ , with superweights, i.e., wi th the weights of the form ( a + b | x | α ) β p | x | m allowing for different choices of α and β . There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters |

Issue Date: | 14-Feb-2018 |

Date of Acceptance: | 14-Sep-2017 |

URI: | http://hdl.handle.net/10044/1/50818 |

DOI: | https://dx.doi.org/10.1090/btran/22 |

ISSN: | 0002-9947 |

Publisher: | American Mathematical Society |

Start Page: | 32 |

End Page: | 62 |

Journal / Book Title: | Transactions of the American Mathematical Society |

Volume: | 5 |

Copyright Statement: | © Copyright 2018 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0) (https://creativecommons.org/licenses/by/3.0/) |

Sponsor/Funder: | Engineering & Physical Science Research Council (EPSRC) The Leverhulme Trust |

Funder's Grant Number: | EP/K039407/1 RPG-2014-002 |

Keywords: | 0101 Pure Mathematics General Mathematics |

Publication Status: | Published |

Appears in Collections: | Pure Mathematics Mathematics Faculty of Natural Sciences |