Schauder theory for Dirichlet elliptic operators in divergence form

Let A be a strongly elliptic operator of order 2m in divergence form with Hölder continuous coefficients of exponent ${\sigma \in (0,1)}$ defined in a uniformly C ^1+σ domain Ω of ${\mathbb{R}^n}$ . Regarding A as an operator from the Hölder space of order m +  σ associated with the Dirichlet data to the Hölder space of order ?m +  σ, we show that the inverse (A ? λ)^?1 exists for λ in a suitable angular region of the complex plane and estimate its operator norms. As an application, we give a regularity theorem for elliptic equations.