Littlewood-Paley theory and dense property of Sobolev spaces [duplicate]

Question

I'm learning the Littlewood-Paley theory by myself and I encounter the following claim:

Pick a smooth function $\chi$ such that: $$\chi(\xi) = \begin{cases} 1 &|\xi| \leq \frac{1}{2}\\ 0 &|\xi| \geq 1 \end{cases}$$ Based on the function $\xi$ picked above, let's define an operator $S_{k} \ (\forall \ k \in \mathbb{Z})$ acting on an arbitrary function $f$ as follows: $$S_{k}f(x) = \left(\chi\Big(\tfrac{\xi}{2^{k}}\Big)\hat{f}(\xi)\right)^{\vee}(x)$$ where $\hat{}$ and $\vee{}$ above denotes Fourier transform and inverse Fourier transform, respectively. Moreover, based on $\xi$, we can also define the Littlewood-Paley projection operators $P_{l} \ (\forall \ l \in \mathbb{Z})$.

Now for any $s \in \mathbb{R}$ and any $k \in \mathbb{Z}^{+} \cup \{0\}$, we have the following inequality (equivalence of norms) for any function $f$ in the Sobolev space $H^s$: $$c^{-1}\|f-S_{k}f\|_{H^{s}}^2 \leq \sum_{l \geq k}2^{2ls}\|P_{l}f\|_{L^2}^2 \leq c\|f-S_{k}f\|_{H^{s}}^2$$ Above $c=c(s) \geq 1$ is some constant dependent on $s$. From the inequality above (equivalence of norms), we can further deduce that $\cap_{t \in \mathbb{R}}H^{t}$ is dense in $H^{s}$.

I roughly get the idea of deducing the dense property from the inequality listed above. However, I'm bit stuck with proving the inequality... it feels that $S_{k}f$ is some other form of "projection", so I guess probably "almost orthogonality" will be useful here? Any help/hint will be appreciated!

Answer 1

By Plancherel

$$ \| f - S_k f\|_{H^s}^2 \approx \int (1 + |\xi|^2)^{s} (1 - \chi(2^{-k}\xi))^2 |\hat{f}|^2 ~d\xi $$

By almost orthogonality you can write

$$ (1 - \chi(2^{-k}\xi))^2 \approx \sum_{\ell = k}^{\infty} (\underbrace{\chi(2^{-\ell-1}\xi) - \chi(2^{-\ell}\xi)}_{\phi_\ell(\xi)})^2 $$

as the telescoping sum. On the support of $\phi_\ell(\xi)$, you have that

$$ |\xi| \approx 2^{\ell} $$

So

$$ \| f - S_k f\|_{H^s}^2 \approx \int \sum_{\ell = k}^\infty (1 + 2^{2s\ell}) \phi_\ell^2 |\hat{f}|^2 ~d\xi $$

As each $\ell \geq k \geq 0$ you have that $1 + 2^{s\ell} \approx 2^{s\ell}$ and so you finally get

$$ \| f- S_k f\|_{H^s}^2 \approx \sum_{\ell = k}^\infty 2^{2sl} \| P_\ell f\|_{L^2}^2 $$