Quantitative unique continuation for elliptic equations with Hölder continuous potentials

We study quantitative unique continuation for second order elliptic equations with lower-order terms of Hölder regularity via a weighted frequency function method. We establish quantitative three-ball inequalities and corresponding vanishing-order bounds for Schrödinger equations with Hölder potentials and Hölder gradient terms, and corresponding results for elliptic equations with variable leading coefficients. Our results are quantitative with explicit dependence of Hölder norms in the three-ball inequalities. These fill in the gap for quantitative unique continuation between bounded potentials and $C^1$ potentials.