  In this study, for any given terminal time $T$, we establish an $L^p$ ($P>2$)
estimations of fully coupled FBSDEs based on the $L^2$ estimations. Yong [24]
proposed that a natural question is whether an adapted $L^2$-solution is an
adapted $L^p$ solution for some $p>2$. In this study, we give a positive answer
to this question. For any given terminal time $T$, based on an observation of
the relation between $L^2$ and $L^p$ estimations of FBSDEs, we prove that a
unique $L^2$-solution of fully coupled FBSDEs is an $L^p$-solution under
standard conditions on the coefficients. Furthermore, we show that the fully
coupled FBSDEs developed in the linear quadratic optimal control problem or
investigated by the "decoupling random field" method admit a unique
$L^p$-solution.
