  We propose a strategy to study massive Quantum Field Theory (QFT) using
conformal bootstrap methods. The idea is to consider QFT in hyperbolic space
and study correlation functions of its boundary operators. We show that these
are solutions of the crossing equations in one lower dimension. By sending the
curvature radius of the background hyperbolic space to infinity we expect to
recover flat-space physics. We explain that this regime corresponds to large
scaling dimensions of the boundary operators, and discuss how to obtain the
flat-space scattering amplitudes from the corresponding limit of the boundary
correlators. We implement this strategy to obtain universal bounds on the
strength of cubic couplings in 2D flat-space QFTs using 1D conformal bootstrap
techniques. Our numerical results match precisely the analytic bounds obtained
in our companion paper using S-matrix bootstrap techniques.
