Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres

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Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds (2005, methodology): “The present paper is the result of the combination of the arguments of [13], [14] and of [2].”
Long time existence for fully nonlinear NLS with small Cauchy data on the circle (2018, result): “We quote for instance the papers by Bambusi [4], Bambusi-Grebert [6] and by Delort-Szeftel [12, 13].”
Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds (2006, background): “Using the remark after (2.2.22), we decompose M ν , N p ,1 [ A 0 ] = p +1 j =1 F j where F j is a closed subspace such that if M ∈ F j (2.2 j 1 p +1 j A 0 sup k (cid:2) = j n k ) n p +1 ≡ 0 we have”