On the interior smoothness of solutions to second-order elliptic equations

We study the interior smoothness properties of solutions to a linear second-order uniformly elliptic equation in selfadjoint form without lower-order terms and with measurable bounded coefficients. In terms of membership in a special function space we combine and supplement some properties of solutions such as membership in the Sobolev space $W^1_{2,\mathrm{loc}}$ and Holder continuity. We show that the membership of solutions in the introduced space which we establish in this article gives some new properties that do not follow from Holder continuity and the membership in $W^1_{2,\mathrm{loc}}$.