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Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space
Journal of Inequalities and Applications volume 2015, Article number: 281 (2015)
Abstract
In this paper, we establish Hardy inequalities of logarithmic type involving singularities on spheres in \(\mathbb{R}^{n}\) in terms of the Sobolev-Lorentz-Zygmund spaces. We prove it by absorbing singularities of functions on the spheres by subtracting the corresponding limiting values.
1 Introduction and the main theorem
The classical Hardy inequalities in one dimension are stated as
and its dual inequality
where \(1< p<\infty\) and \(r>0\); see [1, 2] for instance. The constant \((\frac{p}{r})^{p}\) is best-possible in both inequalities (1.1) and (1.2). A higher dimensional variant of (1.1) and (1.2) is
for all \(f\in W^{1}_{p}(\mathbb{R}^{n})\), where \(n\geq2\) and \(1< p< n\), and the constant \(\frac{p}{n-p}\) in (1.3) is also optimal. For the critical case \(p=n\), the inequality (1.3) makes no sense, and instead the inequality
holds for all \(f\in W^{1}_{n}(\mathbb{R}^{n})\), where \(n\geq2\), \(B_{1}:=\{ x\in\mathbb{R}^{n} ; |x|<1\}\), and the constant C depends only on n (see [3] for instance). There are a number of both mathematical and physical applications of Hardy type inequalities. Among others, we refer the reader to [3–19].
In a recent paper [12], the authors established the logarithmic Hardy type inequality on the two dimensional ball \(B_{R}:=\{ x\in\mathbb{R}^{2} ; |x|< R\}\) with \(R>0\), by taking into account the behavior of \(W^{1}_{2}(B_{R})\) functions on the boundary \(\partial B_{R}=\{x\in\mathbb{R}^{2} ; |x|=R\}\). Indeed, the following inequality was proved.
Theorem
(Theorem 5 in [12])
Let \(n=2\) and \(R>0\). Then the inequality
holds for all \(f\in W^{1}_{2}(B_{R})\).
The purpose of this paper is to extend the inequality (1.4) to the higher dimensional cases \(n\geq1\) in terms of the Lorentz-Zygmund type spaces in \(\mathbb{R}^{n}\). To this end, we first recall the Lorentz-Zygmund spaces.
For \(n\in\mathbb{N}\) and \(1\leq p,q\leq\infty\), the Lorentz spaces are defined by
where
with the usual modification when \(q=\infty\). If a function f is non-negative, radially symmetric and non-increasing with respect to the radial direction, then the norm \(\|f\|_{L_{p,q}(\mathbb{R}^{n})}\) coincides with the Lorentz norm in terms of the rearrangement of f. In fact, it follows that
where \(f^{*}\) denotes the symmetric decreasing rearrangement of f, and \(\omega_{n}\) is the volume of the unit ball in \(\mathbb{R}^{n}\).
Furthermore, the Lorentz-Zygmund spaces on \(B_{R}\) with \(R>0\) are defined by
where \(\lambda\in\mathbb{R}\) and
We then define the Sobolev-Lorentz-Zygmund spaces by
endowed with the norm \(\|\cdot\|_{W^{1}L_{p,q,\lambda}(B_{R})}:=\|\cdot\| _{L_{p,q,\lambda}(B_{R})} +\|\nabla\cdot\|_{L_{p,q,\lambda}(B_{R})}\), and \(W^{1}_{0}L_{p,q,\lambda}(B_{R}):=\overline{C_{0}^{\infty}(B_{R})}^{\|\cdot \|_{W^{1}L_{p,q,\lambda}(B_{R})}}\). Note that the special case \(W^{1}L_{p,p,0}(B_{R})\) coincides with the classical Sobolev space \(W^{1}_{p}(B_{R})\). As a further generalization, the Lorentz-Zygmund spaces involving the double logarithmic weights can be introduced by
where \(\lambda_{1},\lambda_{2}\in\mathbb{R}\) and
The Sobolev-Lorentz-Zygmund spaces \(W^{1}L_{p,q,\lambda_{1},\lambda _{2}}(B_{R})\) and \(W^{1}_{0}L_{p,q,\lambda_{1},\lambda_{2}}(B_{R})\) are defined similarly to above.
We next introduce the Lorentz-Zygmund spaces in \(\mathbb{R}^{n}\) having the scaling properties. The Lorentz-Zygmund spaces \(L_{p,q,\lambda}(\mathbb{R}^{n})\) are defined by
where
Similarly, the spaces \(L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) are defined by
where
Remark
The space \(L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) extends the spaces \(L_{p,q,\lambda}(\mathbb{R}^{n})\) and \(L_{p,q}(\mathbb{R}^{n})\) in the sense that \(L_{p,q,\lambda,0}(\mathbb{R}^{n})=L_{p,q,\lambda }(\mathbb{R}^{n})\) and \(L_{p,q,0,0}(\mathbb{R}^{n})=L_{p,q}(\mathbb{R}^{n})\). Moreover, remark that the space \(L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb {R}^{n})\) has a scaling property in the sense that \(\|\delta_{l} f\|_{L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb {R}^{n})}=l^{\frac{n}{p}}\|f\|_{L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})}\), where \((\delta_{l}f)(x):=f(\frac{x}{l})\) for \(l>0\).
In addition, the Sobolev-Lorentz-Zygmund spaces \(W^{1}L_{p,q,\lambda _{1},\lambda_{2}}(\mathbb{R}^{n})\) are defined in the same manner as above. We refer to [20] for an enlightening exposition concerning these functional spaces.
Finally, in order to state the main theorems in this paper, we need to introduce the Lorentz-Zygmund type spaces \({\mathcal {L}}_{p,q,\lambda}(\mathbb{R}^{n})\) taking into account the behavior of functions on spheres defined by
where
Furthermore, we define the Lorentz-Zygmund type spaces \({\mathcal {L}}_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) by
where
Remark
The spaces \({\mathcal{L}}_{p,q,\lambda}(\mathbb{R}^{n})\) and \({\mathcal{L}}_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) have the same scaling property as in the space \(L_{p,q,\lambda _{1},\lambda_{2}}(\mathbb{R}^{n})\). Namely, it follows that \(\|\delta_{l} f\|_{{\mathcal{L}}_{p,q,\lambda }(\mathbb{R}^{n})}=l^{\frac{n}{p}}\|f\|_{{\mathcal{L}}_{p,q,\lambda }(\mathbb{R}^{n})}\) and \(\|\delta_{l} f\|_{{\mathcal{L}}_{p,q,\lambda_{1},\lambda _{2}}(\mathbb{R}^{n})}=l^{\frac{n}{p}}\|f\|_{{\mathcal{L}}_{p,q,\lambda _{1},\lambda_{2}}(\mathbb{R}^{n})}\), where \((\delta_{l}f)(x):=f(\frac{x}{l})\) for \(l>0\).
We are now in a position to state the main theorems.
Theorem 1.1
Let \(n\in\mathbb{N}\), \(1<\alpha<\infty\) and \(\max\{1,\alpha-1\} <\beta<\infty\). Then the continuous embedding
holds. In particular, for any \(R>0\), the inequality
holds for all \(f\in W^{1}L_{n,\beta,\frac{\beta-\alpha}{\beta }}(\mathbb{R}^{n})\), where the embedding constant \(\frac{\beta}{\alpha-1}\) in (1.5) is best-possible.
Remark
Denoting \(W^{1}L_{2,2,0}(\mathbb{R}^{2})=W^{1}_{2}(\mathbb{R}^{2})\) and restricting the functions in \(W^{1}_{2}(\mathbb{R}^{2})\) on \(B_{R}\), we see that the special case \(n=\alpha=\beta=2\) in (1.5) yields (1.4) obtained in [12].
Our next aim is to consider the limiting case \(\alpha=1\) in (1.5). However, the inequality (1.5) with \(\alpha=1\) makes no sense since the weight \(\vert \log\frac{1}{|x|}\vert ^{-1}|x|^{-n}\) is not locally integrable at the origin. To overcome this difficulty, we need the aid of a logarithmic weight to recover the corresponding double logarithmic Hardy type inequality. Our next theorem now reads as follows.
Theorem 1.2
Let \(n\in\mathbb{N}\), \(1<\alpha<\infty\) and \(\max\{1,\alpha-1\} <\beta<\infty\). Then the continuous embedding
holds. In particular, for any \(R>0\), the inequality
holds for all \(f\in W^{1}L_{n,\beta,\frac{\beta-1}{\beta},\frac {\beta-\alpha}{\beta}}(\mathbb{R}^{n})\), where the embedding constant \(\frac{\beta}{\alpha-1}\) in (1.6) is best-possible.
Remark
Remark that we do not need to subtract the boundary value of functions on \(|x|=eR\) in the integrand on the left-hand side of (1.6) in spite of the fact that the integrand on the right-hand side has singularities on \(|x|=R\), \(|x|=eR\), and \(|x|=e^{2}R\).
This paper is organized as follows. Section 2 is devoted to establishing the inequalities (1.5) in Theorem 1.1 and (1.6) in Theorem 1.2. We shall prove the optimality of the embedding constants in the two inequalities (1.5) and (1.6) in Section 3.
2 Proof of inequalities (1.5) and (1.6)
In this section, we shall prove inequalities (1.5) and (1.6).
Proof of (1.5) in Theorem 1.1
We first prove (1.5) for \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\). We introduce polar coordinates \((r,\omega)=(|x|,\frac{x}{|x|})\in (0,\infty)\times S^{n-1}\) and the Lebesgue measure σ on the unit sphere \(S^{n-1}\). We write the integral on the left-hand side of (1.5) restricted on \(B_{R}\) in polar coordinates and then by integration by parts to obtain
where the boundary value at \(r=R\) vanishes since
and
for \(0< r\leq R\), and \(\beta-\alpha+1>0\) by assumption. By the Hölder inequality, we have
This implies
In the same manner as above, we have
Thus combining (2.1) with (2.2), we obtain (1.5) for \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\).
Now we prove (1.5) for \(f\in W^{1}L_{n,\beta,\frac{\beta-\alpha }{\beta}}(\mathbb{R}^{n})\). For \(f\in W^{1}L_{n,\beta,\frac{\beta-\alpha}{\beta}}(\mathbb {R}^{n})\), we choose a sequence \(\{f_{j}\}_{j\in\mathbb{N}}\subset C_{0}^{\infty}(\mathbb{R}^{n})\) such that \(f_{j}\to f\) in \(W^{1}L_{n,\beta,\frac{\beta-\alpha}{\beta }}(\mathbb{R}^{n})\) as \(j\to\infty\) and almost everywhere by density. Since the inequality (1.5) holds for \(f_{j}-f_{k}\in C_{0}^{\infty}(\mathbb{R}^{n})\), we see that \(\{(f_{j})_{R}^{\#}\}_{j\in\mathbb{N}}\) is a Cauchy sequence in \(L_{\beta}(\mathbb{R}^{n} ; \frac{dx}{|x|^{n}})\), where we define
for \(f\in L_{1,\mathrm{loc}}(\mathbb{R}^{n})\). Then there exists a function \(g_{R}\in L_{\beta}(\mathbb{R}^{n} ; \frac{dx}{|x|^{n}})\) such that \((f_{j})_{R}^{\#}\to g_{R}\) in \(L^{\beta}(\mathbb{R}^{n} ; \frac {dx}{|x|^{n}})\) as \(j\to\infty\). The inclusion relationship
implies that \(f_{j}(R\frac{x}{|x|})\to f(R\frac{x}{|x|})\) almost everywhere, and that \(f_{R}^{\#}=g_{R}\). Therefore, the inequality (1.5) holds for all \(f\in W^{1}L_{n,\beta,\frac{\beta-\alpha}{\beta}}(\mathbb{R}^{n})\). □
In order to prove (1.6) in Theorem 1.2, we first show the following proposition.
Proposition 2.1
Let \(n\in\mathbb{N}\), \(1<\alpha<\infty\) and \(\max\{1,\alpha-1\} <\beta<\infty\). Then, for any \(R>0\), the inequality
holds for all \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\).
Proof
We first consider the integrals in (2.3) restricted on \(B_{R}\). Using polar coordinates and integration by parts, we see
where the boundary value at \(r=R\) vanishes since
and
for \(0< r\leq R\), and \(\beta-\alpha+1>0\) by the assumption. By the Hölder inequality, we have
which implies
Next, we consider the integrals in (2.3) restricted on \(B_{eR}\setminus B_{R}\). Using polar coordinates and integration by parts, we see
where the boundary value at \(r=R\) vanishes since
and
for \(R\leq r< eR\), and \(\beta-\alpha+1>0\) by the assumption. By the Hölder inequality, we have
which implies
Thus combining (2.4) with (2.5), we obtain (2.3). □
We can prove a dual inequality of (2.3) in a similar way as in Proposition 2.1 stated as follows.
Proposition 2.2
Let \(n\in\mathbb{N}\), \(1<\alpha<\infty\) and \(\max\{1,\alpha-1\} <\beta<\infty\). Then, for any \(R>0\), the inequality
holds for all \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\).
We shall show (1.6) in Theorem 1.2 by combining Proposition 2.1 with Proposition 2.2.
Proof of Theorem 1.2
By considering a density argument as used in the proof of Theorem 1.1, it suffices to prove (1.6) for \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\). Applying Proposition 2.2 with R replaced by eR, we obtain
Thus from (2.3) and (2.6), we obtain (1.6) for \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\), and then for \(f\in W^{1}L_{n,\beta,\frac{\beta-1}{\beta},\frac{\beta -\alpha}{\beta}}(\mathbb{R}^{n})\). □
3 Optimality of the embedding constant
In this section, we shall prove that the embedding constant \(\frac {\beta}{\alpha-1}\) is best-possible in the inequalities (1.5) in Theorem 1.1 and (1.6) in Theorem 1.2.
First, we consider the optimality of \(\frac{\beta}{\alpha-1}\) in (1.5). As a direct consequence of (1.5), we obtain
for all \(f\in W^{1}_{0}L_{n,\beta,\frac{\beta-\alpha}{\beta}}(B_{R})\). Therefore, it suffices to prove the optimality of \(\frac{\beta}{\alpha-1}\) in (3.1). Define a sequence of functions \(\{f_{m}\}\) for large \(m\in\mathbb{N}\) by
We can easily check \(f_{m}\in W^{1}_{0}L_{n,\beta,\frac{\beta-\alpha }{\beta}}(B_{R})\). More precisely, we calculate the norm \(\|f_{m}\|_{W^{1}L_{n,\beta,\frac {\beta-\alpha}{\beta}}(B_{R})}\) below. Letting \(\tilde{f}_{m}(r):=f_{m}(x)\) with \(r=|x|\geq0\), we have
Thus a direct calculation yields
where note that the last integral in (3.2) is finite by the assumption \(\beta-\alpha>-1\). On the other hand, we see
Here, by applying the inequality \(\log\frac{R}{r}\geq\frac{R-r}{R}\) for all \(r\leq R\), we can estimate \(C_{R,\alpha,\beta}\) as follows:
where we have used \(\beta-\alpha+1>0\) by the assumption. Summing up (3.2) and (3.3), we obtain
as \(m\to\infty\), which implies that the constant \(\frac{\beta }{\alpha-1}\) in (3.1) is best-possible.
We next consider the optimality of \(\frac{\beta}{\alpha-1}\) in (1.6) in Theorem 1.2. As a direct consequence of (1.6), we obtain
for all \(f\in W^{1}_{0}L_{n,\beta,\frac{\beta-1}{\beta},\frac{\beta -\alpha}{\beta}}(B_{R})\). In order to prove that the constant \(\frac{\beta}{\alpha-1}\) in (3.4) is best-possible, we take a sequence of functions \(\{f_{m}\}\) for large \(m\in\mathbb{N}\) defined by
Then a direct calculation yields
where
Note that the assumption \(\beta-\alpha>-1\) implies \(C_{\alpha,\beta }<+\infty\). Furthermore, we see
where
Utilizing the elementary inequality \(\log (\log\frac {eR}{r} )\geq\frac{R-r}{R}\) for all \(r\leq R\) and the assumption \(\beta-\alpha+1>0\), we easily see that \(C_{R,\alpha,\beta}<+\infty\). Hence, from (3.5) and (3.6), we obtain
as \(m\to\infty\), which implies that the constant \(\frac{\beta }{\alpha-1}\) in (3.4) is best-possible.
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Acknowledgements
This paper has been completely written by SM, TO, and HW without any other person’s substantial contributions. In addition, we have not received any funding for making out a draft of this paper. The authors would like to express their heartfelt thanks to the referee for his/her valuable comments.
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SM and TO gave critical inspiration for the establishment of the Hardy type inequality in this paper. HW proved it rigorously and made the draft. All authors read and approved the final manuscript.
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Machihara, S., Ozawa, T. & Wadade, H. Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space. J Inequal Appl 2015, 281 (2015). https://doi.org/10.1186/s13660-015-0806-1
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DOI: https://doi.org/10.1186/s13660-015-0806-1