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You can't have a convolution of two functions in $L_1(0,c)$ that is not in $L_1(0,c)$, else you violate Young's inequality, or the fact that $L_1(0,c)$ (i.e. the space you refer to as $L^{+}$) is a Banach algebra under convolution, so the answer is no. I assume you are taking $a < c$ and $b < c$ (otherwise you can't work in $L_1(\mathbb{R})$ because $f_a$ and $f_b$ don't belong there).