Global estimates for kernels of Neumann series and Green's functions

Abstract

We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators \[Tf(x) = \int_{\Omega} K(x, y) f(y) d \omega(y)\] on $L^2(\Omega, \omega)$ under the assumptions that $||T||_{L^2(\omega) \rightarrow L^2 (\omega)} <1$ and $d(x,y)=1/K(x,y)$ is a quasi-metric. Let $K_1=K$ and $K_j(x,y) = \int_{\Omega} K_{j-1} (x,z) K(z,y) \, d \omega (z)$ for $j \geq 1$. Then $$ K(x,y) e^{c K_2 (x,y)/K(x,y)} \leq \sum_{j=1}^{\infty} K_j(x,y) \leq K(x,y) e^{C K_2 (x,y)/K(x,y)}, $$ for some constants $c,C>0$. Our estimates yield matching bilateral bounds for Green's functions of the fractional Schr\"{o}dinger operators $(-\triangle)^{\alpha/2}-q$ with arbitrary nonnegative potentials $q$ on $\mathbb{R}^n$ for $0<\alpha<n$, or on a bounded non-tangentially accessible domain $\Omega$ for $0<\alpha \le 2$. In probabilistic language, these results can be reformulated as explicit bilateral bounds for the conditional gauge associated with Brownian motion or $\alpha$-stable L\'evy processes.