Quantitative uniqueness for bi-Laplace equations with $C^2$ lower-order coefficients

We study quantitative unique continuation for bi-Laplace equations $${\mathrm{\Delta }}^{2}u+W(x)\mathrm{\Delta }u+V(x)\cdot \mathrm{\nabla }u=0$$ with $C^2$ lower-order coefficients. Using the weighted-frequency function, we establish an $L^{\mathrm{\infty }}$ three-ball inequality whose exponential factor has coefficient dependence $\mathrm{\parallel }W{\mathrm{\parallel }}_{C^2}^{1\mathrm{/}2}+\mathrm{\parallel }V{\mathrm{\parallel }}_{C^2}^{1\mathrm{/}3}$. Under some normalization, we obtain a vanishing-order bound of size $C(\mathrm{\parallel }W{\mathrm{\parallel }}_{C^2}^{1\mathrm{/}2}+\mathrm{\parallel }V{\mathrm{\parallel }}_{C^2}^{1\mathrm{/}3})$.