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.
Hinweise
Competing interests
We declare that none of the authors have any competing interests in the manuscript.
Authors’ contributions
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.
1 Introduction and the main theorem
The classical Hardy inequalities in one dimension are stated as
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
$$ \biggl\Vert \frac{f}{|x|(1+|\log|x||)} \biggr\Vert _{L_{n}(B_{1})}\leq C \|f \|_{W^{1}_{n}(\mathbb{R}^{n})} $$
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.
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
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
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
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
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
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 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 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.
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
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
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
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
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.
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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Competing interests
We declare that none of the authors have any competing interests in the manuscript.
Authors’ contributions
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.