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In this paper, we prove the existence and uniqueness of $W^{2,p}$($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstaclefunctions. Moreover, we show the optimal regularity of the solution and thelocal $C^1$ regularity of the free boundary. In the study of the regularity ofthe free boundary, we deal with a general problem, the no-sign reduced doubleobstacle problem with an upper obstacle $\psi$, $F(D^2 u,x) =f\chi_{\Omega(u)\cap\{ u< \psi\} } + F(D^2\psi,x) \chi_{\Omega(u)\cap \{u=\psi\}}, u\le \psi\text { in } B_1$, where $\Omega(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nablau =0\}\right)$.

The Regularity Theory for the Double Obstacle Problem for Fully Nonlinear Operator