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Consider the $L^p$ norm defined on some bounded open subset $\Omega\subset\mathbb{R}^n$ with $p>1$. Does there exist a constant $c$ such that $$\Vert f\Vert_{L_p(\Omega)} \le c\Vert f\Vert_{L_\infty(\Omega)} $$ for any function $f\in L_\infty(\Omega)$?

I think this is similar to this question, but the inequality is reversed and I am considering the norms over a bounded domain.

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$$\int_{\Omega}|f|^p \le \int_{\Omega}||f||_{\infty}^p = m(\Omega) ||f||_{\infty}^p$$ hence, taking $p$-th roots, $$||f||_p \le m(\Omega)^{1/p} ||f||_{\infty}$$

Finally, $c=m(\Omega)^{1/p}$ is optimal, since the inequality becomes an equality for constant function $f$.

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