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 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 (Evans in Partial Differential Equations, 2010, Hardy in Inequalities, 1952) is the special case of our results.

Introduction

It is well known that inequalities are important tools in classical analysis [2–6, 13, 14, 26–29, 31–39, 41–43, 45]. One application of inequalities is to study the properties of partial differential equations. Li and his coauthors [15-23] studied the global existence and uniqueness, limit behavior, uniform stability, and blow-up of solutions for partial differential equations by using various inequalities. Liu [11, 24, 25] showed the stability and convergence results of evolution equations and Du [8, 9] studied obstacle problems by using various inequalities.

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 holds if , , and 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 to with suitable and extend the space dimension from to . Hardy’s inequality in [10, 12] is the special case of our results.

Let be a closed ball in with center o and radius , be a vector in , be the unit outward normal to . and 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 is . Let for some . Then there exist functions such that

Lemma 2.2

([10, 12], Hardy’s inequality)

Assume and . Let . Then with the estimate

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 . Noting that for any and integrating by parts, we have Therefore

For any , using the Cauchy inequality and Schwarz inequality, we obtain Fixing such that , we conclude According to the divergence theorem, we have Using the Cauchy inequality and Schwarz inequality, we get Combining (2.5) and (2.6), we obtain the trace inequality 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 of Hardy’s inequality. Now we show our main results.

Theorem 2.1

Assume and , . Then, for , we have with the estimate as follows:

If and , we have

If and , we have

Remark 2.1

Hardy’s inequality (2.1) is the case of and in Theorem 2.1.

Remark 2.2

If , then . denotes a closed circular region with center o and radius , denotes a circle, and denotes curvilinear integration.

Proofs of the main results

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

For any , since which implies

By the global approximation theorem, we may assume . Noting that (3.1) holds, we obtain Hence For any , using the Cauchy inequality and Schwarz inequality, we obtain According to the divergence theorem, we have and using the Cauchy inequality and Schwarz inequality, we get which implies By substituting (3.4) and (3.5) into (3.3), fixing ε such that , we conclude

Therefore, from (3.6), for and :

if , noting that we obtain

if , noting that we obtain

The proof of Theorem 2.1 is completed. □

Conclusions

In this paper, 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 to with suitable and extend the space dimension from to . Hardy’s inequality in [10, 12] is the special case of our results.