According to Riemann surfaces, dynamics and geometry by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by $$ \|\phi\|_p = \left(\int_X \rho^{2-2p} |\phi|^p\right)^{1/p}, \|\phi\|_\infty = \sup_X \frac{|\phi|}{\rho^2}, $$

where $\rho |dz|$ is the hyperbolic metric.

For a Beltrami coefficient $\mu,$ $$ \|\mu\|_p = \left(\int_X \rho^{2} |\mu|^p\right)^{1/p}, \|\mu\|_\infty =\sup_X |\mu|. $$

I am confused about the mysterious exponents like $2-2p$ and I am not sure how the definition of $L^p$ can pass "smoothly" to $L^\infty.$ I also do not know why we are dividing $\rho^2$ in the first expression but not in the second one.

**Why must we put that certain power of $\rho$ into the expression?**