ScalingStacks

Corollary 2.16 . [040N]

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Corollary 2.16.

Let 2<q<42<q<4 and 0<ϵ<q−20<\epsilon<q-2. There is a bounded Green operator for T2T^{2}-invariant functions

Gg(2):{f∈C0,α|f=O⁡(|μ→|a−q)​ for large |μ→|a}→{u∈C2,α|u=O⁡(|μ→|a2−q+ϵ)}.G_{g^{(2)}}:\{f\in C^{0,\alpha}|f=O(|\vec{\mu}|_{a}^{-q})\text{ for large $|\vec{\mu}|_{a}$}\}\to\{u\in C^{2,\alpha}|u=O(|\vec{\mu}|_{a}^{2-q+\epsilon})\}.

such that u=Gg(2)​fu=G_{g^{(2)}}f satisfies Δg(2)​u=f\Delta_{g^{(2)}}u=f.

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