  The Loewner equation provides a correspondence between continuous real-valued
functions $\lambda_t$ and certain increasing families of half-plane hulls
$K_t$. In this paper we study the deterministic relationship between specific
analytic properties of $\lambda_t$ and geometric properties of $K_t$. Our
motivation comes, however, from the stochastic Loewner equation
(SLE$_{\kappa}$), where the associated function $\lambda_t$ is a scaled
Brownian motion and the corresponding domains $\mathbb{H} \backslash K_t$ are
H\"older domains. We prove that if the increasing family $K_t$ is generated by
a simple curve and the final domain $\mathbb{H} \backslash K_T$ is a H\"older
domain, then the corresponding driving function has a modulus of continuity
similar to that of Brownian motion. Informally, this is a converse to the fact
that SLE$_{\kappa}$ curves are simple and their complementary domains are
H\"older, when $\kappa < 4$. We also study a similar question outside of the
simple curve setting, which informally corresponds to the SLE regime $\kappa >
4$. In the process, we establish general geometric criteria that guarantee that
$K_t$ has a Lip$(1/2)$ driving function.
