Is there anyway to prove the following statement?
$$\int_{0}^{T}a^T(\theta)b(\theta)d\theta \le c_1^2 \Rightarrow \int_{0}^{T}a^T(\theta)Kb(\theta)d\theta \le c_2^2$$
where $a(t),b(t)\in \mathbb{R}^n$, $c_1,c_2 \in \mathbb{R}$ are constant, and $K \in \mathbb{R}^{n\times n}$ is a constant positive definite matrix or even if it helps it can be supposed diagonal with different arrays!
I am going to assume that you want to find a $c_2$ for any given $K$ and $c_1$ such that the inequality holds for arbitrary $a$ and $b.$
Such $c_2$ cannot be found for all $K$ and $c_1.$ For example take $n=2,$ $K=\hbox{Diag}(2,1)$ and $c_1=1.$ Then consider the constant functions $a(\theta)=(1,1)^T$ and $b_r(\theta)=(r,-r)^T.$ The integral on the left is $0$ but the integral on the right is $rt,$ which becomes arbitrarily large for $t>0$ and sufficiently large $r.$