  In this paper, for a family of second-order elliptic equations with rapidly
oscillating periodic coefficients, we are interested in a Carleman-type
inequality for these solutions satisfying an additional growth condition in
elliptic periodic homogenization, which implies a three-ball inequality without
an error term at a macroscopic scale. Moreover, if we replace the additional
growth condition by the doubling condition at a macroscopic scale, then the
three-ball inequality without an error term holds at any scale. The proof
relies on the convergence of $H^1$-norm for the solution and the compactness
argument.
