Is there any way to get rid of the $$F_T$$ in this formula? where U>T>t>0
$$ \begin{aligned} & \mathbb E[(\mathbb E[1_{{s_U}>K}|F_T])^n|F_t]\\ \end{aligned}$$
The best approach I have at the moment is to use the Jensen inequality and the tower property, to get a lower bond if n>1 $$ \begin{aligned} & \mathbb E[ (\mathbb E[1_{{s_U}>K}|F_T])^n|F_t]\\ & >=(\mathbb E[\mathbb E[1_{{s_U}<K}|F_T]|F_t])^n\\ & =(\mathbb E[1_{{s_U}>K}|F_t])^n \end{aligned}$$
I'm pretty sure that you can just apply the tower property without using jensens: $$\mathbb{E}\left[\mathbb{E}\left[\mathbb{1}_{S_U > K} | F_T \right]^n | F_t \right] = \mathbb{E}[(\mathbb{1}_{S_U > K})^n | F_t].$$