Sharp Hardy inequalities for the Vladimirov--Taibleson operator on $\mathbb{Q}_p^n$

We prove sharp Hardy inequalities on $\mathbb{Q}_p^n$ for the quadratic form of the Vladimirov--Taibleson operator, the $L^q$ norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for $0<α<n$ is $[(1-p^{-(n-