For 3<=k<=n-1, 1<p<n, the deficit in the Hardy-Sobolev-Maz'ya inequality is bounded below by a constant times the gradient distance to the extremal manifold raised to the power max{2,p}, and this exponent is optimal.
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Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities
For 3<=k<=n-1, 1<p<n, the deficit in the Hardy-Sobolev-Maz'ya inequality is bounded below by a constant times the gradient distance to the extremal manifold raised to the power max{2,p}, and this exponent is optimal.