I am trying to understand this paper, and have some basic question, and hope this is OK for the MO.
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: $$P_N^a:= \psi_N(\sqrt{\mathcal{L}_a})$$
My Basic Questions are: (1) How should I interpret $\psi_N(\sqrt{\mathcal{L}_a})f$ explicitly? (Do we have explicit formula like: $P_Nf= (\psi_N)^{\vee} \ast f$ ?) (2) How should I use Spectral Theorem to define $m(\sqrt{\mathcal{L}_a})f$ for $m:(0, \infty)\to \mathbb C$ some nice?
