- Publication venue
- 'Wiley'
- Publication date
- 16/03/2014
- Field of study
We obtain global pointwise estimates for kernels of the resolvents
(IβT)β1 of integral operators Tf(x)=β«Ξ©βK(x,y)f(y)dΟ(y) on L2(Ξ©,Ο) under the assumptions that
β£β£Tβ£β£L2(Ο)βL2(Ο)β<1 and d(x,y)=1/K(x,y) is a
quasi-metric. Let K1β=K and Kjβ(x,y)=β«Ξ©βKjβ1β(x,z)K(z,y)dΟ(z) for jβ₯1. Then K(x,y)ecK2β(x,y)/K(x,y)β€j=1βββKjβ(x,y)β€K(x,y)eCK2β(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 (ββ³)Ξ±/2βq with arbitrary
nonnegative potentials q on Rn for 0<Ξ±<n, or on a bounded
non-tangentially accessible domain Ξ© for 0<Ξ±β€2. In
probabilistic language, these results can be reformulated as explicit bilateral
bounds for the conditional gauge associated with Brownian motion or
Ξ±-stable L\'evy processes.Comment: 22 page