Let $\Omega$ be an open domain of $\mathbb{R}^n$.
The Newtonian potential of a function $f$ is the convolution $$ Nf(x) = \int_\Omega \Gamma(x-y)f(y)\, dy $$ where $\Gamma$ is the fundamental solution of Laplace equation. I want to show that the Newton potential is a bounded map from $L^p(\Omega$) to itself for $1\leq p < \infty$. I know from lemma 7.12 of Gilbarg-Trudinger Elliptic partial differential equation of second order that for $n>2$ it is true. Now my question is for $n=2$. In this case $N:L^p(\Omega)\to L^p(\Omega)$ become $$ Nf(x) = \int_\Omega \log|x-y|f(y)\, dy. $$ How can I show that this operator is bounded ?