Given a continuous, compactly supported function $f$ on $R^2$, it is known that the logarithmic potential of $f$, that is $$ U_{f}(x):=-\frac{1}{2\pi}\int\log|x-y|f(y)dy $$ has the following decay at infinity $$ U_{f}(x)=-\frac{M}{2\pi}\log|x|+O(1/|x|)\quad as\ |x|\rightarrow\infty. $$ My question is rather simple: can this result be extended when we consider $f\in L^{1}\cap L^{\infty}$.

Thank you very much for any suggestion you can give me, Bruno