  In this note we prove an end-point regularity result on the Lp integrability
of the second derivatives of solutions to non-divergence form uniformly
elliptic equations whose second derivatives are a priori only known to be
integrable. The main assumption on the elliptic operator is the Dini continuity
of the coefficients. We provide a counterexample showing that the Dini
condition is somehow optimal. We also give a counterexample related to the BMO
regularity of second derivatives of solutions to elliptic equations.
