Hello Arc,
The key issue here is that your functions $f_a$ and $f_b$ belong to the same $L_1$ space if and only if $a=b$ (i.e. $L_1(0,a)$). Thus, when you carry out a convolution over $\mathbb{R}$, you get a contradiction to Young's inequality (or the fact that $L_1(\mathbb{R})$ is a Banach algebra). Even if you work in $L_1(0,c)$ (i.e. the space to which you refer as $L^{+}$) with $a\leq c$ and $b\leq c$, you'll still get the same contradiction. So your calculation doesn't say anything about $L_1$ spaces I'm afraid, in particular, about $L_1(0,c)$.