Is anybody aware of an inequality in the following form $$ \Vert f \Vert_{L_p(\Omega)} \leq C(p) \Vert f \Vert_{L_q(\Omega)} $$ where $f$ is a polynomial function of degree $p$ on $\Omega \subset \mathbb{R}^d$. For sure the Holder inequality can be applied here but what I am looking for is whether or not some $p$ dependency is hidden in the constant that appears there?

I'm somehow doubtful that this can be done in general since the constant appears in Holder inequality looks quite sharp but I wanted to make sure of my guess before looking for other solution. Thank you.


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