Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in Rn. If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order