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 \[Tf(x) = \int_Ω K(x, y) f(y) d ω(y)\] on $L^2(\Omega ,\omega )$ under the assumptions that $||T|{|}_{L^2(\omega )\to L^2(\omega )}<1$ 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 $j\ge 1$. Then $$K(x,y)e^{c{K}_2(x,y)/K(x,y)}\le \sum _{j=1}^{\infty }{K}_j(x,y)\le K(x,y)e^{C{K}_2(x,y)/K(x,y)},$$ for some constants $c,C>0$. Our estimates yield matching bilateral bounds for Green's functions of the fractional Schrödinger operators $(-△{)}^{\alpha /2}-q$ 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 Lévy processes.

22 pages

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