Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-Lipschitz gradient. Define the norm of a function $g:\mathbb{R}^d\rightarrow R$:

$$\|g\|_p=\left(\int_{\mathbb{R}^d} |g(x)|^p \pi(x)dx\right)^\frac{1}{p}$$.

Question: For $p\geq q$, is there any inequality like the following: $$\|g\|_q\leq C\|g\|_p$$

where $C$ is a constant. By Jensen's inequality, $C=1$ satisfies the inequality. Is it possible to get a $C$ which uses the fact that the density is strongly log-concave and $f(x)$ has $L$-Lipschitz gradient?

Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-Lipschitz gradient. Define the norm of a function $g:\mathbb{R}^d\rightarrow R$:

$$\|g\|_p=\left(\int_{\mathbb{R}^d} |g(x)|^p \pi(x)dx\right)^\frac{1}{p}$$.

Question: For $p\geq q$, is there any inequality like the following: $$C_0\|g\|_p\leq \|g\|_q\leq C_1\|g\|_p$$

where $C_0,C_1$ are constants. By Jensen's inequality, $C_1=1$ satisfies the upper bound. Is it possible to get $C_0,C_1$ which uses the fact that the density is strongly log-concave and $f(x)$ has $L$-Lipschitz gradient?