Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 1159-1174

CRITICAL HARDY INEQUALITIES

Michael Ruzhansky and Durvudkhan Suragan

Ghent University, Department of Mathematics, Belgium, and
Queen Mary University of London, School of Mathematical Sciences
Mile End Road, London E1 4NS, United Kingdom; Michael.Ruzhansky 'at' ugent.be

Nazarbayev University, Department of Mathematics
53 Kabanbay Batyr Ave, Astana 010000, Kazakhstan; durvudkhan.suragan 'at' nu.edu.kz

Abstract. We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy–Sobolev type inequalities. Most of the obtained estimates are new already for the case of Rⁿ. For example, for any f ∈ C₀^∞(Rⁿ \ {0}) we obtain the range of critical Hardy inequalities of the form

sup_{R > 0}‖(f – f_R)/(|x|^n/p log(R/|x|)‖_Lp(Rⁿ)) ≤ p/(p – 1)‖1/|x|^{n/p
– 1} ∇f‖_Lp(Rⁿ), 1 < p < ∞,

where f_R = f(Rx/|x|), with sharp constant p/(p – 1), recovering the known cases of p = n and p = 2. Moreover, we also show a new type of a critical Hardy inequality of the form

‖f/|x|‖_Ln(Rn) ≤ n ‖(log|x|)∇f‖_Ln(Rn),

for all f ∈ C₀^∞(Rⁿ \ {0}), where the constant n is sharp.

2010 Mathematics Subject Classification: Primary 22E30, 43A80.

Key words: Critical Hardy inequality, homogeneous Lie group, uncertainty principle.

Reference to this article: M. Ruzhansky and D. Suragan: Critical Hardy inequalities. Ann. Acad. Sci. Fenn. Math. 44 (2019), 1159-1174.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4467