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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.

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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.
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