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Hardy type inequalities in \(L^{p}\) with sharp remainders

Journal of Inequalities and Applications20172017:5

https://doi.org/10.1186/s13660-016-1271-1

Received: 10 August 2016

Accepted: 2 December 2016

Published: 3 January 2017

Abstract

Sharp remainder terms are explicitly given on the standard Hardy inequalities in \(L^{p}(\mathbb {R}^{n})\) with \(1< p< n\). Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals.

Keywords

Hardy’s inequalitiesremainders

MSC

26D1026D1546E35

1 Results and discussion

The following Hardy inequalities are now well known:
$$\begin{aligned}& \biggl( \int_{0}^{\infty}x^{-r-1}\biggl\vert \int_{0}^{x}f(y)\,dy\biggr\vert ^{p} \biggr)^{\frac{1}{p}} \leq \frac{p}{r} \biggl( \int_{0}^{\infty}x^{p-r-1}\bigl|f(x)\bigr|^{p} \,dx \biggr)^{\frac{1}{p}}, \end{aligned}$$
(1.1)
$$\begin{aligned}& \biggl( \int_{0}^{\infty}x^{r-1}\biggl\vert \int_{x}^{\infty}f(y)\,dy\biggr\vert ^{p} \biggr)^{\frac{1}{p}} \leq \frac{p}{r} \biggl( \int_{0}^{\infty}x^{p+r-1}\bigl|f(x)\bigr|^{p} \,dx \biggr)^{\frac{1}{p}}, \end{aligned}$$
(1.2)
where \(1\leq p<\infty\), \(r>0\), and f is a real-valued measurable function on \((0,\infty)\),
$$ \biggl\Vert \frac{f}{|x|}\biggr\Vert _{L^{p}(\mathbb {R}^{n})} \leq \frac{p}{n-p} \biggl\Vert \frac{x}{|x|}\cdot\nabla f\biggr\Vert _{L^{p}(\mathbb {R}^{n})}, $$
(1.3)
where \(1\leq p< n\) and \(f\in W_{p}^{1}(\mathbb {R}^{n})\) (see [1, 2] for instance).

We revisit this famous inequality. Particularly, we present equalities which fill the gaps between the right- and left-hand sides of (1.1)-(1.3) with explicit remainder terms for \(p>1\). Those equalities yield (1.1)-(1.3) by dropping remainder terms. Moreover, we give a characterization of functions which leads to vanishing remainders. The study of the Hardy inequalities which is based on the viewpoint of the equality leads to a direct and explicit understanding of the Hardy type inequalities as well as of the nonexistence of nontrivial extremals.

To state our main theorems, we introduce some necessary notation. In this paper, we deal with real-valued functions and we argue with sufficiently smooth functions with compact support in \(\mathbb {R}^{n}\setminus\{0\}\) so that the standard density argument goes through. Let us introduce
$$\begin{aligned}& R_{p}(\xi,\eta) = \biggl(\frac{1}{p}|\eta|^{p}+ \frac{1}{p'}|\xi|^{p}-|\xi|^{p-2}\xi\eta \biggr) \big/|\xi- \eta|^{2}\quad\mbox{if }\xi\neq\eta, \end{aligned}$$
(1.4)
$$\begin{aligned}& R_{p}(\xi,\xi) = \frac{p-1}{2}|\xi|^{p-2}, \end{aligned}$$
(1.5)
for \(p>1\) and ξ, \(\eta\in \mathbb {R}\), where \(1/p'=1-1/p\) and \(R_{p}(\xi,\xi)\) makes sense only if \(p\geq2\) and if \(p<2\) and \(\xi\neq0\). In other words, \(R_{p}: (\xi,\eta)\mapsto R_{p}(\xi,\eta)\) is well defined on \(\mathbb {R}\times \mathbb {R}\) if \(p\geq2\) and on \((\mathbb {R}\times \mathbb {R})\setminus\{(0,0)\}\) if \(1< p<2\). For p with \(1\leq p\leq\infty\), the Banach space which consists of pth integrable Lebesgue measurable functions is denoted by \(L^{p}(\Omega)\). The norm of it is also denoted by \(\|\cdot\|_{L^{p}}\) or \(\|\cdot\|_{p}\) if it does not cause confusion. The Sobolev space of order one introduced by \(L^{p}\) is denoted \(W^{1}_{p}=W^{1}_{p}(\mathbb {R}^{n})\) for \(1\leq p<\infty\).

The basic properties of \(R_{p}\) are summarized in the following proposition.

Proposition 1

Let \(p\in \mathbb {R}\) satisfy \(p>1\). Then \(R_{p}\) satisfies the following properties:

(1) \(R_{p}\) has the integral representation
$$ R_{p}(\xi,\eta)=(p-1) \int_{0}^{1} \bigl\vert \theta\xi+(1-\theta)\eta \bigr\vert ^{p-2}\theta \,d\theta. $$
(1.6)
(2) \(R_{p}\) satisfies the estimates
$$\begin{aligned}& R_{p}(\xi,\eta) \leq \textstyle\begin{cases} \frac{p-1}{2} (\vert \xi \vert \lor \vert \eta \vert )^{p-2}& \textit{if }p\geq 2,\\ \frac{p-1}{2} (\vert \xi \vert \land \vert \eta \vert )^{p-2}&\textit{if }p< 2, \end{cases}\displaystyle \\& R_{p}(\xi,\eta) \geq \textstyle\begin{cases} \frac{p-1}{2} (\vert \xi \vert \land \vert \eta \vert )^{p-2}&\textit{if }p\geq 2,\\ \frac{p-1}{2} (\vert \xi \vert \lor \vert \eta \vert )^{p-2}&\textit{if }p< 2, \end{cases}\displaystyle \end{aligned}$$
where \(a\lor b=\max(a,b)\) and \(a\land b=\min(a,b)\) for a, \(b\geq0\).

(3) Let \(p>2\) and let ξ, \(\eta\in \mathbb {R}\). Then \(R_{p}(\xi,\eta)=0\) if and only if \(\xi=\eta=0\).

(4) Let \(p\leq2\) and let ξ, \(\eta\in \mathbb {R}\setminus\{0\}\). Then \(R_{p}(\xi,\eta)>0\).

(5) \(R_{2}(\xi,\eta)=\frac{1}{2}\) for all ξ, \(\eta\in \mathbb {R}\).

We now state our main results.

Theorem 1

Let n and p satisfy \(1< p< n\). Then the equality
$$\begin{aligned} \biggl\Vert \frac{f}{|x|}\biggr\Vert _{L^{p}(\mathbb {R}^{n})}^{p} =& \biggl(\frac{p}{n-p} \biggr)^{p} \biggl\Vert \frac{x}{|x|} \cdot\nabla f\biggr\Vert _{L^{p}(\mathbb {R}^{n})}^{p} \\ &{}- p \int_{\mathbb {R}^{n}}R_{p} \biggl(\frac{1}{|x|}f,- \frac{p}{n-p}\frac{x}{|x|}\cdot\nabla f \biggr) \biggl\vert \frac{p}{n-p}\frac{x}{|x|}\cdot\nabla f+\frac{1}{|x|}f\biggr\vert ^{2}\,dx \end{aligned}$$
(1.7)
holds for all \(f\in W^{1}_{p}(\mathbb {R}^{n})\). If the second term on the right hand side of (1.7) vanishes, then the left-hand side of (1.7) is finite if and only if \(f=0\).

Remark 1

In fact, we prove that if the second term on the right hand side of (1.7) vanishes, then there exists a function \(\varphi: S^{n-1}\to \mathbb {R}\) on the unit sphere \(S^{n-1}\) such that
$$ f(x)=|x|^{-\frac{n-p}{p}}\varphi \biggl(\frac{x}{|x|} \biggr) $$
(1.8)
almost everywhere in \(\mathbb {R}^{n}\setminus\{0\}\). In this case,
$$ \frac{|f(x)|^{p}}{|x|^{p}} = \frac{\vert \varphi (\frac{x}{|x|} )\vert ^{p}}{|x|^{n}} $$
(1.9)
and the left-hand side of (1.7) is finite if and only if \(\varphi=f=0\).

Remark 2

The special case \(p=2\) is studied in [3].

Theorem 2

Let p and r satisfy \(1< p<\infty\) and \(r>0\). Then:

(1) The equality
$$\begin{aligned} \int_{0}^{\infty}x^{-r-1}\biggl\vert \int_{0}^{x}f(y)\,dy\biggr\vert ^{p}\,dx =& \biggl(\frac{p}{r} \biggr)^{p} \int_{0}^{\infty}x^{p-r-1}\bigl\vert f(x)\bigr\vert ^{p}\,dx \\ & {}-p \int_{0}^{\infty}R_{p} \biggl(x^{-\frac{r+1}{p}} \int_{0}^{x}f, \frac{p}{r}x^{1-\frac{r+1}{p}}f \biggr) \\ &{}\times \biggl\vert x^{-\frac{r+1}{p}} \biggl( \frac{p}{r}xf- \int_{0}^{x}f \biggr) \biggr\vert ^{2} \,dx \end{aligned}$$
(1.10)
holds for all real-valued measurable functions on \((0,\infty)\) with \(xf\in L^{p}(0,\infty; x^{-r-1}\,dx)\). Moreover, there exists \(c\in \mathbb {R}\) which satisfies, for almost everywhere \(x\in(0,\infty)\),
$$ f(x)=cx^{\frac{r}{p}-1} $$
(1.11)
when the last term in the right hand side of (1.10) equals zero. In this case,
$$ x^{-r-1}\biggl\vert \int_{0}^{x}f(y)\,dy\biggr\vert ^{p}=|c|^{p} \biggl(\frac{p}{r} \biggr)^{p}x^{-1} $$
(1.12)
and the left-hand side of (1.10) is finite if and only if \(c=0\).
(2) The equality
$$\begin{aligned}& \int_{0}^{\infty}x^{r-1}\biggl\vert \int_{x}^{\infty}f(y)\,dy\biggr\vert ^{p} \,dx \\& \quad= \biggl(\frac{p}{r} \biggr)^{p} \int_{0}^{\infty}x^{p+r-1}\bigl\vert f(x)\bigr\vert ^{p}\,dx \\& \qquad {}-p \int_{0}^{\infty}R_{p} \biggl(x^{\frac{r-1}{p}} \int_{x}^{\infty}f, \frac{p}{r}x^{1+\frac{r-1}{p}}f \biggr) \biggl\vert x^{\frac{r-1}{p}} \biggl(\frac{p}{r}xf- \int_{x}^{\infty}f \biggr) \biggr\vert ^{2} \,dx \end{aligned}$$
(1.13)
holds for all real-valued measurable functions on \((0,\infty)\) with \(xf\in L^{p}(0,\infty; x^{r-1}\,dx)\). Moreover, there exists \(c\in \mathbb {R}\) which satisfies, for almost everywhere \(x\in(0,\infty)\),
$$ f(x)=cx^{-\frac{r}{p}-1} $$
(1.14)
provided that the last term in the right hand side of (1.13) vanishes. In this case,
$$ x^{r-1}\biggl\vert \int_{x}^{\infty}f(y)\,dy\biggr\vert ^{p} = |c|^{p} \biggl(\frac{p}{r} \biggr)^{p}x^{-1} $$
(1.15)
and the left-hand side of (1.13) is finite if and only if \(c=0\).

Remark 3

The special case \(p=2\) is studied in [3].

We prove the theorems in subsequent sections. The first step of the proof is the same as the standard one. We need the following identity:
$$ \int\frac{|f(x)|^{p}}{|x|^{p}}\,dx = -\frac{p}{n-p} \int\frac{|f(x)|^{p-2}f(x)}{|x|^{p-1}}\frac {x}{|x|}\cdot\nabla f(x)\,dx, $$
(1.16)
which holds for all \(f\in C_{0}^{\infty}(\mathbb {R}^{n})\), provided \(1< p< n\). It can be obtained expressing the integral on the left-hand side by means of the spherical coordinates and using the integration by parts (cf. [4], Proof of Theorem 1.1).
Equation (1.16) together with the Hölder inequality with \(1/p+1/p'=1\), \(1\leq p<\infty\), implies (1.3). In this sense, the standard method depends upon duality. In this paper, we adopt a different view. We rewrite (1.16) in the form
$$ \int|u|^{p}\,dx= \int|u|^{p-2}uv\,dx $$
(1.17)
with \(u=\frac{f}{|x|}\) and \(v=-\frac{p}{n-p}\frac{x}{|x|}\cdot \nabla f\) and modify (1.17) as
$$ \int \bigl( \vert u\vert ^{p}-\vert u\vert ^{p-2}uv\bigr)\,dx=0. $$
(1.18)
Now the equality (1.18) can be understood as representing a cancelation as well as an oscillation or an orthogonality. This point of view for equation (1.18) can be stated in the following way.

Lemma 1

Let \(L^{p}(\Omega,\mu)\) with \(1< p<\infty\) be the Banach space of pth integrable real-valued functions on a measure space \((\Omega,\mu)\) endowed with a norm \(\|\cdot\|_{p}\). Then the following three assertions are equivalent for any u, \(v\in L^{p}(\Omega,\mu)\):
  1. (1)
    We have
    $$ \Vert u\Vert _{p}^{p}= \int_{\Omega} \vert u\vert ^{p-2}uv\,d\mu. $$
    (1.19)
     
  2. (2)
    We have
    $$ \Vert u\Vert _{p}^{p}=\Vert v\Vert _{p}^{p}- \int_{\Omega} \bigl( \vert v\vert ^{p}+(p-1)\vert u \vert ^{p}-p\vert u\vert ^{p-2}uv \bigr)\,d\mu. $$
    (1.20)
     
  3. (3)
    We have
    $$ \Vert u\Vert _{p}^{p}=\Vert v\Vert _{p}^{p}-p \int_{\Omega}R_{p}(u,v) \vert u-v\vert ^{2} \,d\mu. $$
    (1.21)
     

Proof of Lemma 1

The assertions are trivial for \(u=v\). If \(u\neq v\), then the relation
$$\begin{aligned}& \Vert v\Vert _{p}^{p}-\Vert u\Vert _{p}^{p}+p \int_{\Omega}\bigl(\vert u\vert ^{p}-\vert u\vert ^{p-2}uv\bigr)\,d\mu \\& \quad= \int_{\Omega} \bigl(\vert v\vert ^{p}+(p-1)\vert u \vert ^{p}-p\vert u\vert ^{p-2}uv\bigr)\,d\mu = p \int_{\Omega}R_{p}(u,v)\vert u-v\vert ^{2} \,d\mu \end{aligned}$$
immediately yields the conclusion. □

The subsequent sections are organized as follows. Proposition 1 will be proved in Section 2. Section 3 is devoted to the verification of Theorem 1. The proof of Theorem 2 is given in Section 4. There is a large literature on Hardy type inequalities and related subjects. See [132] and the references therein for instance.

2 Proof of Proposition 1

First of all, we remark that \(R_{2}(\xi,\eta)=1/2\), by definition. This proves Part (5) as well as Parts (1), (2), and (4) for \(p=2\). By a direct calculation, (1.6) holds if \(\xi=\eta\). Let \(\xi\neq\eta\). We obtain
$$\begin{aligned}& \frac{1}{p}\vert \eta \vert ^{p}+\frac{1}{p'}\vert \xi \vert ^{p}-\vert \xi \vert ^{p-2}\xi\eta \\& \quad= \biggl(1-\frac{1}{p} \biggr) \bigl(\vert \xi \vert ^{p}-\vert \eta \vert ^{p}\bigr)-\eta\bigl(\vert \xi \vert ^{p-2}\xi -\vert \eta \vert ^{p-2} \eta\bigr) \\& \quad= (p-1) \int_{0}^{1}\bigl\vert \theta\xi+(1-\theta)\eta \bigr\vert ^{p-2}\bigl(\theta\xi +(1-\theta)\eta\bigr) \,d\theta(\xi- \eta) \\& \qquad {}-(p-1) \int_{0}^{1} \bigl\vert \theta\xi+(1-\theta)\eta \bigr\vert ^{p-2}\,d\theta\eta(\xi -\eta) \\& \quad= (p-1) \int_{0}^{1} \bigl\vert \theta\xi+(1-\theta)\eta \bigr\vert ^{p-2}\theta \,d\theta (\xi-\eta)^{2}, \end{aligned}$$
which yields (1.6). Then Part (2) follows immediately from Part (1). If \(p>2\) and \(R_{p}(\xi,\eta)=0\), then by the integral representation (1.6) we have \(\theta\xi+(1-\theta)\eta=0\) for any θ with \(0<\theta<1\). This implies \(\xi=\eta=0\). If \(p<2\) and \(R_{p}(\xi,\eta)=0\), then \(|\theta\xi+(1-\theta)\eta|=\infty\) for any θ with \(0<\theta<1\), which is absurd. This proves Proposition 1.

3 Proof of Theorem 1

By a standard density argument, it is enough to prove Theorem 1 for \(f\in C_{0}^{\infty}(\mathbb {R}^{n})\). Applying (1.16), (1.7) is then a direct consequence of Lemma 1 with \(u=\frac{f}{|x|}\) and \(v=-\frac{p}{n-p}\frac{x}{|x|}\cdot\nabla f\). Now suppose that the second term on the right hand side of (1.7) vanishes. Then by Parts (3) and (4) of Proposition 1, we easily see that f satisfies the equation
$$\frac{p}{n-p}\frac{x}{|x|}\cdot\nabla f+\frac{1}{|x|}f=0, $$
which is equivalent to
$$\frac{x}{|x|}\cdot\nabla \bigl(|x|^{\frac{n-p}{p}}f \bigr)=0. $$
This implies (1.8), which in turn implies the rest of the statements of the theorem.

4 Proof of Theorem 2

By integration by parts, we have
$$\int_{0}^{\infty}x^{-r-1}\biggl\vert \int_{0}^{x}f\biggr\vert ^{p}\,dx = \frac{p}{r} \int_{0}^{\infty}x^{-r}\biggl\vert \int_{0}^{x}f\biggr\vert ^{p-2} \int_{0}^{x}f\cdot f\,dx, $$
so that (1.10) follows from Lemma 1 by setting \(u=x^{-\frac{r+1}{p}}\int_{0}^{x}f\) and \(v=x^{1-\frac{r+1}{p}}f\). The rest of the statements of Part (1) follow if we notice that
$$\frac{p}{r}xf- \int_{0}^{x}f=\frac{p}{r}x^{1+\frac{p}{r}} \frac{d}{dx} \biggl(x^{-\frac{p}{r}} \int_{0}^{x} f \biggr). $$
Part (2) follows by the same argument.

5 Conclusions

In this paper, we examined the sharp remainder terms of the Hardy inequality for \(L^{p}\)-functions. From these sharp remainder terms, we can derive several consequences including the explicit form of the extremal function for the inequality which reveals the nature of the nonexistence of extremals in the \(L^{p}\)-setting. Our analysis only requires some elementary calculus with some insight in the structure of the remainder term and is also applicable to other critical type inequalities such as the Hardy inequalities in \(L^{n}\).

Declarations

Acknowledgements

The authors wish to extend their gratitude to the anonymous referees for valuable comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Graduate School of Science and Engineering, Ehime University, Ehime, Japan
(2)
Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Osaka, Japan
(3)
Department of Applied Physics, Waseda University, Tokyo, Japan

References

  1. Folland, GB: Real Analysis, 2nd edn. Pure and Applied Math. Wiley, New York (1999) MATHGoogle Scholar
  2. Reed, M, Simon, B: Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness. Academic Press, San Diego (1975) MATHGoogle Scholar
  3. Machihara, S, Ozawa, T, Wadade, H: On the Hardy type inequalities. Preprint (2015) Google Scholar
  4. Ioku, N, Ishiwata, M, Ozawa, T: Sharp remainder of a critical Hardy inequality. Arch. Math. (Basel) 106(1), 65-71 (2014) MathSciNetView ArticleMATHGoogle Scholar
  5. Adimurthi, A, Chaudhuri, N, Ramaswamy, M: An improved Hardy-Sobolev inequality and its application. Proc. Am. Math. Soc. 130, 489-505 (2002) MathSciNetView ArticleMATHGoogle Scholar
  6. Aldaz, JM: A stability version of Hölder’s inequality. J. Math. Anal. Appl. 343, 842-852 (2008) MathSciNetView ArticleMATHGoogle Scholar
  7. Barbatis, G, Filippas, S, Tertikas, A: Series expansion for \(L^{p}\) Hardy inequalities. Indiana Univ. Math. J. 52, 171-190 (2003) MathSciNetView ArticleMATHGoogle Scholar
  8. Bogdan, K, Dyda, B, Kim, P: Hardy inequalities and non-explosion results for semigroups, arXiv:1412.7717v2 [math.AP] 28 Dec. 2014
  9. Brezis, H, Marcus, M: Hardy’s inequalities revisited. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 25, 217-237 (1997) MathSciNetMATHGoogle Scholar
  10. Brezis, H, Vázquez, JL: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madr. 10, 443-469 (1997) MathSciNetMATHGoogle Scholar
  11. Cianchi, A, Ferone, A: Hardy inequalities with non-standard remainder terms. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 889-906 (2008) MathSciNetView ArticleMATHGoogle Scholar
  12. Cianchi, A, Ferone, A: Best remainder norms in Sobolev-Hardy inequalities. Indiana Univ. Math. J. 58, 1051-1096 (2009) MathSciNetView ArticleMATHGoogle Scholar
  13. Detalla, A, Horiuchi, T, Ando, H: Missing terms in Hardy-Sobolev inequalities and its application. Far East J. Math. Sci.: FJMS 14, 333-359 (2004) MathSciNetMATHGoogle Scholar
  14. Detalla, A, Horiuchi, T, Ando, H: Sharp remainder terms of Hardy-Sobolev inequalities. Math. J. Ibaraki Univ. 37, 39-52 (2005) MathSciNetView ArticleMATHGoogle Scholar
  15. Dolbeault, J, Volzone, B: Improved Poincaré inequalities. Nonlinear Anal. 75, 5985-6001 (2012) MathSciNetView ArticleMATHGoogle Scholar
  16. Edmunds, DE, Triebel, H: Sharp Sobolev embeddings and related Hardy inequalities: the critical case. Math. Nachr. 207, 79-92 (1999) MathSciNetView ArticleMATHGoogle Scholar
  17. Filippas, S, Tertikas, A: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186-233 (2002) MathSciNetView ArticleMATHGoogle Scholar
  18. Frank, RL, Seiringer, R: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255, 3407-3430 (2008) MathSciNetView ArticleMATHGoogle Scholar
  19. Fujiwara, K, Ozawa, T: Exact remainder formula for the Young inequality and applications. Int. J. Math. Anal. 7, 2723-2735 (2013) MathSciNetView ArticleGoogle Scholar
  20. Fujiwara, K, Ozawa, T: Stability of the Young and Hölder inequalities. J. Inequal. Appl. 2014, 162 (2014) MathSciNetView ArticleGoogle Scholar
  21. García Azorero, JP, Peral, I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 144, 441-476 (1998) MathSciNetView ArticleMATHGoogle Scholar
  22. Ghoussoub, N, Moradifam, A: Bessel pairs and optimal Hardy and Hardy-Rellich inequalities. Math. Ann. 349, 1-57 (2011) MathSciNetView ArticleMATHGoogle Scholar
  23. Ghoussoub, N, Moradifam, A: On the best possible remaining term in the Hardy inequality. Proc. Natl. Acad. Sci. USA 105, 13746-13751 (2008) MathSciNetView ArticleMATHGoogle Scholar
  24. Herbst, IW: Spectral theory of the operator \((p^{2}+m^{2})^{1/2}-Ze^{2}/r\). Commun. Math. Phys. 53, 285-294 (1977) MathSciNetView ArticleMATHGoogle Scholar
  25. Ioku, N, Ishiwata, M: A scale invariant form of a critical Hardy inequality. Int. Math. Res. Not. 2015(18), 8830-8846 (2015) MathSciNetView ArticleMATHGoogle Scholar
  26. Machihara, S, Ozawa, T, Wadade, H: Hardy type inequalities on balls. Tohoku Math. J. 65, 321-330 (2013) MathSciNetView ArticleMATHGoogle Scholar
  27. Machihara, S, Ozawa, T, Wadade, H: Generalizations of the logarithmic Hardy inequality in critical Sobolev-Lorentz spaces. J. Inequal. Appl. 2013, 381 (2013) MathSciNetView ArticleMATHGoogle Scholar
  28. Mitidieri, E: A simple approach to Hardy inequalities. Math. Notes 67, 479-486 (2000) MathSciNetView ArticleMATHGoogle Scholar
  29. Muckenhoupt, B: Hardy’s inequality with weights. Stud. Math. 44, 31-38 (1972) MathSciNetMATHGoogle Scholar
  30. Takahashi, F: A simple proof of Hardy’s inequality in a limiting case. Arch. Math. 104, 77-82 (2015) MathSciNetView ArticleMATHGoogle Scholar
  31. Triebel, H: Sharp Sobolev embeddings and related Hardy inequalities: the sub-critical case. Math. Nachr. 208, 167-178 (1999) MathSciNetView ArticleMATHGoogle Scholar
  32. Zhang, J: Extensions of Hardy inequality. J. Inequal. Appl. 2006, Article ID 69379 (2006) MathSciNetView ArticleMATHGoogle Scholar

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