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 T f ( 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 non-negative potentials q on R n for < α > n , or on a bounded non-tangentially accessible domain Ω for < 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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