An example of an expectation operator that is uniformly bounded Define the expectation operator $P_j : L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ as
$$
P_j f(x) := m_I := {1 \over |I|} \int_I f(t) \, dt
$$
where $I$ is of form $[2^{-j}k, 2^{-j}(k+1))$ (i.e., a dyadic interval) s.t. $x \in I$ (for each $x \in \mathbb{R}$, there is a unique $I$ for which $x \in I$).
According to an unproven lemma in a textbook on Harmonic Analysis, we have



Question: What's an example of a function $f \in L^2(\mathbb{R})$ (preferably one that only takes real values) where this inequality is strict? That is, where
$$
\|P_jf\|_2 < \|f\|_2 ?
$$
 A: Let $I_{j, k} := [2^{-j}k, 2^{-j}(k+1)) $. Observe we have
\begin{align}
P_j f(x) = \sum^\infty_{k=-\infty} \left(\frac{1}{|I_{j, k}|}\int_{I_{j, k}}f(t)\ dt\right)\chi_{I_{j, k}(x)}
\end{align}
then it follows
\begin{align}
\|P_jf\|^2_2=&\ \int^\infty_{-\infty}\left| \sum^\infty_{k=-\infty} \left(\frac{1}{|I_{j, k}|}\int_{I_{j, k}}f(t)\ dt\right)\chi_{I_{j, k}(x)} \right|^2\ dx\\
=&\ \int^\infty_{-\infty} \sum^\infty_{k=-\infty} \frac{1}{|I_{j, k}|^2}\left(\int_{I_{j, k}} f(t)\ dt\right)^2 \chi_{I_{j, k}}(x)\ dx\\
=&\ \sum^\infty_{k=-\infty} \frac{1}{|I_{j, k}|} \left(\int_{I_{j, k}} f(t)\ dt\right)^2 \leq \sum^\infty_{k=-\infty} \int_{I_{j, k}}|f(t)|^2\ dt =\|f\|_2^2.
\end{align}
Since the inequality is purely a consequence of Cauchy-Schwarz inequality, then it's not hard to concoct up an example where the inequality is strict.
Edit:  Consider
\begin{align}
f(x) = 2x\cdot \chi_{I_{j, 0}}(x),
\end{align}
then we see that
\begin{align}
P_jf(x) = 2^{-j}\chi_{I_{j, 0}}(x).
\end{align}
Then it follows
\begin{align}
\|P_j f\|_2=2^{-3j/2}  \ \ \ \text{ and } \ \ \ \|f\|_2 = \sqrt{\frac{4}{3}}2^{-3j/2}.
\end{align}
