Riesz potential is defined as
$$I_s(f) = K_s*f$$
for Schwartz function $f$ with $K_s(x) = c_s|x|^{-n+s}$.
By weak Young's inequality, $I_s$ can be extended to $L^p \rightarrow L^q$ bounded operator with $\frac{1}{p} = \frac{1}{q} + \frac{s}{n}$ and $p,q \in (1, \infty)$.
Here's an argument showing that $I_s$ fails to extend to a $L^1 \rightarrow L^{\frac{n}{n-s}}$ bounded operator: Suppose not. Take a smooth approximation to the identity $\psi_\epsilon$. Then as $\epsilon \rightarrow 0$, we have $I_s(\psi_\epsilon) \rightarrow K*\delta_0 = K$, and so $K \in L^{\frac{n}{n-s}}$, which is a contradiction.
In this argument, I am not sure how one is using the boundedness of $I_s$ to get $I_s(\psi_\epsilon) \rightarrow K$. It is clearly not true that $\psi_\epsilon \rightarrow \psi_0$ in $L^1$ for some $\psi_0 \in L^1$ such that $I_s(\psi_0) = K$.
I would appreciate if you could provide some hints or reference to any relevant theorems.
Ignore the constant $c_s$. We have that $I_s(\psi_{\epsilon}):=K_s*\psi_{\epsilon}\to K_s*\delta_0=K_s$ in the weak* topology of distributions (by general results about convolutions). Here, we’re not using boundedness of $I_s:L^1\to L^q= L^{\frac{n}{n-s}}$, but rather we’re simply using the fact that $K_s\in L^1_{\text{loc}}$.
It is important to note that the above convergence happens in the topology of distributions. Now, suppose that $I_s:L^1\to L^q$ is a bounded operator. For each $\phi$ in $\mathcal{D}(\Bbb{R}^n)$ we have \begin{align} \left|\int_{\Bbb{R}^n}K_s(x)\phi(x)\,dx\right|&=|\langle K_s,\phi\rangle|\\ &=\lim_{\epsilon\to 0^+}|\langle K_s*\psi_{\epsilon},\phi\rangle|\\ &\leq\limsup_{\epsilon\to0^+}\|K_s*\psi_{\epsilon}\|_{L^q}\|\phi\|_{L^{q’}}\tag{Holder}\\ &\leq\limsup_{\epsilon\to 0^+}C\|\psi_{\epsilon}\|_{L^1}\|\phi\|_{L^{q’}}\tag{$I_s:L^1\to L^q$ bounded}\\ &=C\|\psi\|_{L^1}\|\phi\|_{L^{q’}}, \end{align} where at the end I supposed that $\psi_{\epsilon}=\frac{1}{\epsilon^n}\psi\left(\frac{x}{\epsilon}\right)$, so their $L^1$ norms are all the same. This is sufficient (since $\mathcal{D}(\Bbb{R}^n)$ is nice and dense in $L^{q’}$) to conclude that $K_s$ belongs to $L^q$ (and $\|K_s\|\leq C\|\psi\|_{L^1}$). But a direct calculation shows that \begin{align} \|K_s\|_{L^q}^q&=\int_{\Bbb{R}^n}|K_s(x)|^q\,dx=\int_{\Bbb{R}^n}\frac{1}{|x|^n}\,dx=\infty, \end{align} which is a contradiction.
Side remark: the case $q=\infty$ (so $p=\frac{n}{s}$) is dual to the above situation, so we also can’t have boundedness there either.