Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients (Q1883423)

This paper deals with the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients. Let \[ Au(x)=\sum_{|\alpha|,|\beta|\leq m}D^{\alpha}(a_{\alpha\beta}(x)D^{\beta}u(x)) \] be a formally self-adjoint, strong elliptic operator in \({\mathbb R}^{n}\). Suppose that all the coefficients of \(A\) are bounded and the coefficients of top order are Hölder continuous of exponent \(\tau\in (0,1]\). Denote by \(a(x,\xi)\) the principal symbol of \(A\). Put \[ \omega_{A}(x)=(2\pi)^{-n}\int_{a(x,\xi)<1}\,d\xi. \] For a function \(F\) of \((x,y)\in{\mathbb R}^{n}\times{\mathbb R}^{n}\) and \(h\in{\mathbb R}^{n}\), we define \(\Delta_{1,h}F(x,y)=F(x+h,y)-F(x,y)\) and \(\Delta_{2,h}F(x,y)=F(x,y+h)-F(x,y)\). Let \(e(t,x,y)\) stand for the spectral function of the self-adjoint realization of \(A\) in \(L^{2}({\mathbb R}^{n})\). The main results of this paper are as follows. (1) Let \(|\alpha|<m\) and \(|\beta|<m\). The derivatives \(\partial_{x}^{\alpha}\partial_{y}^{\beta}e(t,x,y)\) are Hölder continuous of exponent \(\theta\) for any \(\theta\in (0,1)\). There are \(C_{1}\) and \(C_{2}\) such that \[ |\partial^{\alpha}_{x}\partial^{\beta}_{y}e(t,x,y)|\leq C_{1}t^{(n+|\alpha|+|\beta|)/2m}, \] \[ |\Delta_{1,h}\partial^{\alpha}_{x}\partial^{\beta}_{y}e(t,x,y)|+ |\Delta_{2,h}\partial^{\alpha}_{x}\partial^{\beta}_{y}e(t,x,y)|\leq C_{2}t^{(n+|\alpha|+|\beta|+\theta)/2m}|h|^{\theta} \] hold for \(x,y,h\in{\mathbb R}^{n}\) and \(t\geq 1\). (2) For any \(\theta\in (0,\tau)\) there is \(C\) such that \[ |e(t,x,x)-\omega_{A}(x)t^{n/2m}|\leq Ct^{(n-\theta)/2m} \] for \(x\in{\mathbb R}^{n}\) and \(t\geq 1\).