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Let $\{f_k(x)\}_{k=0}^\infty$ be a Littlewood-Paley decompositon, that is, $$ f_k \in C_c^\infty $$ $$ \sum_{k=0}^\infty f_k (x) = 1,$$ $$ \text{supp} f_0 \subset \{ |x| \leq 2 \},$$ $$ \exists f \in C_c^\infty \; \text{such that}\; \text{supp} f \subset \{ 2^{-1} \leq |x| \leq 2 \} \; \text{satisfying}\; f_k (x) = f(x/2^k).$$ Then I hope to show that there exist $C, C'>0$ such that $$ C' \sum_{k=0}^\infty \| f_k(g) \|_{L^2}^2 \le \| g \|_{L^2}^2 \le C \sum_{k=0}^\infty \| f_k(g) \|_{L^2}^2$$ for $g \in L^2 (\mathbb R)$. Here $C_c^\infty$ means $C^\infty$ functions with compact support and $\text{supp}$ means the support.

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