Second order $L_p$ estimates for subsolutions of fully nonlinear equations

We obtain new $L_p$ estimates for subsolutions to fully nonlinear equations. Based on our $L_p$ estimates, we further study several topics such as the third and fourth order derivative estimates for concave fully nonlinear equations, critical exponents of $L_p$ estimates and maximum principles, and the existence and uniqueness of solutions to fully nonlinear equations on the torus with free terms in the $L_p$ spaces or in the space of Radon measures.