Existence of a function in $L^{p}$ with a certain property

Question

Is there a function $f \in L^{p}(\mathbb{R}^{n})$ such that $\|If\|_{L^{p}(\mathbb{R}^{n})} = \infty$ where $If(x) = \int_{\mathbb{R}^{n}}\frac{f(y)}{|x - y|^{n}}\, dy$? Such a case isn't covered by the Hardy-Littlewood-Sobolev inequality, so perhaps one doesn't exist as I expect the inequality to be optimal in some sense.

Answer 1

Convolution with a positive kernel that is not locally integrable works about as well as you might imagine. Take $f = \chi_B$ (characteristic function of a ball); then $If \equiv \infty$ on $B$.