I am reading Tao's lecture note on Littlewood-Paley decomposition.
Let $\phi(\xi)$ be a bump function supported on $\{\xi\in\mathbb{R}^n:|\xi|\leq 2\}$ and equals $1$ on $\{\xi\in\mathbb{R}^n:|\xi|\leq 1\}$. Let $\varphi(\xi)=\phi(\xi)-\phi(2\xi)$, we define the Littlewood-Paley projection operator $P_k,P_{\leq k}$ by
$\hat{P_k(\xi)}=\varphi(\frac{\xi}{2^k})\hat{f(\xi)}, \hat{P_{\leq k}(\xi)}=\phi(\frac{\xi}{2^k})\hat{f(\xi)}$.
Thus $\hat{P_{\leq k}(f)}=\int f(x+2^{-k}y)\hat{\phi}(y)dy$. I don't understand what the following two sentences mean and how it relates to uncertainty principle
The function $P_{\leq k}f$ is an average of $f$ localized to physical scales $\lesssim 2^{-k}$ and we expect $P_{\leq k}f$ to be essentilly constant at scales $\ll 2^{-k}$ which is consistent with uncertainty principle
Purely heuristic, but can be formalize. The uncertainty principle essentially states: $\Delta x \Delta \xi \geq C>0$. Since $P_{\leq k} f$ localizes the frequencies of $f$ to $|\xi| \lesssim 2^{k}$ then the corresponding $x$ values are concentrated in the region $|x|\gtrsim 2^{-k}$. In particular, when $|x|\ll 2^{-k}$, there are no corresponding frequencies, i.e. $P_{\leq k}f$ is essentially constant.