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In this paper we deal with a class of inequalities which interpolate theKato's inequality and the Hardy's inequality in the half space. Starting fromthe classical Hardy's inequality in the half space $\rnpiu=\R^{n-1}\times(0,\infty)$, we show that, if we replace the optimal constant$\frac{(n-2)^2}{4}$ with a smaller one $\frac{(\beta-2)^2}{4}$, $2\le \beta<n$, then we can add an extra trace-term equals to that one that appears in theKato's inequality. The constant in the trace remainder term is optimal and ittends to zero when $\beta$ goes to $n$, while it is equal to the optimalconstant in the Kato's inequality when $\beta=2$.

Sharp Hardy inequalities in the half space with trace remainder term