# Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

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?

The powers of $$\rho$$ are necessary to make the integrals well-defined. In probably excessive detail:

1. Because $$X$$ is a Riemann surface, the integrands here need to be $$(1,1)$$-forms. This is just the familiar fact (change of variables theorem) that integrals over (oriented) manifolds are only well-defined on objects $$\omega$$ which are given in local coordinates by functions $$\omega(x)$$ satisfying the transformation rule $$\omega(x) = \det(D\psi_x) \omega(y)$$ whenever $$y = \psi(x)$$ a smooth change of coordinates. If $$M = X$$ is a Riemann surface, the coordinate changes $$w = \psi(z)$$ are holomorphic, so that $$\det(D\psi) = |\psi'|^2$$ (Cauchy-Riemann), and thus the above rule can be rewritten as $$\phi(z) = |\psi'(z)|^2 \phi(w) = \psi'(z) \overline{\psi'(z)} \ \phi(w).$$ This condition can be summarized in the language of Riemann surface $$(n, m)$$-tensors (see Lyubich's book, p. 99) by saying that $$\phi$$ can be integrated over $$X$$ iff it is a $$(1,1)$$-form, i.e. an object with local form $$\phi = \phi(z) \ dz \ d\bar{z} = \phi(z) \ |dz|^2.$$

2. Quadratic differentials are $$(2, 0)$$-forms. Beltrami forms are $$(-1, 1)$$-forms. Take $$\phi$$ a quadratic differential. Then $$|\phi|^p$$ is a $$(p, p)$$-form. In local coordinates: $$\phi = \phi(z) \ dz^2 \implies |\phi|^p = |\phi(z) dz^2|^p = |\phi(z)|^p |dz|^{2p} = |\phi(z)|^p dz^p d\bar{z}^p.$$ In particular, $$|\phi|$$ is a $$(1, 1)$$-form, so can be integrated over $$X$$. This shows that the $$L^1$$-norm on quadratic differentials is canonically defined, even when $$X$$ is not hyperbolic (and thus has no preferred conformal metric), in the way you'd expect: $$||\phi||_1 = \int_X |\phi|.$$ When $$p \neq 1$$, however, $$|\phi|^p$$ is not a $$(1,1)$$-form, so its integral over $$X$$ is not defined, and thus the naive definition of the $$L^p$$-norm for quadratic differentials will not work! Fortunately, this issue can be remedied in the presence of a conformal metric $$\rho = \rho(z) |dz|$$, because we can adjust the tensor type of our candidate integrand using powers of $$\rho$$. In particular, multiplying by $$\rho^{2-2p}$$ gives $$\rho^{2-2p} |\phi|^p = \rho(z)^{2-2p} |dz|^{2-2p} |\phi(z)|^p |dz|^{2p} = \rho(z)^{2-2p} |\phi(z)|^p |dz|^{2},$$ which is a $$(1,1)$$-form, and so can be integrated over $$X$$. Therefore we can define an $$L^p$$-norm on quadratic differentials by the expression you gave above: $$||\phi||_p = \left( \int_X \rho^{2-2p} |\phi|^p \right)^{1/p}.$$ Moreover, though this definition appears to depend on an arbitrary choice of $$\rho$$, when $$X$$ is a hyperbolic Riemann surface there is a canonical choice, namely the unique conformal metric of constant curvature -1. So, in the hyperbolic case with $$\rho$$ taken to be this distinguished metric, it makes sense to refer to the above definition as the $$L^p$$-norm on quadratic differentials. The Beltrami case works similarly.

The construction here is exactly analogous to the problem of defining $$L^p$$-norms for spaces of functions on a smooth manifold $$M$$. The $$L^\infty$$-norm of a function is, of course, defined as usual by $$||f||_\infty = \displaystyle\text{ess sup}_M |f|.$$ But without further structure, the $$L^p$$-norms for $$p < \infty$$ are not well-defined, precisely because functions cannot be integrated over $$M$$. After fixing a Riemannian metric on $$M$$, however, the $$L^p$$-norm of a function can be defined in the obvious way by integrating against the associated volume form: $$||f||_p = \left( \int_M |f|^p \ \text{dVol} \right)^{1/p}$$ Taking this approach in our context leads to the same result as above. That is, we could have alternatively begun by modifying $$\phi$$ by $$\rho$$ to obtain a function on $$X$$, then applied the Riemannian definition given above to our special case. In detail, the local computation $$\dfrac{|\phi|}{\rho^2} = \dfrac{|\phi(z)||dz|^2}{\rho(z)^2 |dz|^2} = \dfrac{|\phi(z)|}{\rho(z)^2},$$ shows that $$\dfrac{|\phi|}{\rho^2}$$ is a function on $$X$$. Then we can define $$||\phi||_\infty = \displaystyle\text{ess sup}_X \dfrac{|\phi|}{\rho^2}$$ and, using the volume form $$\rho^2 = \rho(z)^2 |dz|^2$$ associated to the conformal metric $$\rho$$, define $$||\phi||_p = \left( \int_X \left( \dfrac{|\phi|}{\rho^2} \right)^p \ \rho^2 \right)^{1/p} = \left( \int_X \rho^{2-2p} |\phi|^p\right)^{1/p}$$ for $$p < \infty$$.