For $j,k \in \mathbb{Z}$, $m \in \mathbb{Z}$ and $c > 0$ define $$ \psi_{\lambda}(x) := \psi_{j,k,m}(x_1,x_2) := 2^{3j/4} \psi(S_k A_{2^j} x - cm), $$ where $\psi \in L^2(\mathbb{R}^2)$ and $$ A_{2^j} := \begin{pmatrix} 2^{j} & 0 \\ 0 & 2^{j/2} \end{pmatrix} \qquad \text{and} \qquad S_k := \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} $$ In "Compactly supported shearlets are optimally sparse" by Prof. Kutyniok and Wang Q.-Lim on page 1576 (section 2.3) it says that if we assume $\| \psi \|_1 < 1$ we have $$ | \langle f, \psi_{\lambda} \rangle | \le 2^{-3j/4}, $$ where $f \in \mathcal{C}^2(\mathbb{R}^2)$ with $\| f \|_{\mathcal{C}^2} = \sum_{| \alpha | \le 2} \| D^{\alpha} f \|_{\infty} \le 1$.
My try: We have \begin{align} | \langle f, \psi_{\lambda} \rangle | & = 2^{3j/4} \left|\int_{\mathbb{R}^2} f(x_1, x_2) \psi(S_k A_{2^j} x - cm) dx \right| \\ & \le 2^{3j/4} \| f \|_{\mathcal{C}^2} \| \psi(S_k A_{2^j} \cdot - cm) \|_{L^1(\mathbb{R}^2)} \tag{?} \\ & \le 2^{3j/4} \underbrace{\| f \|_{\mathcal{C}^2}}_{\le 1} \cdot 2^{-3j/2} \underbrace{\| \psi \|_{1}}_{< 1} \le 2^{-3j/4}. \end{align} I am very unsure of step (?). Is it correct? If no, can we take a similar approach?
What you've written is fine. From the comments, the key clarification is that since $\|f\|_\infty \leq \|f\|_{C^2}$ we can conclude that $$\int_{\mathbb{R}^2} |f(x_1,x_2) \psi(S_k A_{2^j} x - cm)| dx \leq \int_{\mathbb{R}^2} \|f\|_\infty |\psi(S_k A_{2^j} x - cm)| dx \leq \|f\|_{C^2} \|\psi(S_kA_{2^j} \cdot - cm) \|_1.$$