Regularity and analyticity of solutions in a direction for elliptic equations

In this paper, we study the regularity and analyticity of solutions to linear elliptic equations with measurable or continuous coefficients. We prove that if the coefficients and inhomogeneous term are Hölder-continuous in a direction, then the second-order derivative in this direction of the solution is Hölder-continuous, with a different Hölder exponent. We also prove that if the coefficients and the inhomogeneous term are analytic in a direction, then the solution is analytic in that direction.