I don't know how to prove (or disprove) this.
Let $(m_\varepsilon)_\varepsilon$ be a sequence of non-negative functions which can not vanish at any point, there exists $M>0$ such that $\forall\varepsilon$ we have $\int_{\mathbb{R}^N}m_\varepsilon dx=M$, $\|m_\varepsilon\|_{L^\infty(\mathbb{R}^N)}\le C_1$ (uniformly).
Assume also that there exists $C>0$ not depending on $\varepsilon$ such that $\forall\varepsilon$ $$I_\alpha\ast m_\varepsilon(0)\ge C>0$$
Then, there exist $R>0$ and $C_2$ not depending on $\varepsilon$ such that $$\int_{B_R(0)}m_\varepsilon(x)dx\ge C_2>0$$
where $I_\alpha(x)=\frac{1}{|x|^{N-\alpha}}$ and $\alpha\in(0,N)$.
Any ideas? Thanks