The Hausdorff measure of the level sets of Brownian motion on the Sierpinski carpet

Let L(t)(x) {x is an element of F, t > 0} be the local time of Brownian motion B on the Sierpinski carpet F, and phi(h) = h(beta)(log\logh\)(1-beta), For Allh is an element of (0, 1/4], beta is a constant. In this paper, we show that for each x is an element of F. cL(t)(x) less than or equal to phi - m{s : s less than or equal to t, B(s) = x} less than or equal to CL(t)(x), a.e. For Allt > 0. for some constants c and C is an element of (0, infinity).