Directional integration by parts on a fundamental parallelepiped

Let $E$ be a finite-dimensional real normed space, equipped with a Borel measurable structure and an additive Haar measure $\mu$, let $b$ be a basis of $E$ indexed by a finite type $\iota$, and let $\ell : E \to \mathbb{R}$ be a continuous linear form. Let $v \in E$ satisfy $\ell(v) \neq 0$, and let $H_0, H_1, H_2, \dots : E \to \mathbb{C}$ be a sequence of continuous functions such that for every $j$ and every $x \in E$ the function $t \mapsto H_j(x + t v)$ of a real variable has derivative $H_{j+1}(x)$ at $t = 0$, and such that for every $j$, every $x \in E$ and every index $i$ one has $e^{2\pi i \ell(x + b_i)} H_j(x + b_i) = e^{2\pi i \ell(x)} H_j(x)$, i.e. each product $x \mapsto e^{2\pi i \ell(x)} H_j(x)$ is invariant under translation by each basis vector. Then for every natural number $M$,

where $\mathcal{F} =$ ZSpan.fundamentalDomain b is the fundamental parallelepiped $\{\sum_i t_i b_i : 0 \le t_i < 1\}$ of the lattice spanned by $b$, and the inverse is taken in $\mathbb{R}$ (so the factor is $1$ when $M = 0$).

This is the standard decay estimate for Fourier coefficients of smooth functions on a torus, here in a directional form: differentiation is performed only along the single vector $v$, and only the products $e^{2\pi i \ell} H_j$, not the character and the functions $H_j$ separately, are assumed periodic under the lattice. It is used to bound Whittaker coefficients of automorphic forms, via AutomorphicForm.exists_norm_whittakerCoefficient_le_mul_of_hasDerivAt_unipotentGL2_of_forall_norm_le.