Let $\Omega$ be a nonzero integrable function on $S^{n−1}$ with mean value zero. Let $f \geq 0$ be nonzero and integrable over $\mathbb{R}^n$. Prove that $T_\Omega( f )$ is not in $L^1(\mathbb{R}^n)$.
The definition of $T_\Omega( f ) = \lim_{\epsilon \to 0, N \to \infty}{}\int_{\epsilon \leq |y| \leq N} f(x-y) \frac{\Omega(y/|y|)}{|y|^ n} dy$.
I would like to know how to show this statement. First I wanted to show $\widehat{T_\Omega(f) } $ cannot be continuous at zero.But I don't know how to prove it.
Also I think that in this excersise we need the hypothesis that $\Omega \not = 0 \quad a.s$, but I'm not sure.