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\title[Optimal weak estimates for Riesz potentials]{Optimal weak estimates for Riesz potentials}

\author{\firstname{Liang} \lastname{Huang}}
\address{School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China}
\email{huangliang10@163.com}

\author{\firstname{Hanli} \lastname{Tang}\IsCorresp}
\address{Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China}
\email{hltang@bnu.edu.cn}

\keywords{\kwd{Riesz potentials}
\kwd{sharp constant}
\kwd{optimal estimate}}
\thanks{This project is supported by National Key Research and Development Program of China (Grant No.~2020YFA0712900) and the Natural Science Basic Research in Shaanxi Province of China (Grant No.~2022JQ-055, 2023-JC-QN-0056)}
\CDRGrant[National Key Research and Development Program of China]{2020YFA0712900}
\CDRGrant[Natural Science Basic Research in Shaanxi Province of China]{2022JQ-055}
\CDRGrant[Natural Science Basic Research in Shaanxi Province of China]{2023-JC-QN-0056}

\subjclass[2020]{42B20}

\begin{abstract}
In this note we prove a sharp reverse weak estimate for Riesz potentials
\[
\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\geq \gamma_sv_{n}^{\frac{n-s}{n}}\|f\|_{L^1}~\text{for}~0<f\in {L^1(\mathbb{R}^n)},
\]
where $\gamma_s=2^{-s}\pi^{-\frac{n}{2}}\frac{\Gamma(\frac{n-s}{2})}{\Gamma(\frac{s}{2})}$. We also consider the behavior of the best constant $\mathcal{C}_{n,s}$ of weak type estimate for Riesz potentials, and we prove $\mathcal{C}_{n,s}=O(\frac{\gamma_s}{s})$ as $s\rightarrow 0$.
\end{abstract}

\begin{document}
\maketitle

\section{introduction}

The Riesz potentials(fractional integral operators) $I_{s}$, which play an important part in Analysis, are defined by
\[
I_{s}(f)(x)=\gamma_s\int_{\mathbb{R}^n}\frac{f(x-y)}{|y|^{n-s}}\dy,
\]
where $0<s<n$ and $\gamma_s=2^{-s}\pi^{-\frac{n}{2}}\frac{\Gamma(\frac{n-s}{2})}{\Gamma(\frac{s}{2})}$. Such operators were first systematically investigated by M.Riesz~\cite{R}. The $(L^p,L^q)$-boundedness of Riesz potentials were proved by G.Hardy and J.~Littlewood~\cite{HL} when $n=1$ and by S.~Sobolev~\cite{S} when $n>1$. The $(L^1,L^{\frac{n}{n-s},\infty})$ -boundedness were obtained by A.~Zygmund~\cite{Z}. More precisely, they established the following theorem.

%\textbf{Theorem~A. }
\begin{theo}\label{thmA}
Let $0<s<n$ and let $p,q$ satisfy $1\leq p<q<\infty$ and $\frac{1}{p}-\frac{1}{q}=\frac{s}{n}$, then when $p>1$,
\[
\|I_{s}(f)\|_{L^q(\mathbb{R}^n)}\leq{C(n,p,s)}\|f\|_{L^{p}(\mathbb{R}^n)}.
\]
And when $p=1$,
\[
\|I_s(f)\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}=\sup_{\lambda>0}\lambda|\{x\in{\mathbb{R}^n}:|I_{s}f|>\lambda\}|^{\frac{n-s}{n}}\leq{C(n,s)\|f\|_{L^{1}(\mathbb{R}^n)}}.
\]
\end{theo}

The best constant in the $(L^p,L^q)$ inequality when $p=\frac{2n}{n+s}$, $q=\frac{2n}{n-s}$ was precisely calculated by E.~Lieb~\cite{L} (see also~\cite{FL}), and E.~Lieb and M.~Loss also offered an upper bound of $C(n,p,s)$(see~\cite[Chapter~4]{LL}).

Although the best constant of $(L^p,L^q)$ estimate for Riesz potentials has been studied for decades, to the best of the authors' knowledge there is no result about the best constant of $(L^1,L^{\frac{n}{n-s},\infty})$ estimate for Riesz potentials. In this paper, we will provide some estimates for the best constant of the weak type inequality.

In~\cite{T} (see multilinear case in~\cite{TW}), the second author setted up the following limiting weak-type behavior for Riesz potentials,
\begin{align*}
\lim_{\lambda\rightarrow{0}}\lambda|\{x\in{\mathbb{R}^n}:|I_{s}f|>\lambda\}|^{\frac{n-s}{n}}=\gamma_s v_{n}^{\frac{n-s}{n}}\|f\|_{L^{1}(\mathbb{R}^n)} \quad\text{for }~0<f\in L^{1}(\mathbb{R}^n),
\end{align*}
which implies a reverse weak estimate
\begin{align}\label{sharp reverse inequality}
\|I_s(f)\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}\geq \gamma_s v_{n}^{\frac{n-s}{n}}\|f\|_{L^{1}(\mathbb{R}^n)}\quad\text{for }~0<f\in L^{1}(\mathbb{R}^n),
\end{align}
where $v_n$ is the volume of the unit ball in $\mathbb{R}^n$. So a natural question that arises here is whether the constant $\gamma_s v_{n}^{\frac{n-s}{n}}$ is sharp? In the paper, we will give an affirmative answer.

Let $\mathcal{C}_{n,s}$ be the best constant such that the $(L^1,L^{\frac{n}{n-s},\infty})$ estimate holds for Riesz potentials,
i.e.
\[
\mathcal{C}_{n,s}=\sup_{f\in{L^{1}(\mathbb{R}^n)}}\frac{\|I_s(f)\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}}{\|f\|_{L^{1}(\mathbb{R}^n)}}.
\]
Then from~\eqref{sharp reverse inequality}, one can directly obtain a lower bound for $\mathcal{C}_{n,s}$,
\[
\mathcal{C}_{n,s}\geq \gamma_s v_{n}^{\frac{n-s}{n}}.
\]

Our another goal in this paper is to provide upper and lower bounds of $\mathcal{C}_{n,s}$ and to study the behavior of $\mathcal{C}_{n,s}$ as $s\rightarrow 0$. Our approach depends on the weak $L^{\frac{n}{n-s}}$ norm $\interleave \cdot \interleave_{L^{\frac{n}{n-s},\infty}}$ which is defined by
\[
\interleave f \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}=\sup_{0<|E|<\infty}|E|^{-\frac{1}{r}+\frac{n-s}{n}}\left(\int_E|f|^r\dx\right)^{\frac{1}{r}},\quad 0<r<\frac{n}{n-s}.
\]
The norm $\interleave \cdot \interleave_{L^{\frac{n}{n-s},\infty}}$ is equivalent to $\|\cdot\|_{L^{\frac{n}{n-s},\infty}}$. In fact there holds(see Exercise~1.1.12 in~\cite{G})
\begin{align}\label{equa of weak norms}
\|f\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}\leq \interleave f \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}\leq (\frac{n}{n-r(n-s)})^{\frac{1}{r}}\|f\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}.
\end{align}

Closely related to the Riesz potentials is the centered fractional maximal function, which is defined by
\[
M_{s}f(x)=\sup_{r>0}\frac{1}{|B(x,r)|^{1-\frac{s}{n}}}\int_{B(x,r)}|f(y)|\dy,~0<s<n.
\]
$M_s$ satisfies the same $(L^p,L^q)$ and $(L^1,L^{\frac{n}{n-s},\infty})$ inequality as $I_s$ does, see~\cite{A} and~\cite{MW}. For any positive function $f$ it is easy to see $M_s(f)\leq 1/\gamma(s)v_n^{\frac{s-n}{n}} I_s(f)$. Although the reverse inequality dose not hold in general, B.Muckenhoupt and R.Wheeden~\cite{MW} proved that the two quantities are comparable in $L^p$ norm($1< p<\infty$) when $f$ is nonnegative.

Now let us state our main results. First of all we consider the weak estimate of $I_{s}(f)$ and $M_{s}(f)$ under the norm $\interleave \cdot \interleave_{L^{\frac{n}{n-s},\infty}}$. Surprisingly identities for the weak type estimate of Riesz potentials and fractional maximal function can be established, which implies the two quantities are comparable in $L^{\frac{n}{n-s},\infty}$ (quasi)norm when $f\in L^1(\mathbb{R}^n)$ is nonnegative.

\begin{theo}\label{theorem1.1}
Let $0<s<n$ and $f\in{L^1(\mathbb{R}^n)}$. When $1\leq r< \frac{n}{n-s}$,
\[
\interleave I_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)} \leq \gamma_s v_{n}^{\frac{n-s}{n}}\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1(\mathbb{R}^n)},
\]
and
\[
\interleave M_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)} = \left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1(\mathbb{R}^n)}.
\]
Moreover if\/ $0<f\in{L^1(\mathbb{R}^n)}$, then
\[
\interleave I_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}=\gamma_s v_{n}^{\frac{n-s}{n}}\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1(\mathbb{R}^n)}.
\]
\end{theo}

\begin{rema}
In fact, from the proof one can obtain the reverse weak estimate holds when $0< r< \frac{n}{n-s}$. More precisely when $0< r< \frac{n}{n-s}$,
\[
\interleave I_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)} \geq \gamma_s v_{n}^{\frac{n-s}{n}}\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1(\mathbb{R}^n)},\quad\text{if }~0<f\in{L^1(\mathbb{R}^n)},
\]
and
\[
\interleave M_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)} \geq \left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1(\mathbb{R}^n)},\quad\text{if }~f\in{L^1(\mathbb{R}^n)}.
\]
\end{rema}

Then we prove the following sharp reverse weak estimates for Riesz potentials.

\begin{theo}\label{theorem1.2}
Let\/ $0<f\in{L^1(\mathbb{R}^n)}$, then
\[
\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}(\mathbb{R}^n)}\geq \gamma_sv_{n}^{\frac{n-s}{n}}\|f\|_{L^1(\mathbb{R}^n)}.
\]
And the equality holds when $f=(\frac{a}{b+|x-x_0|^2})^{\frac{n+s}{2}}$, where $a,b>0$ and $x_0\in{\mathbb{R}^n}$.
\end{theo}

As a corollary of Theorem~\ref{theorem1.1} and Theorem~\ref{theorem1.2}, we can obtain the following sharp reverse inequality.

\begin{coro}\label{corollary1}
Let $f\in L^{1}(\mathbb{R}^{n})$, then
\[
\|M_{s}f\|_{L^{\frac{n}{n-s},\infty}}\geq\|f\|_{L^1}.
\]
And the equality holds when $f(x)=h(|x-x_0|)$ where $h$ is a radial decreasing function.
\end{coro}

At last we offer an upper and a lower bound for $\mathcal{C}_{n,s}$, which implies that the behavior of the best constant $\mathcal{C}_{n,s}$ for small $s$ is optimal, i.e. $\mathcal{C}_{n,s}=O(\frac{\gamma_s}{s})=O(1)$ as $s\rightarrow 0$.

\begin{theo}\label{theorem1.3}
When $n>2$ and $0<s<\frac{n-2}{4}$,
\[
\gamma_s v_n^{\frac{n-s}{n}}\frac{n-2-4s}{2s(n-2-s)}\leq \mathcal{C}_{n,s}\leq \gamma_s v_n^{\frac{n-s}{n}}\frac{n }{s}.
\]
\end{theo}
\begin{rema}
Besides using the rearrangement inequality to obtain an upper bound $\gamma_s v_n^{\frac{n-s}{n}}\frac{n }{s}$, we can take the heat-diffusion semi-group as a tool (see the Appendix), which was used by E.~Stein and J.~Str\"omberg in~\cite{SS} to study the $(L^1,L^{1,\infty})$ bound for centered maximal function, to obtain another upper bound which is equal to $O(\gamma_s v_n^{\frac{n-s}{n}}\frac{n }{s})=O(1)$ as $(s,n)\rightarrow(0,\infty)$.
\end{rema}

\section{The identity for \texorpdfstring{$I_s(f)$}{Is(f)} and \texorpdfstring{$M_s(f)$}{Ms(f)} in \texorpdfstring{$\interleave \cdot \interleave_{L^{\frac{n}{n-s},\infty}}$}{|||. |||/n-s, infty} }

In this section, we will prove Theorem~\ref{theorem1.1}. Without loss of generality let us assume $\|f\|_{L^1(\mathbb{R}^n)}=1$. Since $I_s(f)\leq I_s(|f|)$ and $r\geq 1$, using Minkowski's inequality one have for any measurable set $E$ with $|E|<\infty$,
\begin{align}\label{2.1}
& |E|^{-\frac{1}{r}+\frac{n-s}{n}}\left[\int_E|I_sf(x)|^r\dx\right]^{\frac{1}{r}}
\leq \gamma_s |E|^{-\frac{1}{r}+\frac{n-s}{n}}\int_{\mathbb{R}^n}\left[\int_E\frac{\dx}{|x-y|^{(n-s)r}}\right]^{\frac{1}{r}}|f(y)|\dy.
\end{align}
Then by Hardy Littlewood rearrangement inequality, there holds
\begin{align}\label{2.2}
\int_E\frac{\dx}{|x-y|^{(n-s)r}}\leq \int_{E^*}\frac{\dx}{|x|^{(n-s)r}}= v_{n}^{\frac{n-s}{n}r}\frac{n}{n-(n-s)r}|E|^{1-\frac{n-s}{n}r},
\end{align}
where $E^*$ is the symmetric rearrangement of the set $E$, i.e. $E^*$ is an open ball centered at the origin whose volume is $|E|$. Therefore by~\eqref{2.1} and~\eqref{2.2} one can obtain
\[
\interleave I_{s}(f) \interleave_{L^{\frac{n}{n-s},\infty}} \leq \gamma_s v_{n}^{\frac{n-s}{n}}\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1}.
\]

Next, let us prove when $0\leq f\in {L^1(\mathbb{R}^n)}$ and $0< r< \frac{n}{n-s}$,
\begin{align}\label{reverse estimate for Is}
\interleave I_{s}(f)\interleave_{L^{\frac{n}{n-s},\infty}}\geq \gamma_s v_{n}^{\frac{n-s}{n}}\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\|f\|_{L^1}.
\end{align}
For any $\epsilon>0$, choose $R$ large enough such that $\int_{B_R(0)}f(y)dy=1-\epsilon$. Let $E=B_{lR}(0)$. Since
\[
\int_{B_R(0)}\frac{f(y)}{|x-y|^{n-s}}\dy\geq \int_{B_R(0)}\frac{f(y)}{(|x|+R)^{n-s}}\dy=(1-\epsilon)(|x|+R)^{s-n},
\]
then
\begin{align*}
\interleave I_{s}(f)\interleave_{L^{\frac{n}{n-s},\infty}} &\geq \gamma_s |E|^{-\frac{1}{r}+\frac{n-s}{n}}\left[\int_E\left(\int_{B_R(0)}\frac{f(y)}{|x-y|^{n-s}}\dy\right)^{r}\dx\right]^{\frac{1}{r}}\\
& \geq \gamma_s |E|^{-\frac{1}{r}+\frac{n-s}{n}}(1-\epsilon)\left[\int_{E}\frac{\dx}{(|x|+R)^{(n-s)r}}\right]^{\frac{1}{r}}\\
& =\gamma_s v_n^{\frac{n-s}{n}}n^{\frac{1}{r}}(1-\epsilon)l^{-\frac{n}{r}+n-s}\left[\int_0^l\frac{t^{n-1}}{(t+1)^{(n-s)r}}\dt\right]^{\frac{1}{r}}.
\end{align*}
By the fact that this inequality holds for any $l>0$, then letting $l\rightarrow\infty$, one obtain
\[
\interleave I_{s}(f)\interleave_{L^{\frac{n}{n-s},\infty}}\geq \gamma_s v_{n}^{\frac{n-s}{n}}(1-\epsilon)\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}},
\]
which implies~\eqref{reverse estimate for Is}. And we finish the proof of the identity for Riesz potential.

For fractional maximal function $M_s$, since
\begin{align*}
M_s(f)(x)&\geq \frac{1}{v^{\frac{n-s}{n}}_n(|x|+R)^{n-s}}\int_{|y-x|\leq R+|x|}|f(y)|\dy\\
& \geq \frac{1}{v^{\frac{n-s}{n}}_n(|x|+R)^{n-s}}\int_{|y|\leq R}|f(y)|\dy=\frac{1-\epsilon}{v^{\frac{n-s}{n}}_n(|x|+R)^{n-s}},
\end{align*}
then one can use the same method to get
\[
\interleave M_{s}(f)\interleave_{L^{\frac{n}{n-s},\infty}} \geq \left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}\quad\text{when }~0<r<\frac{n}{n-s}.
\]
On the other hand,
\[
\interleave M_{s}(f)\interleave_{L^{\frac{n}{n-s},\infty}}\leq \interleave 1/\gamma(s)v_n^{\frac{s-n}{n}} I_s(|f|)\interleave_{L^{\frac{n}{n-s},\infty}}=\left(\frac{n}{n-(n-s)r}\right)^{\frac{1}{r}}.
\]
Thus one can obtain the desired identity for $M_s$.

\section{The sharp reverse weak estimate for \texorpdfstring{$I_s$}{Is} and \texorpdfstring{$M_s$}{Ms}} In this section, first we prove the sharp reverse weak estimate for Riesz potentials $I_s$. By~\eqref{equa of weak norms} and Theorem~\ref{theorem1.1}, there holds
\[
\|I_{s}(f)\|_{L^{\frac{n}{n-s},\infty}}\geq \gamma_sv_{n}^{\frac{n-s}{n}}\|f\|_{L^1},~0<f\in{L^{1}(\mathbb{R}^n)}.
\]

Next, we will prove that the equality can be attained by the function $g(x)=\bigl(\frac{a}{b+|x-x_0|^2}\bigr){}^{\frac{n+s}{2}}$, where $a,b>0$ and $x_0\in{\mathbb{R}^n}$. Since the translation and dilation of $g$ do not change the ratio $\|I_{s}(g)\|_{L^{\frac{n}{n-s},\infty}}/\|g\|_{L^1}$, we only need to consider $g(x)=\bigl(\frac{2}{1+|x|^2}\bigr){}^{\frac{n+s}{2}}$. In our calculus we will use the stereographic projection, so we will introduce some notations about the stereographic\linebreak projection~here.

The inverse stereographic projection $\mathcal{S}: \mathbb{R}^n \to \mathbb{S}^n \setminus \{S\}$, where $S = - e_{n+1}$ denotes the southpole, is given by
\begin{equation*}\label{eq:stereo}
(\mathcal{S}(x))_i= \frac{2 x_i}{1 + |x|^2}, \quad i = 1,\dots,n, \quad (\mathcal{S}(x))_{n+1} = \frac{1-|x|^2}{1+|x|^2}.
\end{equation*}
Correspondingly, the stereographic projection is given by $\mathcal{S}^{-1}: \mathbb{S}^n \setminus \{S\} \to \mathbb{R}^n$,
\[
(\mathcal{S}^{-1}(\xi))_i = \frac{\xi_i}{1 + \xi_{n+1}}, \quad i = 1,\dots,n.
\]
And the Jacobian of the (inverse) stereographic projection are
\[
\mathcal{J}_{\mathcal{S}}(x)=\left(\frac{2}{1+|x|^2}\right)^{n}\quad\text{and}\quad \mathcal{J}_{\mathcal{S}^{-1}}(\xi)=(1+\xi_{n+1})^{-n}.
\]

By a change of variables,
\begin{align}\label{norm of g}
\|g\|_{L^1}&=\int_{\mathbb{R}^n}\left(\frac{2}{1+|x|^2}\right)^{\frac{n+s}{2}}\dx=\int_{\mathbb{S}^n}\left(\frac{2}{1+|\mathcal{S}^{-1}(\xi)|^2}\right)^{\frac{s-n}{2}}\dxi\nonumber\\
& =\int_{\mathbb{S}^n}(1+\xi_{n+1})^{\frac{s-n}{2}}\dxi=\bigl|\mathbb{S}^{n-1}\bigr|\int_{-1}^{1}(1+t)^{\frac{s-2}{2}}(1-t)^{\frac{n-2}{2}}\dt\nonumber\\
& =\pi^{n/2}2^{\frac{s+n}{2}}\frac{\Gamma(s/2)}{\Gamma(\frac{s+n}{2})}.
\end{align}
Denote
\[
c_{n,s}=\pi^{n/2}2^{\frac{s+n}{2}}\frac{\Gamma(s/2)}{\Gamma(\frac{s+n}{2})}.
\]
Since
\[
|\mathcal{S}^{-1}(\xi)-\mathcal{S}^{-1}(\eta)|^2=\mathcal{J}_{\mathcal{S}^{-1}}(\xi)^{\frac{1}{n}}|\xi-\eta|^2\mathcal{J}_{\mathcal{S}^{-1}}(\eta)^{\frac{1}{n}},\quad \text{for any }~\xi, \eta\in{\mathbb{S}^n},
\]
and
\[
\int_{\mathbb{S}^n}\frac{\deta}{|\xi-\eta|^{n-s}}=\frac{2^s\pi^{n/2}\Gamma(s/2)}{\Gamma(\frac{n+s}{2})}=\frac{c_{n,s}}{2^{\frac{n-s}{2}}}, \quad\text{for any }~\eta\in{\mathbb{S}^n}~\text{(see~\cite[D.4]{G})},
\]
one can obtain
\begin{align*}
I_s(g)(x)&=\gamma(s)\int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-s}}\left(\frac{2}{1+|y|^2}\right)^{\frac{n+s}{2}}\dy\\
& =\gamma(s)\int_{\mathbb{S}^n}\frac{1}{|\mathcal{S}^{-1}(\xi)-\mathcal{S}^{-1}(\eta)|^{n-s}}\left(\frac{2}{1+|\mathcal{S}^{-1}(\eta)|^2}\right)^{\frac{s-n}{2}}\deta\\
& =\gamma(s)\int_{\mathbb{S}^n}\frac{1}{|\xi-\eta|^{n-s}|\mathcal{J}_{\mathcal{S}^{-1}}(\xi)|^{\frac{n-s}{2n}} |\mathcal{J}_{\mathcal{S}^{-1}}(\eta)|^{\frac{n-s}{2n}}}\left(\frac{2}{1+|\mathcal{S}^{-1}(\eta)|^2}\right)^{\frac{s-n}{2}}\deta\\
& =\gamma(s)\int_{\mathbb{S}^n}\frac{\deta}{|\xi-\eta|^{n-s}}(1+\xi_{n+1})^{\frac{n-s}{2}}=\gamma(s)\frac{c_{n,s}}{(1+|x|^2)^{\frac{n-s}{2}}}.
\end{align*}
Thus for any $\lambda>0$,
\begin{align}\label{estimate for I_s(g)}
|\{I_s(g)>\lambda\}|=v_{n}\left(\left(\frac{\gamma(s)c_{n,s}}{\lambda}\right)^{\frac{2}{n-s}}-1\right)^{\frac{n}{2}}.
\end{align}
Therefore combining~\eqref{norm of g} and~\eqref{estimate for I_s(g)} one has
\[
\frac{\|I_{s}(g)\|_{L^{\frac{n}{n-s},\infty}}}{\|g\|_{L^1}}=v_{n}^{\frac{n-s}{n}}\sup_{\lambda>0}\left(\gamma(s)^{\frac{2}{n-s}}-\left(\frac{\lambda}{c_{n,s}}\right)^{\frac{2}{n-s}}\right)^{\frac{n-s}{2}}=\gamma(s)v_{n}^{\frac{n-s}{n}}.
\]

Next let us prove the sharp reverse weak estimate for $M_s$. By the identity in Theorem~\ref{theorem1.1} for $M_s$ and~\eqref{equa of weak norms} one can find for any $f\in{L^1}$,
\begin{align}\label{reverse estimate for Ms}
\|M_s(f)\|_{L^{\frac{n}{n-s},\infty}}\geq \|f\|_{L^1}.
\end{align}
On the other hand, since $M_s(f)\leq 1/\gamma(s)v_n^{\frac{s-n}{n}} I_s(f)$ and we already proved that the function $g=\bigl(\frac{a}{b+|x-x_0|^2}\bigr){}^{\frac{n+s}{2}}$ satisfies $\|I_{s}(g)\|_{L^{\frac{n}{n-s},\infty}}=\gamma(s)v_{n}^{\frac{n-s}{n}}\|g\|_{L^1}$, then by~\eqref{reverse estimate for Ms} the following equality holds
\begin{align}\label{equality for Ms}
\|M_s(g)\|_{L^{\frac{n}{n-s},\infty}}= \|g\|_{L^1}.
\end{align}

In fact, one can prove that~\eqref{equality for Ms} holds for any $L^1$ function $f(x)=h(|x-x_0|)$, where $h$ is a radial decreasing function, by using an approach from~\cite{AL}. First assume $\|f\|_{L^1}=1$. Let $\delta_{x_0}$ denote the Dirac delta mass placed at $x_0$. It is easy to check that
\[
M(\delta_{x_0})(x)=\frac{1}{|B(x,|x|)|},
\]
where $M$ is the centered Hardy--Littlewood maximal function. Hence, for every $\lambda>0$, there holds
\[
\lambda|\{x:M(\delta_{x_0})(x)>\lambda\}|^{\frac{n}{n-s}}=1.
\]
Since $h$ is a radial decreasing function with $\|h\|_{L^1}=1$, then by Lemma~2.1 in~\cite{AL}, one has
\[
M(f)(x)\leq M(\delta_{x_0})(x) \text{ for every } x\in\mathbb{R}^{n}.
\]
Then for any $r>0$ and $x\in \mathbb{R}^n$,
\[
\frac{1}{|B(x,r)|^{1-\frac{s}{n}}}\int_{B(x,r)}f(y)\dy\leq \left(\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)\dy\|f\|^{\frac{s}{n-s}}_{L^1}\right)^{\frac{n-s}{n}}\leq (M(\delta_{x_0})(x))^{\frac{n-s}{n}},
\]
which implies that
\begin{align}\label{estimate for Ms}
\|M_{s}f\|_{L^{\frac{n}{n-s},\infty}}\leq 1=\|f\|_{L^1}.
\end{align}
Combining this inequality with~\eqref{reverse estimate for Ms}, one can obtain the desired result for $M_s$.

What is noteworthy at the end of the section is that this result is also true for the centered Hardy--Littlewood maximal function, that is because, using the same method, one can prove~\eqref{estimate for Ms} when $s=0$, i.e.~\eqref{estimate for Ms} is true for the centered Hardy--Littlewood maximal function. On the other hand, using the limiting weak type behavior for the maximal function in~\cite{J}, \eqref{reverse estimate for Ms} is also true for the centered Hardy--Littlewood maximal function.


\section{The upper and lower bounds of \texorpdfstring{$\mathcal{C}_{n,s}$}{Cn,s}}

In this section, we will provide an upper and a lower bound for $\mathcal{C}_{n,s}$. Using Theorem~\ref{theorem1.1} and~\eqref{equa of weak norms}, we can get an upper bound
\[
\mathcal{C}_{n,s}\leq \gamma_s\frac{n }{s}v_n^{\frac{n-s}{n}}.
\]

To obtain the lower bound, we will use the following formula (see~\cite[Section~5.10]{LL}). Let $0<\alpha<n$, $0<s<n$ and $\alpha+s<n$, then
\begin{equation}\label{equation 4.1}
\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-s}}\frac{1}{|y|^{n-\alpha}}\dy=C_{n,\alpha,s}\frac{1}{|x|^{n-s-\alpha}}
\end{equation}
with
\[
C_{n,\alpha,s}=\pi^{\frac{n}{2}}\frac{\Gamma(\frac{s}{2})\Gamma(\frac{\alpha}{2})\Gamma(\frac{n-s-\alpha}{2})}{\Gamma(\frac{n-s}{2})\Gamma(\frac{n-\alpha}{2})\Gamma(\frac{s+\alpha}{2})}.
\]

Now assume $n-2>4s$. Choose $f(y)=\frac{1}{|y|^{n-2}}\chi_{(|y|\leq 1)}$ and let us prove
\[
\|I_s f\|_{\frac{n}{n-s},\infty}\geq \gamma_s\frac{v_n^{\frac{n-s}{n}}}{s}\frac{n-2-4s}{2(n-2-s)}\|f\|_{L^1}.
\]

Since $|x|\leq \frac{1}{2}$, $|y|>1$ implies $|y-x|\geq \frac{|y|}{2}$, using~\eqref{equation 4.1} with $\alpha=2$ one have
\begin{align}\label{equation 4.2}
\frac{1}{\gamma_s}I_{s}(f)(x)&=\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-s}}f(y)\dy=\int_{|y|\leq 1}\frac{1}{|x-y|^{n-s}}\frac{1}{|y|^{n-2}}\dy\nonumber\\
&=\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-s}}\frac{1}{|y|^{n-2}}\dy-\int_{|y|> 1}\frac{1}{|x-y|^{n-s}}\frac{1}{|y|^{n-2}}\dy\nonumber\\
&\geq\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n-s}}\frac{1}{|y|^{n-2}}\dy-\int_{|y|> 1}\frac{2^{n-s}}{|y|^{2n-2-s}}\dy\nonumber\\
&=\frac{c}{|x|^{n-s-2}}-d,
\end{align}
where
\[
c=\frac{4\pi^{n/2}}{(n-s-2)\Gamma(n/2-1)s}\quad\text{ and}\quad d=\frac{2^{n-s+1}\pi^{n/2}}{(n-s-2)\Gamma(n/2)}.
\]

Choose $\lambda_0=\gamma_s(2^{n-s-2}c-d)$, since $\frac{c}{d}=\frac{n-2}{s}\frac{1}{2^{n-s}}>\frac{1}{2^{n-s-2}}$, then $\lambda_0$ is positive. Thus by~\eqref{equation 4.2}, there holds
\begin{equation}\label{equation 4.3}
|\{I_s f>\lambda_0\}|\geq \left|\left\{|x|\leq 1/2, \frac{c}{|x|^{n-s-\alpha}}-d>\frac{\lambda_0}{\gamma_s}\right\}\right|=v_n(\frac{1}{2})^n.
\end{equation}
Using the fact $\|f\|_{L^{1}(\mathbb{R}^{n})}=\frac{\omega_{n-1}}{2}$ and~\eqref{equation 4.3} one can obtain
\[
\frac{\|I_s f\|_{\frac{n}{n-s},\infty}}{\|f\|_{L^1}}\geq \frac{\lambda_0 |\{I_s f>\lambda_0\}|^{\frac{n-s}{n}}}{\|f\|_{L^1}}=\lambda_0v^{\frac{n-s}{n}}_n\frac{\Gamma(n/2)}{2^{n-s}\pi^{n/2}} =\gamma_s\frac{v_n^{\frac{n-s}{n}}}{s}\frac{n-2-4s}{2(n-2-s)}.
\]
So we complete the proof of Theorem~\ref{theorem1.3}.


\section*{Appendix}

In this Appendix, we give an alternative approach to prove the $(L^1,L^{\frac{n}{n-s},\infty})$ estimate for Riesz potentials, and at the same time this approach also provide an upper bound for $\mathcal{C}_{n,s}$, which have the same behavior with $\gamma_s v_n^{(n-s)/n}n/s$ as $(s,n)\rightarrow(0,\infty)$. First, we state a lemma (see~\cite[Section~3]{SS}, also see the Hopf abstract maximal ergodic theorem in~\cite{DS}) about the weak estimate of the average of the heat-diffusion semi-group $T^{t}(f)=P_t \ast f$, where
\[
P_t=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^2}{t}}.
\]

\begin{lemm}\label{lamma 4.1}
For any $f\in L^1(\mathbb{R}^n)$, there holds
\[
\left|\left\{x\in \mathbb{R}^n:\sup_{s>0}\frac{1}{s}\int_0^sP_tf(x)\dt>\lambda\right\}\right|\leq \frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)},\quad \lambda>0.
\]
\end{lemm}

Now let prove the $(L^1,L^{\frac{n}{n-s},\infty})$ estimate for Riesz potentials $I_{s}(f)$, which also can be presented by the following formula related to $T^{t}(f)$,
\[
I_{s}(f)(x)=\frac{1}{\Gamma(s/2)}\int^{\infty}_{0}t^{\frac{s}{2}-1}P_{t}\ast f(x)\dt.
\]
We divide the integral into two parts
\[
\int^{\infty}_{0}t^{\frac{s}{2}-1}P_{t}\ast f(x)\dt=J_{1}(f)(x)+J_{2}(f)(x),
\]
where
\[
J_{1}(f)(x)=\int^{R}_{0}t^{\frac{s}{2}-1}P_{t}\ast f(x)\dt,
\]
\[
J_{2}(f)(x)=\int^{\infty}_{R}t^{\frac{s}{2}-1}P_{t}\ast f(x)\dt,
\]
for some $R$ to be determined later.

Denote $\mathcal{M}^{0}f(x)=\sup_{r>0}\frac{1}{r}\int^{r}_{0}P_{t}\ast f(x)dt$, then we have
\begin{align}\label{A-1}
J_{1}(f)(x)&=\sum^{\infty}_{i=1}\int^{2^{-i+1}R}_{2^{-i}R}t^{\frac{s}{2}-1}P_{t}\ast f(x)\dt\nonumber\\
&\leq\sum^{\infty}_{i=1}\int^{2^{-i+1}R}_{2^{-i}R}(2^{-i}R)^{\frac{s}{2}-1}P_{t}\ast f(x)\dt\nonumber\\
&\leq\sum^{\infty}_{i=1}(2^{-i}R)^{\frac{s}{2}-1}2^{-i+1}R\left(\frac{1}{2^{-i+1}R}\int^{2^{-i+1}R}_{0}P_{t}\ast f(x)\dt\right)\nonumber\\
&\leq2R^{\frac{s}{2}}\frac{2^{-\frac{s}{2}}}{1-2^{-\frac{s}{2}}}\mathcal{M}^{0}f(x).
\end{align}
On the other hand, by direct computation, we obtain that
\begin{align}\label{A-2}
J_{2}(f)(x)&\leq\int^{\infty}_{R}t^{\frac{s}{2}-1}\|P_{t}\|_{L^{\infty}}\|f\|_{L^{1}}\dt\nonumber\\
&\leq\frac{2}{n-s}(4\pi)^{-\frac{n}{2}}R^{\frac{s}{2}-\frac{n}{2}}\|f\|_{L^{1}}.
\end{align}
Combining~\eqref{A-1} and~\eqref{A-2}, we obtain that
\begin{equation}\label{A-3}
I_{s}(f)(x)\leq\frac{1}{\Gamma(s/2)}\left(2R^{\frac{s}{2}}\frac{2^{-\frac{s}{2}}}{1-2^{-\frac{s}{2}}}\mathcal{M}^{0}f(x)+\frac{2}{n-s}(4\pi)^{-\frac{n}{2}}R^{\frac{s}{2}-\frac{n}{2}}\|f\|_{L^{1}}\right)
\end{equation}
for all $R>0$. The choice of
\[
R=\left(\frac{(4\pi)^{-\frac{n}{2}}\|f\|_{L^{1}}}{\frac{s}{2^{\frac{s}{2}-1}-1}\mathcal{M}^{0}f(x)}\right)^{\frac{2}{n}}
\]
minimizes the right side of the expression in~\eqref{A-3}. Thus
\begin{equation}\label{A-4}
I_{s}(f)(x)\leq\tau_{s}(\mathcal{M}^{0}f(x))^{\frac{n-s}{n}}\|f\|^{\frac{s}{n}}_{L^{1}},
\end{equation}
where
\[
\tau_{s}=2(4\pi)^{-\frac{s}{2}}(2^{\frac{s}{2}}-1)^{\frac{s-n}{n}}\frac{n}{n-s}\left(\frac{1}{s}\right)^{\frac{s}{n}}\frac{1}{\Gamma(s/2)}.
\]
Now using Lemma~\ref{lamma 4.1} one can see that
\begin{align*}
\lambda|\{I_{s}f>\lambda\}|^{\frac{n-s}{n}}&\leq\lambda\left|\left\{\tau_{s}\left(\mathcal{M}^{0}f(x)\right)^{\frac{n-s}{n}}\|f\|^{\frac{s}{n}}_{L^{1}}>\lambda\right\}\right|^{\frac{n-s}{n}}\\
&\leq\lambda\left[\left(\frac{\tau_{s}\|f\|^{\frac{s}{n}}_{L^{1}}}{\lambda}\right)^{\frac{n}{n-s}}\|f\|_{L^{1}}\right]^{\frac{n-s}{n}}\\
&\leq\tau_{s}\|f\|_{L^{1}}.
\end{align*}
Notice that
\[
2^{\frac{s}{2}}-1>\frac{\ln2}{2}s\quad\text{ for }\ s>0,
\]
thus,
\[
\tau_{s} \leq\frac{2}{\ln2}\left(\frac{1}{4\pi}\right)^{-\frac{s}{2}}\frac{1}{\Gamma(\frac{s}{2}+1)}\frac{n}{n-s}.
\]
So by this approach, one can obtain that $\mathcal{C}_{n,s}\leq \frac{2}{\ln2}(\frac{1}{4\pi})^{-\frac{s}{2}}\frac{1}{\Gamma(\frac{s}{2}+1)}\frac{n}{n-s}$ and it is easy to check that when $(s,n)\rightarrow(0,\infty)$,
\[
\frac{2}{\ln2}\left(\frac{1}{4\pi}\right)^{-\frac{s}{2}}\frac{1}{\Gamma(\frac{s}{2}+1)}\frac{n}{n-s}=O\left(\gamma_s v_n^{\frac{n-s}{n}}\frac{n }{s}\right).
\]

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