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 and a cutoff Sobolev inequality with scaling function $\Psi$, then any non-local $p$-form induced by a jumping kernel with scaling function $\Upsilon$, where $\Upsilon$ lies strictly above $\Psi$ at small scales, defines a regular $p$-energy satisfying a non-local Poincaré inequality and a non-local cutoff Sobolev inequality.

The corresponding scaling function $\Xi$ is explicitly determined by $\Psi$ and $\Upsilon$.

Our results also cover examples whose jumping kernels have light polynomial tails at infinity.

These results provide a nonlinear extension of the classical subordination principle beyond the Dirichlet form framework.