Suppose we have four functions $u_{1},\ldots,u_{4}\in L^{4}(\mathbb{R}\times\mathbb{R}^{2})$ and set $u:=u_{1}\cdots u_{4}$. Now consider the the space-time integral $$|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|$$
I want to take use Plancherel's theorem in space-time to obtain an equivalent expression to the one, which is easier for me to work with. The issue is that I want to get replace the indicator function $\chi_{[0,T]}$ with a bump function $\psi$ adapted to $[0,T]$ since the temporal Fourier transform of $\psi$ is an $L^{1}$-normalized $L^{1}$-function, unlike the temporal Fourier tranform of $\chi_{[0,T]}$ (which is a $L^{1}$-normalized modulated sinc function). In short, I am looking for an estimate of the following form:
$$|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|\lesssim |\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}\psi(t)u(t,x)dxdt| \tag{1}$$
In (1), it is essential that the implied constant be independent of the interval $[0,T]$ and $u$. This requirement is because I have to work with lots of such expressions for different intervals in $\mathbb{R}$ and different product functions $u$.
I've thought a bit about this goal, which may be hopeless, but if $$\dfrac{|\int_{[0,T]^{c}}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}{|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}\leq\frac{1}{2}$$
Then by the triangle inequality, $$|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|\leq 2|\int_{\mathbb{R}}\int_{\mathbb{R}^{2}}u(t,x)dxdt|, \tag{2}$$ which is acceptable for my purposes, even better actually. A completely analogous argument shows that if $$\dfrac{|\int_{[0,T]^{c}}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}{|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}\geq 2,$$ then (2) also holds. The issue is when
$$\dfrac{|\int_{[0,T]^{c}}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}{|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}\sim 1$$ The worst case I can think of is essentially $$\dfrac{|\int_{[0,T]^{c}}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}{|\int_{0}^{T}\int_{\mathbb{R}^{2}}u(t,x)dxdt|}=1, \quad \int_{\mathbb{R}}\int_{\mathbb{R}^{2}}u(t,x)dxdt=0 \tag{3}$$ I am at a complete loss on how to proceed on this case.
Edit: In case it helps, I should add that I only care about the case when $T\gg 1$, as I can handle the case where $T$ is bounded by some absolute constant using a different strategy than the one sought in this post.