1 Introduction and the main theorem
The classical Hardy inequalities in one dimension are stated as
$$ \int_{0}^{\infty}x^{-r-1}\biggl\vert \int_{0}^{x}f(y)\,dy\biggr\vert ^{p}\,dx \leq \biggl(\frac{p}{r} \biggr)^{p}\int _{0}^{\infty}x^{p-r-1}\bigl\vert f(x)\bigr\vert ^{p}\,dx $$
(1.1)
and its dual inequality
$$ \int_{0}^{\infty}x^{r-1}\biggl\vert \int_{x}^{\infty}f(y)\,dy\biggr\vert ^{p}\,dx \leq \biggl(\frac{p}{r} \biggr)^{p}\int _{0}^{\infty}x^{p+r-1}\bigl\vert f(x)\bigr\vert ^{p}\,dx, $$
(1.2)
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
$$ \biggl\Vert \frac{f}{|x|}\biggr\Vert _{L_{p}(\mathbb{R}^{n})}\leq \frac{p}{n-p} \|\nabla f\|_{L_{p}(\mathbb{R}^{n})} $$
(1.3)
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
$$ L_{p,q}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ f\in L_{1,\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr) ; \|f\|_{L_{p,q}(\mathbb{R}^{n})}< + \infty \bigr\} , $$
where
$$ \|f\|_{L_{p,q}(\mathbb{R}^{n})}:= \biggl(\int_{\mathbb{R}^{n}} \bigl(|x|^{\frac{n}{p}}\bigl\vert f(x)\bigr\vert \bigr)^{q} \frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}} $$
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
$$ \biggl(\int_{\mathbb{R}^{n}} \bigl(|x|^{\frac{n}{p}}\bigl\vert f(x) \bigr\vert \bigr)^{q}\frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}= \omega_{n}^{\frac {1}{q}-\frac{1}{p}} \biggl(\int_{0}^{\infty}\bigl(t^{\frac {1}{p}}f^{*}(t) \bigr)^{q}\frac{dt}{t} \biggr)^{\frac{1}{q}}, $$
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
$$ L_{p,q,\lambda}(B_{R}):= \bigl\{ f\in L_{1,\mathrm{loc}}(B_{R}) ; \|f\|_{L_{p,q,\lambda}(B_{R})}< +\infty \bigr\} , $$
where
\(\lambda\in\mathbb{R}\) and
$$ \|f\|_{L_{p,q,\lambda}(B_{R})}:= \biggl( \int_{B_{R}} \biggl( |x|^{\frac{n}{p}}\biggl\vert \log\frac{R}{|x|}\biggr\vert ^{\lambda}\bigl\vert f(x)\bigr\vert \biggr)^{q} \frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. $$
We then define the Sobolev-Lorentz-Zygmund spaces by
$$ W^{1}L_{p,q,\lambda}(B_{R}):= \bigl\{ f\in L_{p,q,\lambda}(B_{R}) ; \nabla f\in L_{p,q,\lambda}(B_{R}) \bigr\} $$
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
$$ L_{p,q,\lambda_{1},\lambda_{2}}(B_{R}):= \bigl\{ f\in L_{1,\mathrm{loc}}(B_{R}) ; \|f\|_{L_{p,q,\lambda_{1},\lambda _{2}}(B_{R})}< +\infty \bigr\} , $$
where
\(\lambda_{1},\lambda_{2}\in\mathbb{R}\) and
$$ \|f\|_{L_{p,q,\lambda_{1},\lambda_{2}}(B_{R})}:= \biggl( \int_{B_{R}} \biggl( |x|^{\frac{n}{p}}\biggl\vert \log\frac{eR}{|x|}\biggr\vert ^{\lambda_{1}} \biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\lambda_{2}}\bigl\vert f(x)\bigr\vert \biggr)^{q}\frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. $$
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
$$ L_{p,q,\lambda}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ f\in L_{1,\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr) ; \|f\|_{L_{p,q,\lambda}(\mathbb {R}^{n})}< + \infty \bigr\} , $$
where
$$ \|f\|_{L_{p,q,\lambda}(\mathbb{R}^{n})}:=\sup_{R>0} \biggl(\int _{\mathbb{R}^{n}} \biggl(|x|^{\frac{n}{p}}\biggl\vert \log \frac {R}{|x|}\biggr\vert ^{\lambda}\bigl\vert f(x)\bigr\vert \biggr)^{q}\frac {dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. $$
Similarly, the spaces
\(L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) are defined by
$$ L_{p,q,\lambda_{1},\lambda_{2}}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ f\in L_{1,\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr) ; \|f\|_{L_{p,q,\lambda_{1},\lambda _{2}}(\mathbb{R}^{n})}< + \infty \bigr\} , $$
where
$$ \|f\|_{L_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})}:= \sup_{R>0} \biggl(\int _{\mathbb{R}^{n}} \biggl(|x|^{\frac{n}{p}} \biggl\vert \log \frac{R}{|x|}\biggr\vert ^{\lambda_{1}} \biggl\vert \log\biggl\vert \log\frac{R}{|x|}\biggr\vert \biggr\vert ^{\lambda _{2}}\bigl\vert f(x) \bigr\vert \biggr)^{q}\frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. $$
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
$$ {\mathcal{L}}_{p,q,\lambda}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ f\in L_{1,\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr) ; \|f\|_{{\mathcal{L}}_{p,q,\lambda }(\mathbb{R}^{n})}< + \infty \bigr\} , $$
where
$$ \|f\|_{{\mathcal{L}}_{p,q,\lambda}(\mathbb{R}^{n})}:=\sup_{R>0} \biggl(\int _{\mathbb{R}^{n}} \biggl(|x|^{\frac{n}{p}}\biggl\vert \log \frac {R}{|x|}\biggr\vert ^{\lambda}\biggl\vert f(x)-f\biggl(R \frac{x}{|x|}\biggr)\biggr\vert \biggr)^{q}\frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. $$
Furthermore, we define the Lorentz-Zygmund type spaces
\({\mathcal {L}}_{p,q,\lambda_{1},\lambda_{2}}(\mathbb{R}^{n})\) by
$$ {\mathcal{L}}_{p,q,\lambda_{1},\lambda_{2}}\bigl(\mathbb{R}^{n}\bigr):= \bigl\{ f\in L_{1,\mathrm{loc}}\bigl(\mathbb{R}^{n}\bigr) ; \|f\|_{{\mathcal{L}}_{p,q,\lambda _{1},\lambda_{2}}(\mathbb{R}^{n})}< + \infty \bigr\} , $$
where
$$\begin{aligned} \|f\|_{{\mathcal{L}}_{p,q,\lambda_{1},\lambda_{2}}(\mathbb {R}^{n})} :=&\sup_{R>0} \biggl(\int _{\mathbb{R}^{n}} \biggl(|x|^{\frac {n}{p}}\biggl\vert \log \frac{eR}{|x|}\biggr\vert ^{\lambda_{1}} \biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\lambda _{2}} \biggl( \chi_{B_{eR}}(x)\biggl\vert f(x)-f\biggl(R\frac{x}{|x|}\biggr)\biggr\vert \\ &{}+\chi_{B_{eR}^{c}}(x)\biggl\vert f(x)-f\biggl(e^{2}R \frac{x}{|x|}\biggr)\biggr\vert \biggr) \biggr)^{q} \frac{dx}{|x|^{n}} \biggr)^{\frac{1}{q}}. \end{aligned}$$
We are now in a position to state the main theorems.
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.
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.
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
$$ \biggl(\int_{B_{R}}\frac{|f(x)|^{\beta}}{ \vert \log\frac {R}{|x|}\vert ^{\alpha}} \frac{dx}{|x|^{n}} \biggr)^{\frac{1}{\beta}} \leq\frac{\beta}{\alpha-1} \biggl( \int _{B_{R}}|x|^{\beta-n}\biggl\vert \log\frac{R}{|x|} \biggr\vert ^{\beta -\alpha}\bigl\vert \nabla f(x)\bigr\vert ^{\beta}\,dx \biggr)^{\frac{1}{\beta}} $$
(3.1)
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
$$ f_{m}(x):= \textstyle\begin{cases} (\log(mR) )^{\frac{\alpha-1}{\beta}} &\text{when }|x|\leq\frac{1}{m}, \\ (\log\frac{R}{|x|} )^{\frac{\alpha-1}{\beta}}& \text{when }\frac{1}{m}\leq|x|\leq\frac{R}{2}, \\ (\log2)^{\frac{\alpha-1}{\beta}}\frac{2}{R}(R-|x|) &\text{when }\frac{R}{2}\leq|x|\leq R. \end{cases} $$
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
$$ \tilde{f}_{m}'(r)= \textstyle\begin{cases} 0 &\text{when }r< \frac{1}{m}, \\ -\frac{\alpha-1}{\beta}r^{-1} (\log\frac{R}{r} )^{\frac{\alpha-1}{\beta}-1}& \text{when }\frac{1}{m}< r< \frac {R}{2}, \\ -(\log2)^{\frac{\alpha-1}{\beta}}\frac{2}{R}& \text{when }\frac {R}{2}< r< R. \end{cases} $$
Thus a direct calculation yields
$$\begin{aligned}& \int_{B_{R}}|x|^{\beta-n}\biggl\vert \log \frac{R}{|x|} \biggr\vert ^{\beta-\alpha}\bigl\vert \nabla f_{m}(x)\bigr\vert ^{\beta}\,dx \\& \quad = n \omega_{n}\int _{0}^{R}\biggl\vert \log\frac{R}{r}\biggr\vert ^{\beta-\alpha }\bigl\vert \tilde{f}_{m}'(r) \bigr\vert ^{\beta}r^{\beta-1}\,dr \\& \quad = n \omega_{n} \biggl(\frac{\alpha-1}{\beta} \biggr)^{\beta}\int _{\frac{1}{m}}^{\frac{R}{2}} \biggl(\log\frac{R}{r} \biggr)^{-1}r^{-1}\,dr \\& \qquad {}+n \omega_{n}( \log2)^{\alpha-1} \biggl(\frac{2}{R} \biggr)^{\beta}\int _{\frac{R}{2}}^{R} \biggl(\log\frac{R}{r} \biggr)^{\beta-\alpha }r^{\beta-1}\,dr \\& \quad = n \omega_{n} \biggl(\frac{\alpha-1}{\beta} \biggr)^{\beta}\bigl( \log \bigl(\log(mR) \bigr)-\log(\log2) \bigr) \\& \qquad {}+n \omega_{n}2^{\beta}( \log2)^{\alpha-1}\int_{0}^{\log 2}s^{\beta-\alpha}e^{-\beta s} \, ds \\& \quad =: n \omega_{n} \biggl(\frac{\alpha-1}{\beta} \biggr)^{\beta}\bigl( \log \bigl(\log(mR) \bigr)-\log(\log2) \bigr)+n \omega_{n} C_{\alpha,\beta}, \end{aligned}$$
(3.2)
where note that the last integral in (
3.2) is finite by the assumption
\(\beta-\alpha>-1\). On the other hand, we see
$$\begin{aligned}& \int_{B_{R}}\frac{|f_{m}(x)|^{\beta}}{ \vert \log\frac {R}{|x|}\vert ^{\alpha}}\frac{dx}{|x|^{n}} \\& \quad = n \omega_{n}\int_{0}^{R}\frac{|\tilde{f}_{m}(r)|^{\beta}}{ \vert \log\frac {R}{r}\vert ^{\alpha}} \frac{dr}{r} \\& \quad = n \omega_{n} \bigl(\log(mR) \bigr)^{\alpha-1}\int _{0}^{\frac {1}{m}} \biggl(\log\frac{R}{r} \biggr)^{-\alpha}r^{-1}\,dr \\& \qquad {}+n \omega_{n}\int_{\frac{1}{m}}^{\frac{R}{2}} \biggl( \log \frac{R}{r} \biggr)^{-1}r^{-1}\,dr \\& \qquad {}+n \omega_{n}(\log2)^{\alpha-1} \biggl(\frac{2}{R} \biggr)^{\beta}\int_{\frac{R}{2}}^{R}(R-r)^{\beta}\biggl(\log\frac{R}{r} \biggr)^{-\alpha }r^{-1}\,dr \\& \quad =: \frac{n \omega_{n}}{\alpha-1}+n \omega_{n} \bigl(\log \bigl(\log (mR) \bigr)- \log(\log2) \bigr)+n \omega_{n} C_{R,\alpha,\beta}. \end{aligned}$$
(3.3)
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:
$$\begin{aligned} C_{R,\alpha,\beta} \leq&2^{\beta+1}(\log2)^{\alpha-1}R^{-\beta +\alpha-1}\int _{\frac{R}{2}}^{R}(R-r)^{\beta-\alpha}\,dr \\ =&2^{\beta+1}(\log2)^{\alpha-1}R^{-\beta+\alpha-1}\int _{0}^{\frac {R}{2}}s^{\beta-\alpha}\, ds=\frac{2^{\alpha}(\log2)^{\alpha-1}}{\beta -\alpha+1}, \end{aligned}$$
where we have used
\(\beta-\alpha+1>0\) by the assumption. Summing up (
3.2) and (
3.3), we obtain
$$\begin{aligned}& \int_{B_{R}}|x|^{\beta-n}\biggl\vert \log \frac{eR}{|x|}\biggr\vert ^{\beta -1}\biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\beta -\alpha}\bigl\vert \nabla f_{m}(x)\bigr\vert ^{\beta}\,dx \\& \quad {} \times \biggl( \int_{B_{R}}\frac{|f_{m}(x)|^{\beta}}{ \vert \log \vert \log\frac{eR}{|x|}\vert \vert ^{\alpha} \vert \log\frac{eR}{|x|}\vert } \frac{dx}{|x|^{n}} \biggr)^{-1}\to \biggl(\frac{\alpha-1}{\beta} \biggr)^{\beta} \end{aligned}$$
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
$$\begin{aligned}& \biggl(\int_{B_{R}}\frac{|f(x)|^{\beta}}{ \vert \log \vert \log\frac {eR}{|x|}\vert \vert ^{\alpha} \vert \log\frac{eR}{|x|}\vert } \frac{dx}{|x|^{n}} \biggr)^{\frac{1}{\beta}} \\& \quad \leq\frac{\beta}{\alpha-1} \biggl( \int_{B_{R}}|x|^{\beta-n} \biggl\vert \log\frac{eR}{|x|}\biggr\vert ^{\beta -1}\biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\beta-\alpha}\bigl\vert \nabla f(x)\bigr\vert ^{\beta}\,dx \biggr)^{\frac{1}{\beta}} \end{aligned}$$
(3.4)
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
$$ f_{m}(x):= \textstyle\begin{cases} (\log (\log(meR) ) )^{\frac{\alpha-1}{\beta }} &\text{when }|x|\leq\frac{1}{m}, \\ (\log (\log\frac{eR}{|x|} ) )^{\frac{\alpha -1}{\beta}} &\text{when }\frac{1}{m}\leq|x|\leq\frac{R}{2}, \\ (\log (\log(2e) ) )^{\frac{\alpha-1}{\beta }}\frac{2}{R}(R-|x|)& \text{when }\frac{R}{2}\leq|x|\leq R. \end{cases} $$
Then a direct calculation yields
$$\begin{aligned}& \int_{B_{R}}|x|^{\beta-n}\biggl\vert \log \frac{eR}{|x|} \biggr\vert ^{\beta-1}\biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\beta-\alpha}\bigl\vert \nabla f_{m}(x)\bigr\vert ^{\beta}\,dx \\& \quad =n \omega_{n} \biggl(\frac{\alpha-1}{\beta } \biggr)^{\beta}\bigl( \log \bigl(\log \bigl(\log(meR) \bigr) \bigr)-\log \bigl(\log \bigl( \log(2e) \bigr) \bigr) \bigr)+n \omega_{n} C_{\alpha,\beta}, \end{aligned}$$
(3.5)
where
$$ C_{\alpha,\beta}:= (2e)^{\beta} \bigl(\log \bigl(\log(2e) \bigr) \bigr)^{\alpha-1} \int_{0}^{\log (\log(2e) )}s^{\beta-\alpha}e^{\beta(s-e^{s})} \, ds. $$
Note that the assumption
\(\beta-\alpha>-1\) implies
\(C_{\alpha,\beta }<+\infty\). Furthermore, we see
$$\begin{aligned}& \int_{B_{R}}\frac{|f_{m}(x)|^{\beta}}{ \vert \log \vert \log \frac{eR}{|x|}\vert \vert ^{\alpha} \vert \log\frac {eR}{|x|}\vert }\frac{dx}{|x|^{n}} \\& \quad =\frac{n \omega_{n}}{\alpha-1}+n \omega _{n} \bigl( \log \bigl(\log \bigl( \log(meR) \bigr) \bigr) -\log \bigl(\log \bigl(\log(2e) \bigr) \bigr) \bigr) +n \omega_{n} C_{R,\alpha,\beta}, \end{aligned}$$
(3.6)
where
$$ C_{R,\alpha,\beta} := \bigl(\log \bigl(\log(2e) \bigr) \bigr)^{\alpha-1} \biggl(\frac {2}{R} \biggr)^{\beta}\int_{\frac{R}{2}}^{R}(R-r)^{\beta}\biggl(\log \biggl(\log\frac {eR}{r} \biggr) \biggr)^{-\alpha} \biggl( \log\frac{eR}{r} \biggr)^{-1}r^{-1}\,dr. $$
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
$$\begin{aligned}& \int_{B_{R}}|x|^{\beta-n}\biggl\vert \log \frac{eR}{|x|}\biggr\vert ^{\beta -1}\biggl\vert \log\biggl\vert \log\frac{eR}{|x|}\biggr\vert \biggr\vert ^{\beta -\alpha}\bigl\vert \nabla f_{m}(x)\bigr\vert ^{\beta}\,dx \\& \quad {}\times \biggl( \int_{B_{R}}\frac{|f_{m}(x)|^{\beta}}{ \vert \log \vert \log\frac {eR}{|x|}\vert \vert ^{\alpha} \vert \log\frac{eR}{|x|} \vert } \frac{dx}{|x|^{n}} \biggr)^{-1}\to \biggl(\frac{\alpha-1}{\beta} \biggr)^{\beta} \end{aligned}$$
as
\(m\to\infty\), which implies that the constant
\(\frac{\beta }{\alpha-1}\) in (
3.4) is best-possible.
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.