Some boundary fractal properties of the convolution transform of measures by an approximate identity

We consider the convolution transforms of measures on ℝ ^d defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution function and the local Lipschitz exponent of the measure. In particular, the results can be applied to the Poisson and Gauss-Weierstrass kernels. We can then obtain some singular boundary behavior of positive harmonic or parabolic functions on ℝ ₊ ^d+1 by multifractal analysis of measures. Research supported by the NNSF of China