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The Riesz potential is defined by

$$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$

Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. For this result, I need a specific proof.

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In Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?, Prof. Jean Van Schaftingen gave a specific proof of this result. The key estimate in his proof is $$ \int_{B_r} \int_{B_r}\Big\vert \frac{1}{\vert z - x\vert^{d - \alpha}} - \frac{1}{\vert z - y\vert^{d - \alpha}} \Big\vert \,\mathrm{d}x\,\mathrm{d}y \le \frac{C r^{2 d + 1}}{(r + \vert z \vert)^{d - \alpha + 1}}, $$ once this estimate is obtained, then we can use the Holder inequality to complete this proof.

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