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



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