Boundary spectral estimates for semiclassical Gevrey operators

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey- s regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size {O}(h1-1/s) where the resolvent is at most fractional exponentially large in h , as the semiclassical parameter h→ 0⁺ . This is a natural Gevrey analogue of a result by N. Dencker, J. Sjöstrand, and M. Zworski in the C∞ and analytic cases.