Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz Space).
We know that $\widehat{\nabla f}(\xi)= 2 \pi i \xi \hat{f} (\xi). $ We define $$\widehat{|\nabla| f^{s}} (\xi) = (2 \pi |\xi|)^s \hat{f} (\xi), (s\in \mathbb R).$$
We may now define that $$(- \Delta) ^{s/2}f= |\nabla | f^s, (s\in \mathbb R).$$
We define Little wood-Paley projection associated to $-\Delta$
as follows: Let
$\phi:[0, \infty) \to \infty$
be a smooth functin such that
$\phi(\lambda) =1$ for $0 \leq \lambda \leq 1$ and $ \phi(\lambda)=0$ for $ \lambda \geq 2.$
For each dyadic number $N \in 2^{\mathbb Z},$ we define $\phi_N(\lambda) = \phi(\lambda/N)$ and $\psi_N(\lambda) = \phi_N(\lambda)- \phi_{N/2}(\lambda).$ We define the Little wood-Paley projection as follos:
$$\widehat{P_Nf} = \psi_N \hat{f}.$$
(And so $P_Nf= (\psi_N)^{\vee} \ast f$)
Now we define $\mathcal{L}_a$ as the Friedrich extension of the operator $-\Delta + \frac{a}{|x|^2}$ (initially defined on $C_c^{\infty}(\mathbb R^d\setminus \{0\})$) See Section 1.1 for details
Authors (in the same paper page 16) define the Little wood-Paley projections associated to $\mathcal{L}_a$ as follows: $$P_N^a:= \psi_N(\sqrt{\mathcal{L}_a}),$$
and associated to heat kernel $$\tilde{P}_n^a= e^{-\mathcal{L}_a/N^2}-e^{-4\mathcal{L}_a/ N^2}$$
[We may use Spectral Theorem (and page 8) to define $m(\mathcal{L}_a)]$
My Basic Questions: (1) Can we say $P_n^a=m(\mathcal{L}_a) \tilde{P}_N^a$ ? (In other words, how these Lilltewood-Paley projections are related?) (2) Is $P_n^ae^{-it\mathcal{L}_a} f=e^{-it\mathcal{L}_a} P_N^a f$ ? Is $\nabla e^{-it\mathcal{L}_a} f = e^{-it\mathcal{L}_a}\nabla f$? (In other words, when Schroedinger propagator commutes with Littlewood Paley projections)
Note. See also Theorem 1.1 here
