Estimates with $L^n$ norm

33 Views Asked by At

this is my question:

Show that $\epsilon ||f||_{L^n(\Omega')}\leq \epsilon^{\frac{x}{n}}$, where $x$ is a positive number, using these two facts:

  • $\int_{\Omega}|f|^x\leq C $

  • $|f|\le\dfrac{C}{\epsilon}$ in $\Omega'$

with $\Omega'$ subset of $\Omega$.

I think that the trick is something like Holder inequality, but i can't state something useful.