Can one control $\int(f'(x))^2$ by $\int f'(x)+f(x)$?

Question

For a function $f(x)$ continuously differentiable and defined on [a,b] with $f(a)=f(b)=0$, can one control $\{\int_a^b[f'(x)]^2dx\}^{1/2}$ by for example $\int_a^b|f'(x)|dx+\{\int_a^b|f(x)|^pdx\}^{1/p}$ for some $p\geq1$? The important issue is to control a higher power of $f'(x)$ via a lower power of itself, with possible involvement by the function $f(x)$. Thank you!

Answer 1

The answer is negative. An integral of the form $\int_a^b |f^{(k)}|^p$ may be controlled by $\int_a^b |f^{(j)}|^q$ when either

• $j>k$ and $q\ge 1$ (using the fundamental theorem of calculus)
• $j=k$ and $q\ge p$ (using Hölder's inequality)

In your case, $f(x)=\sqrt{x+\epsilon}$ on $[0,1]$ provides a counterexample: as $\epsilon\to 0$, $\int (f')^2$ blows up while the other integrals stay bounded.