We consider an equation with left and right fractional derivatives and with the boundary condition $y(0)=\lim\limits_{x\to 0^+}y(x)=0$, $y(b)=\lim\limits_{x\to b^-}y(x)=0$ in the space $\mathcal{L}^1(0, b)$ and in the subspace of tempered distributions. The asymptotic behavior of solutions in the end points $0$ and $b$ have been specially analyzed by using Karamata's regularly varying functions.