We have $X(t)=[X_1(t)\ X_2(t)\ X_3(t)\ \dots\ X_n(t)]$ and $Y(t)=[Y_1(t)\ Y_2(t)\ Y_3(t)\ \dots\ Y_n(t)]$ are two stochastic process such that: $$\sup E[Y_1^2] \leq K,$$ on $[t_0, T]$ with $K$ a positive integer.
Is that true
$$E\int^T_{t_0} Y_1^{2} F(X(t))^2 \, dt \leq \sup E[Y_1^2]E\int^T_{t_0} F(X(t))^2 \, dt$$
Important note: $Y_1(t)$ is a function of $X_1$.
In general I don't see any reason why this should be true. However, if $Y_1(t)^2$ and $F(X(t))$ are independent for Lebesgue a.e. $t$ (it would be sufficient to assume $Y_1$ and $X$ are independent, of course), then it does hold.
You may say by Tonelli's Theorem (Fubini's Theorem for nonnegative integrands) $$ E\int_{t_0}^T Y_1(t)^2 F(X(t))^2~dt = \int_{t_0}^T E[Y_1(t)^2 F(X(t))^2]~dt $$ which by independence is $$ =\int_{t_0}^T E[Y_1(t)^2] E[F(X(t))^2]~dt $$ and by monotonicity of the integral is $$ \leq \int_{t_0}^T (\sup_s E[Y_1(s)^2]) E[F(X(t))^2]~dt $$ $$ = \sup_t E[Y_1(t)^2]\int_{t_0}^T E[F(X(t))^2]~dt $$ and again by Tonelli's Theorem this is $$ =\sup_t E[Y_1(t)^2]E\int_{t_0}^T F(X(t))^2~dt. $$