Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev theorem on fractional integration states that for $1<p<d/\alpha$ we have $$ \|I_\alpha(f)\|_{L^q(\mathbb R^d)} \le C_{d,\alpha, p} \|f\|_{L^p(\mathbb R^d)}, \quad\text{where}\ \ \ q=\frac{dp}{d-p\alpha}. \tag 1$$
My question is whether there is a local version of this result.
More precisely, let $B_\rho$ denote a ball with radius $\rho>0$. Since $I_\alpha(f)$ is a non-local integral, it is clear that $$ \|I_\alpha(f)\|_{L^q(B_\rho)} \le C_{d,\alpha, p} \|f\|_{L^p(B_\rho)} $$ cannot hold in general (just take a function $f$ supported outside $B_\rho$). On the other hand, the left hand side here tends to 0 as $\rho \to 0$, so estimating the $L^p$ norm in $B_\rho$ by the norm in $\mathbb R^d$ does not seem very sharp as such an estimate loses the information that the left hand side is small for small $\rho$.
So is there an estimate which somehow gives the sharp behavior in this setting of local balls $B_\rho$? For simplicity, we can also assume that $f$ is supported in $B_r$ for $r>>\rho$.
