Does Poisson Kernel for fractional Laplacian $(-\Delta)^s$ exists for smooth bounded domain with two boundaries. If so, is there a upper and lower bound for the Poisson kernel.
Sure it does! Estimates of the Poisson kernel (and closely related bounds for the Green function) were studied already in the 90s by two independent groups of people: Bogdan, Byczkowski and Kulczycki, and Chen and Song. You can find the estimate you need in Theorem 1.5 in:
- Z.-Q. Chen, R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes. Math Ann 312, 465–501 (1998), DOI:10.1007/s002080050232.