on $\left\{ \xi \in \mathbb{R}^{3}: \, |\xi| \leq 2 \right\}$. Let $\psi$ denote the function

$$\begin{array}{ll}
\psi(\xi) & := \phi(\xi) - \phi(2 \xi)
\end{array}$$

If $(M,M_{1},M_{2}) \in 2^{\mathbb{N}}$ are dyadic numbers such that $M_{2}>M_{1}$ we define the Paley-Littlewood operators in the Fourier domain by

-   if $M > 1$

    $$\begin{array}{ll}
    \widehat{P_{M} f}(\xi) & := \psi \left( \frac{\xi}{M} \right) \hat{f}(\xi) \
    \end{array}$$

-   if $M=1$

    $$\begin{array}{ll}
    \widehat{P_{1} f }(\xi) & := \phi \left( \xi \right) \hat{f}(\xi)
    \end{array}$$

and

$$\begin{array}{ll}
\widehat{P_{\leq M} f}(\xi) & := \phi \left( \frac{\xi}{M} \right) \hat{f}(\xi) \\
\widehat{P_{> M} f}(\xi) & := \widehat{f}(\xi) - \widehat{P_{\leq M} f}(\xi) \\
\widehat{P_{ << M } f}(\xi) & : = \widehat{P_{ \leq \frac{M}{128} }f}(\xi) \\
\widehat{P_{ \gtrsim  M } f}(\xi) & : = \widehat{P_{ > \frac{M}{128}}f}(\xi) \\
P_{M_{1} < . \leq M_{2}} f  &  := P_{ \geq M_{2}} f -  P_{< M_{1}} f \\
\end{array}$$

Since $ \phi(\xi) + \sum_{M \in  2^{\mathbb{N}^{*}}} \psi \left( \frac{\xi}{M} \right)=1$ we have

$$\begin{array}{ll}
f & = \sum_{M \in 2^{\mathbb{N}}} P_{M} f
\end{array}$$

Notice also that

$$\begin{array}{ll}
f & = P_{\lesssim M}f + P_{\gtrsim M} f
\end{array}$$

Throughout this paper we constantly use Strichartz-type estimates (see [@ginebvelo; @nakamura; @triroyradnlkg] )

[**“Strichartz estimates for Klein-Gordon equations in $L_{t}^{q} 