I am reading an outstanding paper by Bateman and Katz, improving the best known bounds on the cap set problem (Roth's theorem over $\mathbb{F}_3^N$).

The paper contains some technical lemmas for which I believe there must be an excellent geometric intuition -- which I am afraid I am missing.

Excerpting from, and simplifying, Section 8 of their paper, let $A \subset Y := \mathbb{F}_3^n$

be some subset, and also write $A(x)$ for the characteristic function of $A$. Let $H$ be a subspace of $Y$ and let $H^{\perp}$ be its annihilator. For $h \in H$, write $A_{H, h} := A \cap (H^{\perp} + h)$. Then we have a version of Plancherel

$$\sum_{h \in H} |\widehat{A}(h)|^2 = |H| 3^{-2N} \sum_{h \in H} |A_{H, h}|^2,$$

and further, if $K$ is a subspace of $Y$ containing $H$,

$$\sum_{0 \neq k \in K} |\widehat{A}(k)|^2 = \sum_{0 \neq h \in H} |\widehat{A}(h)|^2 + \frac{1}{|H|} \sum_{h \in H} \sum_{0 \neq k \in K/H} |\widehat{A}_{H, h}(k)|^2.$$

There are other interesting related formulas as well. The authors remark that the latter equality "can simply be thought of as Plancherel for a 'local Fourier transform' of $A$. Here, we localize to the translates of $H^{\perp}$."

I can verify the identities readily enough, but I feel like there should be some excellent geometric intuition to be had here, with which all of these equalities are obvious. Is there anything that can be said which will render these equalities transparent? Perhaps some elaboration of the authors' remark?

Thank you!