Sharp Hardy inequalities in the half space with trace remainder term

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In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half space =ⁿ⁻¹×(0,∞), we show that, if we replace the optimal constant (n-2)²/4 with a smaller one (β-2)²/4, 2≤ β<n, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when β goes to n, while it is equal to the optimal constant in the Kato's inequality when β=2.