Heat Kernel Estimates for Truncated Stable-like Processes and Weighted Poincaré Inequality

In this paper, we investigate pure jump symmetric processes in R whose jumping kernel is comparable to the one for truncated rotationally symmetric α-stable process where jumps of size larger than a fixed number κ > 0 is removed. We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for such jump-type processes. Moreover, we show that bounded functions that are parabolic with respect to the jump process are jointly Hölder continuous. Along the way, some heat kernel lower bound estimate is obtained for more general truncated jump processes that were studied in [BBCK]. We also present a new form of weighted Poincaré inequality of fractional order which played a crucial role in obtaining the lower bound heat kernel estimate. AMS 2000 Mathematics Subject Classification: Primary 60J75 , 60J35, Secondary 31C25 , 31C05. Running title: Heat Kernel Estimate for Truncated Stable Process.