The Loewner equation and Lipschitz graphs
Solutions \(g_t(z)\) to the chordal Loewner differential equation \[ \partial_tg_t(z)=\frac{2}{g_t(z)-\lambda_t},\;\;\;g_0(z)=z,\;\;\;\lambda_t \in\mathbb R, \] satisfying the hydrodynamic normalization at infinity, map decreasing families of simply connected domains \(\mathbb H\setminus K_t\) onto the upper half-plane \(\mathbb H=\{z\in\mathbb C:\text{Im}\,z>0\}\). For simple curves \(\gamma\subset\mathbb H\cup\{0\}\), \(\gamma_0=0\), the family \(K_t=\gamma[0,t]\) yields continuous functions \(\lambda_t\) while the converse is not true in general. The stochastic setting means that \(\lambda_t=\sqrt{\kappa}B_t\) and \(B_t\) is a standard one-dimensional Brownian motion. The authors adopt the method developed in the stochastic version to the deterministic setting and obtain a new and elementary proof of the following known theorem. Theorem 1.1. If the driving function \(\lambda\) has Hölder-1/2 norm less than 4, then the chordal Loewner equation generates a simple curve \(\gamma\). Moreover, \(\gamma\) is a quasiconformal arc that meets \(\mathbb R\) non-tangentially. The authors' approach gives a sufficient condition for the driving functions to generate the graph of a Lipschitz function. Theorem 1.2. There exists a constant \(C_0>0\) such that, for every continuous function \(\lambda\) satisfying \[ \int_0^t\frac{N_{s,t}^{\lambda}}{(t-s)^{3/2}}ds\leq C_0\;\;\text{for all}\;0<t<T, \] where \(N_{s,t}^{\lambda}=\sup\{|\lambda_r-\lambda_s|:s\leq r\leq t\}\), the Loewner equation generates a graph of a Lipschitz function. The key point is Theorem 3.1 regarding the trajectories \(z_t=x_t+iy_t=f_t(i)-\lambda_t\) of the upward Loewner equation \[ \partial_tf_t(z)=\frac{-2}{f_t(z)-\xi_t},\;\;\;f_0(z)=z, \] for \(z\in\mathbb H\) and real-valued continuous function \(\xi_t\). Let \(\sigma:=\|\xi\|_{1/2}<4\) and denote \(A_c\) the cone \(\{x+iy:|x|\leq cy\}\). Theorem 3.1. There is a constant \(c_{\sigma}\) such that, if \(z_0=iy\), then \(z_t=f_t(z_0+\xi_0)-\xi_t\) stays in the cone \(A_{c_{\sigma}}\) for all \(t\). Moreover, \[ \sqrt{\frac{4t}{1+c_{\sigma}^2}+y^2}\leq y_t\leq\sqrt{4t+y^2} \] for all \(t\geq0\), and \(c_{\sigma}\leq\sigma/\sqrt{4-\sigma^2}\) for \(\sigma<2.\)
scientific article; zbMATH DE number 6897392