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

Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Given an integral operator T on a σ-finite measure space (Ω, ω) with kernel K : Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Given an integral operator T on a σ-finite measure space (Ω, ω) with kernel K : Tf (x) = Ω K (x, y)f (y) dω(y) Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Given an integral operator T on a σ-finite measure space (Ω, ω) with kernel K : Tf (x) = Ω K (x, y)f (y) dω(y) We assume throughout that K ≥ 0 is symmetric (K (x, y) = K (y , x)) and measurable. If T L 2 (ω)→L 2 (ω) < 1, consider the Neumann series Background Joint work with Fedor Nazarov and Igor Verbitsky (preprint) Start with general functional analysis theorem: Given an integral operator T on a σ-finite measure space (Ω, ω) with kernel K : Tf (x) = Ω K (x, y)f (y) dω(y) We assume throughout that K ≥ 0 is symmetric (K (x, y) = K (y , x)) and measurable. If T L 2 (ω)→L 2 (ω) < 1, consider the Neumann series (I − T) −1 = ∞ j=0 T j. Question Let K j be the kernel of T j : K 1 = K and K j (x, y) = Ω K j−1 (x, z)K (z, y) dω(z).