Let $K > 0$. Let $\mathcal{C} = \mathcal{C}^1([0,K], [-1,1] )$ be the space of $C^1$ functions $f : [0, K] \rightarrow [-1,1]$, such that $f(0)=0$. Let's use the norm:
$$||f||_2 = \sqrt{\int_0^K f(t)^2 d t}$$
and also the total variation:
$$||f||_T = \int_0^K |f'(t)| d t$$
Question: are there constants $a$ and $b$ such that:
$$||f||_T \leq a ||f||_2 \qquad \text{and} \qquad ||f||_2 \leq b ||f||_T$$
for all functions $f \in \mathcal{C}$ ?
Discrete analogous question:
Let $N > 0$ and $\mathcal{D}$ be the space of all sequences $(y_n)_{0 \leq n \leq N}$, with $-1 \leq y_n \leq 1$ for all $n$, and $y_0 = 0$. We use the norms:
$$||(y_n)||_2 = \sqrt{\sum_{n=0}^N y_(n)^2}$$
and
$$||(y_n)||_T = \sum_{n=1}^N |y(n)-y(n-1)|$$
Are there constants $c$ and $d$ such that:
$$||(y_n)||_T \leq c ||(y_n)||_2 \qquad \text{and} \qquad ||(y_n)||_2 \leq d ||(y_n)||_T$$
for all sequences $(y_n) \in \mathcal{D}$ ?
Note:
Ideally, the constants $a$, $b$ should be independant of $K$. The same for the discrete case.
If not true in general on $\mathcal{C}$, would it be possible to find such inequalities on a subspace $\mathcal{C}^*$ of functions that have a limited bandwidth in high frequencies? (let's say the spectrum has no or few composant of frequency $f \geq f_0$; in my audio application: $f_0$ is $20 000$ Hz).
For the second inequality:
$$\int_0^K f(t)^2\,dt = \int_0^K \left (\int_0^t f'(s)\,ds\right)^2\,dt$$ $$ \le \int_0^K \left (\int_0^t |f'(s)|\,ds\right)^2\,dt\le \int_0^K \|f\|_T^2\,dt = K\|f\|_T^2.$$
Take square roots to see $\|f\|_2 \le \sqrt K \|f\|_T.$