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With the help of a radially invariant vector field, we derive inequalities ofthe Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded stardomains. Our results are not obtainable from the classical inequalities for$W^{1,p}_{0}$ functions. Unlike in $W^{1,p}_{0}$, our inequalities admitmaximizers that we describe explicitly.

On the boundary terms in Hardy's inequalities for $W^{1,p}$ functions