Let $P$ be an orthogonal projector on $C^\infty([0,1])$.
For $0<p<\infty$, we define for $f \in C^\infty$ the norm (quasi-norm if $p<1$) $\lVert f \rVert_p$ in the usual way. Moreover, we define $$\lVert P\rVert_p := \sup \{\lVert Pf\lVert_p \ | \ \lVert f \rVert_p \leq 1 \}.$$
Question: Do we have, for every $0<p<\infty$, $\lVert P \rVert_p <\infty$? Actually, is it even true that $Pf$ is in $L_p$ if $f$ is in $L_p$?
Case 1: $p<1$
Consider the function
\begin{equation*} f(t) = \begin{cases} L^{1/p}, & t<L^{-1} \\ 0, &t \geq L^{-1} \end{cases} \end{equation*} with $L_p$ quasinorm $1$. Its Fourier coefficients are \begin{equation*} c_N = \left< f, e^{2\pi i N \cdot}\right> = \frac{-i}{2\pi} \frac{L^{1/p}}{N} \left(1 - e^{-2\pi i N/L} \right). \end{equation*} Moreover, the $L_p$ quasinorm of the exponential is \begin{equation*} \|c_Ne^{2\pi i N \cdot}\|_{L_p} = \frac{1}{\pi} \frac{L^{1/p}}{N} | \sin(\pi N /L) | \end{equation*}
Let $P$ be the projector mapping $e^{-2\pi i N_0\cdot}$ to $0$, for some fixed $N_0$. If we let $L\rightarrow \infty$, we can see that the projector is not bounded on $L_p$ for $p<1$. Indeed,
\begin{align*} \|Pf\|_{L_p([0,1])} &= \|f-c_{N_0}e^{2\pi i N_0 \cdot}\|_{L_p([0,1])} \\ &\geq \|c_{N_0}e^{2\pi i N_0 \cdot}\|_{L_p([1/2,1])} \\ &=\frac{1}{2\pi} \frac{L^{1/p}}{N} | \sin(\pi N /L) |\\ & \rightarrow \infty. \end{align*}
Remark:
Of course $f$ is not smooth, but we can obtain the same result with a smooth approximation to $f$.
Case 2: $p=1,\infty$
Some $L_2$ projectors are bounded. For example, Shadrin proved the conjecture of de Boor that spline projectors are bounded on $L_\infty$ (cf. Shadrin, Acta Mathematica, 2001). Also, every idempotent measure corresponds to a bounded projector for $p=1,\infty$.
However, there are also unbounded projectors. For simplicity, consider the domain to be the circle $\mathbb{T}$. The idempotent measures have Fourier transforms that take the values 0 or 1. Helson (cf. Helson, Proc. AMS, 1953) has shown that if the location of the zeros satisfy the following property P1, then the tempered distribution is not a finite measure, and hence the projector is unbounded:
P1: the sequence in $\mathbb{Z}$ cannot be made periodic by adding or dropping a finite number of elements.
Rudin generalized this result to $\mathbb{T}^d$, showing that an idempotent measure $\mu$ should have a Fourier transform $\widehat{\mu}$ supported on the coset ring of $\mathbb{Z}^d$, (cf. Rudin, Pacific J. Math., 1959).