Caloric measure on domains bounded by Weierstrass-type graphs

Abstract

Let ! be the caloric measure on a domain bounded by a Weierstrass-type graph. We prove that ! is a quasi-Bernoulli measure. Therefore, the multifractal formalism is available for such a measure. We can then give necessary and su-cient conditions which ensure that ! is supported by a set of small dimension and prove that this is the generic case. These results improve and generalize an earlier work due to R. Kaufman and J.M. Wu. Let f: R ! R be a real function of class C 1=2 (i.e., Holder of exponent 1=2) and › = f(x;t) 2 R 2 : x > f(t)g. For every point M0 = (x0;t0) 2 ›, let ! M0 be the harmonic measure at M0 with respect to the heat operator C = @=@ti@ 2 =2@x 2 . This measure is called the caloric measure at M0 . It is supported by the set of points (x;t)2 @› such that tt0 and may also be deflned by ! M0 (E) = Px0 i T < +1 and (BT;t0iT )2 E ¢ ;