Suppose I have an $n$-dimensional probability space $X$ (e.g. a product space or an $n$-dimensional compact group $G$) for which there exists a concentration of measure result for Lipschitz functions $f: X \to \mathbb{R}$ that depends on the Lipschitz constant $L$.
Suppose further that $f$ has large Lipschitz constant $L_1$ only "in one direction" (e.g. on one coordinate of the product space or "along one direction" on $G$; the latter is purposefully vague ) and small Lipschitz constants $L_2$ in other directions, $L_1 \gg L_2$. Can one show concentration result with concentration constant depending on some sort of average of $L_1$ and $L_2$ instead of the "worst case" constant $L_1$?