Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense).

Then, these functions span a subspace $V$ of $L^2$. Now, consider two $L^2$ functions $g$ and $h$ with non-intersecting support. Assume that both $g$ and $h$ only overlap with $k$ of those functions $f_n$. It may happen now that although $\langle g,h \rangle =0$ due to the non-intersecting support of the two, $\langle P_V g,P_V h \rangle \neq 0$ where $P_V$ is the projection onto $V$. However, we expect that "morally" $\langle P_V g,P_V h \rangle$ should still be small if they are far apart and the $(f_n)$ are sufficiently localized.

Of course one could trivially estimate $\langle P_V g,P_V h \rangle \le \left\lVert g\right\rVert \left\lVert h \right\rVert,$ but this estimate is very bad in general as it does not capture the fact that most of the time, this should be something close to zero, since $\langle g,h \rangle =0.$

I ask: Does anybody see a better estimate that captures this $\langle P_V g,P_V h \rangle \neq 0$ being small? Especially, one would expect that the larger $V$ is, the closer should we get to zero.

Let me rephrase the questions like this: Assuming you are not allowed to evaluate $P_Vg$ or $P_Vh$ but only $g,h$ and all $(f_n)$ and inner-products among those objects. Can we get from this a better bound?