Let us denote the Riesz potential in $\mathbb R^d$ by $$ I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$

By the classical Hardy-Littlewood-Sobolev theorem on fractional integration we have for $1 < p <d/\alpha$ that $$ \|I_\alpha(f)\|_{L^q} \le C_{d, \alpha, p} \|f\|_{L^p}, \quad\text{where}\ \ \tag 1 q=\frac{dp}{d-p\alpha}. $$

I am looking for a reference (with a proof) for the borderline case $p=d/\alpha$: $$I_\alpha:L^{d/\alpha} \to \rm{BMO}. \tag 2$$

I have checked Grafakos's and Stein's harmonic analysis books, but I don't think the proof is in any of them.

A related result, which would probably be enough for me, is stated as an exercise 8.11. on page 62-63 of

http://www.ms.uky.edu/~rbrown/courses/ma773/notes.pdf

but at least a quick look seems to indicate that the constant in (1) is of the form $C_{d,\alpha} q$ (for $p$ close to $d/\alpha$), so that the corresponding power series in the exercise does not converge after raising the estimate to power $q$ (the factor k^k dominates k! of the denominator). Probably I am missing something.

In any case, the best would be to find a self-contained proof of (2) directly.