The refinement and generalization of Hardy’s inequality in Sobolev space

In this paper, we refine the proof of Hardy’s inequality in (Evans in Partial Differential Equations, 2010, Hardy in Inequalities, 1952) and extend Hardy’s inequality from two aspects. That is to say, we extend the integral estimation function from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{u}{|x|}$\end{document}u|x| to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{u}{|x|^{\sigma }}$\end{document}u|x|σ with suitable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma >0$\end{document}σ>0 and extend the space dimension from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 3$\end{document}n≥3 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq 2$\end{document}n≥2. Hardy’s inequality in (Evans in Partial Differential Equations, 2010, Hardy in Inequalities, 1952) is the special case of our results.

In recent decades, there have been many results on the extension and refinement of inequality [7,10,12,30,40,44]. Qin [30] summarized a large number of inequalities and applications, but Hardy's inequality was not included. The authors [7,40] generalized the summation form Hardy inequality, Zhang [44] extended Hardy inequalities using Littlewood-Paley theory and nonlinear estimates method in Besov spaces, and the results improved and extended the well-known results in [1].
The first edition of classic textbook [10] does not contain Hardy's inequality, we see that the very significant Hardy's inequality r)), n ≥ 3, and r > 0 in the second edition of [10]. The proof of Hardy's inequality given in [10,12] is very ingenious, but it is not easy to master for the reader. Therefore, we refine the proof of Hardy's inequality for readers to grasp the essence of the proof and extend Hardy's inequality in Sobolev space from two aspects. That is to say, we extend the integral estimation function from u |x| to u |x| σ with suitable σ > 0 and extend the space dimension from n ≥ 3 to n ≥ 2. Hardy's inequality in [10,12] is the special case of our results.
Let B(o, r) be a closed ball in R n with center o and radius r > 0, . , x n r ) be the unit outward normal to ∂B(o, r). W k,p (Ω) and H 1 (Ω) denote the Sobolev spaces. We write In Sect. 2, we first recall Hardy's inequality, refine the proof for completeness, and state our main results. The proofs of the main results are given in Sect. 3.

Main results
Now, we present the global approximation theorem and Hardy's inequality in Sobolev space.

Lemma 2.1 ([10], Global approximation theorem) Assume that Ω is bounded and ∂Ω
For readers to grasp the essence of the proof, we give the refined proof below.
Proof By the global approximation theorem Lemma 2.1, we may assume u ∈ C ∞ (B(o, r)).
Noting that D( 1 |x| ρ ) = -ρ x |x| ρ+2 for any ρ > 0 and integrating by parts, we have For any ε > 0, using the Cauchy inequality and Schwarz inequality, we obtain Fixing ε > 0 such that n -2 -2ε > 0, we conclude According to the divergence theorem, we have Using the Cauchy inequality and Schwarz inequality, we get Employing this inequality (2.7) in (2.4) finishes the proof of (2.1).
Under the circumstance, we extend the space dimension n and parameter σ in u |x| σ of Hardy's inequality. Now we show our main results. Theorem 2.1 Assume n ≥ 2 and r > 0, u ∈ H 1 (B(o, r)). Then, for σ < n 2 , we have u |x| σ ∈ L 2 (B(o, r)) with the estimate as follows: If σ ≤ 1 and σ < n 2 , we have If σ > 1 and σ < n 2 , we have

Proofs of the main results
In this section we show the proofs of the main results Theorem 2.1.