Shifted Hardy inequalities on the unit ball and polydisk

We refine and extend shifted Hardy inequalities in several complex variables. A sharp estimate for homogeneous holomorphic polynomials on the sphere in Cn, combined with Yang's one-variable inequality and a slice identity, yields ball inequalities for H1(Bn), including a Bergman-scale family with weights. We also prove product inequalities on Dn and mixed-norm extensions for radial measures, where coefficient weights are moments of the measure. Motivated by Yang's shifted Hilbert form, we define a linear transform, derive its integral representation, compute its exact Hp(Bn)→Hp(Bn) norm, and obtain an H1(Bn)→H∞(Bn) bound.