It is known that for a Lipschitz function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ that if $X \sim \mathcal{N}(0,I_n)$ then $$ \| f(X) - \mathbb{E}f(X)\|_{\psi_2} \leq C \|f\|_{Lip} $$ where $\| \|_{\psi_2}$ refers to the subgaussian norm (i.e. the smallest constant $C$ such that $\mathbb{E} \exp(X^2/C^2) \leq 2$). Is it possible to replace the $\mathbb{E}f(X)$ by $(\mathbb{E} f(X)^3)^{1/3}$ and still get a concentration inequality of the same form.
I have no problem replacing the expectation by the median as it is well known that the expectation and median are close for subgaussian random variables. I would like to prove the same phenomenon for Lp norms.