Hölder regularity of the SLE trace

S. Rohde and O. Schramm have recently shown that the SLE trace is Hölder continuous (2005). However, their results are not optimal for all values of and only yield a Hölder exponent near for near 0. In this paper, we give improved lower bounds on the optimal Hölder exponent for two natural parametrizations of the SLE trace. Our estimates give a Hölder exponent near 1 for near 0, as expected. The work of I. Binder and B. Duplantier (2002) suggests that our results are optimal for the two parametrizations considered.