I am delving into an analysis involving a specific integral inequality that arises within the context of Fourier analysis, particularly focusing on expressions involving a series of integrations over expanding nested sets. The core inequality I am wrestling with is presented as follows: Let $\varphi_0$ and $\varphi$ be $C^\infty$ complex-valued functions respectively on $\mathbb{R}^n$ and $\mathbb{R}^n \setminus \{0\}$ that satisfy the Tauberian conditions: \begin{equation}\label{taub} |\varphi_0(x)| > 0\, \text{ if } \, |x| \leq 2 \hspace{1cm} \text{and} \hspace{1cm} |\varphi(x)| > 0 \, \text{ if } \, \frac{1}{2} \leq |x| \leq 2. \end{equation} Define $\varphi_j(x) = \varphi(2^{-j}x)$ for $x \in \mathbb{R}^n \setminus \{0\}$ and $j \in \mathbb{N}$.
Consider a function $f \in B^s_{p,q}$ along with associated dyadic partition of unity $\{\rho_m(x)\}_{m=0}^{\infty}$;$\psi(x)\in\mathscr{S}$ with $\text{supp}\psi\subset\{y: |y|\leq2^{K+1}\}$ and $\psi(x)=1$ se $|x|\leq2^K$. I am interested in proving the following inequality: \begin{equation} \int (1+|2^j y|)^{-b}\left| \mathscr{F}^{-1}(\varphi_j \psi(2^{-j} \cdot) \hat{f})(x-y) \right|^r dy\leq c \sum_{l=0}^\infty 2^{-ld} \int_{\{y : |y| \leq 2^{-j+l}\}} \left| \mathscr{F}^{-1}(\varphi_j \psi(2^{-j} \cdot) \hat{f})(x-y) \right|^r dy, \end{equation} where $c$ is a constant, $d > 0$, and $K$ is a chosen natural number.
My challenge lies in formally justifying each step that leads to this inequality, especially elucidating how the decay term $2^{-ld}$ integrates within the broader context to ensure convergence and proper accounting for the nested integrations. The assumptions regarding the functions $\varphi_j$ and $\rho_m$ are standard within the analysis context, focusing on their roles in decomposing the function $f$ across different scales.
I would greatly appreciate any insights, rigorous mathematical justifications, or references that could aid in establishing the validity of this inequality.
Thank you very much for your assistance!