Cutoff Sobolev inequalities for local and non-local $p$-energies on metric measure spaces

For $p>1$, we study subordination phenomena for local and non-local regular $p$-energies on metric measure spaces. Under suitable geometric assumptions, we show that if a local regular $p$-energy satisfies a Poincaré inequality together with a cutoff Sobolev inequality with scaling function $\mathrm{\Psi }$, then all associated stable-like non-local $p$-energies with scaling functions strictly below $\mathrm{\Psi }$ are regular and satisfy the corresponding non-local cutoff Sobolev inequalities. Moreover, if a stable-like non-local regular $p$-energy with scaling function $\mathrm{\Psi }$ satisfies the corresponding non-local cutoff Sobolev inequality, then the same conclusion holds for all associated stable-like non-local $p$-energies with scaling functions below $\mathrm{\Psi }$. These results provide a non-linear extension of the classical subordination principle beyond the Dirichlet form framework.