Heat kernel estimates for stable-like processes on <f>d</f>-sets

The notion of <f>d</f>-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of <f>d</f>-sets. In this paper stable-like processes on <f>d</f>-sets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the range of stable-like processes are also obtained. [Copyright &y& Elsevier]