Improved Hardy inequalities and weighted Hardy type inequalities with spherical derivatives
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Improved Hardy inequalities and weighted Hardy type inequalities with spherical derivatives Nguyen Tuan Duy1 · Nguyen Lam2 · Le Long Phi3 Received: 17 June 2020 / Accepted: 6 November 2020 © Universidad Complutense de Madrid 2020
Abstract The first purpose of this paper is to set up several enhancements of the Hardy type inequalities on certain subspaces of the Sobolev spaces. The second aim is to explore the role of the spherical derivative in such improvements and to prove some weighted versions of the Hardy inequality with spherical derivative. As applications of our results, we establish several Hardy’s inequalities and the Hardy inequalities with spherical derivatives on half-spaces. Keywords Hardy inequality · Spherical derivative · Best constant · Bessel pair Mathematics Subject Classification 26D10 · 46E35 · 35A23
1 Introduction Hardy’s inequalities are one of the most used inequalities in analysis. They have been studied intensively and extensively in the literature due to their important roles in many areas of mathematics such as mathematical physics, spectral theory, analysis of linear and non-linear PDE, harmonic analysis and stochastic analysis. The interested
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Le Long Phi [email protected] Nguyen Tuan Duy [email protected] Nguyen Lam [email protected]
1
Faculty of Economics and Law, University of Finance-Marketing, 2/4 Tran Xuan Soan St., Tan Thuan Tay Ward, Dist. 7, Ho Chi Minh City, Vietnam
2
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland, Corner Brook, NL A2H5G4, Canada
3
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
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N. T. Duy et al.
reader is referred to the monographs [3,28,31,32,38,40], to name just a few, that are standard references on the subject. In this note, we concern the Hardy inequality of the following form in R N , N ≥ 3: 1 Let u be in D 1,2 R N , the completion with respect to R N |∇u|2 d x 2 of C0∞ R N . |u|2 1 R N . Moreover Then |x| 2 ∈ L
|∇u| d x ≥ 2
N −2 2
2
RN
RN
|u|2 d x. |x|2
(1)
2 The constant N 2−2 in (1) has been well-investigated in the literature. The fact that N −2 2 is optimal but is never achieved in (1) by nontrivial functions in D 1,2 R N 2 is well-understood. Also several questions about the possible improvements of (1) have been raised. For instance, we may ask for the strengthened versions of the Hardy inequalities where extra nonnegative terms can be added to the RHS of (1). On the whole space R N , Ghoussoub and Moradifam showed in [28] that there is no strictly positive V ∈ V 1 ((0, ∞)) such that the inequality
|∇u|2 d x −
N −2 2
2
RN
RN
|u|2 |x|
dx ≥ 2
V (|x|) u 2 d x RN
holds for all u ∈ C0∞ R N . However, the situation on bounded domain is different. Indeed, let Ω be a bounded domain in R N , N ≥ 3, with 0 ∈ Ω. Then Brezis and Vázquez, in order to investigate the stability of singular solutions of certain nonlinear elliptic equations, proved in [8] that for all u ∈ W01,2 (Ω):
|∇u
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