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In this note we prove an end-point regularity result on the Lp integrabilityof the second derivatives of solutions to non-divergence form uniformlyelliptic equations whose second derivatives are a priori only known to beintegrable. The main assumption on the elliptic operator is the Dini continuityof the coefficients. We provide a counterexample showing that the Dinicondition is somehow optimal. We also give a counterexample related to the BMOregularity of second derivatives of solutions to elliptic equations.

Some remarks on the Lp regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition