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

We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators $Tf(x) = \int_Ω K(x, y) f(y) d ω(y)$ on $L^2(Ω, ω)$ under the assumptions that $||T||_{L^2(ω) \rightarrow L^2 (ω)} <1$ and $d(x,y)=1/K(x,y)$ is a quasi-metric. Let $K_1=K$ and $K_j(x,y) = \int_Ω K_{j-1} (x,z) K(z,y) \, d ω(z)$ 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 Schrödinger operators $(-\triangle)^{α/2}-q$ with arbitrary nonnegative potentials $q$ on $\mathbb{R}^n$ for $0<α<n$, or on a bounded non-tangentially accessible domain $Ω$ for $0<α\le 2$. In probabilistic language, these results can be reformulated as explicit bilateral bounds for the conditional gauge associated with Brownian motion or $α$-stable Lévy processes.

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