Global estimates for kernels of Neumann series and Green's functions (original) (raw)

Abstract

We obtain global pointwise estimates for kernels of the resolvents (I-T)^-1 of integral operators Tf(x) = ∫_Ω K(x, y) f(y) d ω(y) on L^2(Ω, ω) under the assumptions that ||T||_L^2(ω) → 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) = ∫_Ω K_j-1 (x,z) K(z,y) d ω (z) for j ≥ 1. Then K(x,y) e^c K_2 (x,y)/K(x,y)≤∑_j=1^∞ K_j(x,y) ≤ 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 (-)^α/2-q with arbitrary nonnegative potentials q on R^n for 0<α<n, or on a bounded non-tangentially accessible domain Ω for 0<α< 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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