How is there a functional inequality that allows me to bound $$\int_x\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$$ in terms of a Sobolev norm of $f$, say?
If $f$ is equal to its Taylor series at $0$, we have $f(0)=f(x)-xf'(x)+\frac{x^2}{2}f''(x)-...$, so $$\frac{f(x)-f(0)}{x}=f'(x)-\frac{x}{2}f''(x)+\frac{x^2}{3!}f'''(x)-...,$$ but I can't see that this helps very much as this contains infinitely many derivatives of $f$.