I want to ask how to prove the following result:
Suppose $u\in C(\overline{B_{1}})$, if there exists a universal constant M>0, such that for any $x_{0}\in \overline{B_{1}} $, we have a quadratic polynomial $P_{x_{0}}$ such that $$|u(x)-P_{x_{0}}(x)|\leq M|x-x_{0}|^{2+\alpha},\qquad in\ B_{1/4}(x_{0})\cap\overline{B_{1}} $$ Then, $u\in C^{2,\alpha}(\overline{B_{1}})$, and $$ [D^2u]_{C^{0,\alpha}\ \ (\overline{B_{1}})}\leq C_{n}M. $$
The above result states that uniformly pointwise $C^{2,\alpha}$ in $\overline{B_{1}}$ is equivalent to $C^{2,\alpha}(\overline{B_{1}})$ in the classical sense. It has background in the theory of elliptic PDE. Caffarelli introduced a method of pointwise approximation by polynomials to prove the interior regularity of solutions to fully nonlinear elliptic equations. Moreover, this method can be used to prove boundary regularity.