Let $f:\mathbb R\to\mathbb [0,\infty)$ be a non-negative, twice-differentiable function. Suppose that
- $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$,
- $\int_{-\infty}^{\infty}|f''(x)|\,\mathrm dx<\infty$.
Does it necessarily follow that the first derivative is integrable as well: $$\int_{-\infty}^{\infty}|f'(x)|\,\mathrm dx<\infty$$ in Lebesgue’s sense (i.e., absolutely integrability, not merely the existence of improper integrals)?
More generally, if $f:\mathbb R\to\mathbb [0,\infty)$ is a non-negative function differentiable $k\in\mathbb N$ times ($k\geq 2$), and
- $\int_{-\infty}^{\infty}f(x)\,\mathrm dx<\infty$,
- $\int_{-\infty}^{\infty}|f^{(k)}(x)|\,\mathrm dx<\infty$,
then is it true that $$\int_{-\infty}^{\infty}|f^{(\ell)}(x)|\,\mathrm dx<\infty\quad\text{forall $\ell\in\{1,\ldots,k-1\}$}?$$
N.B.: continuity is not assumed on $f^{(k)}$.