We give an alternative proof of a theorem by Brothers and Ziemer concerning extremal functions in the Pólya-Szegö rearrangements inequality for Dirichlet type integrals.

Abstract. 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 spacewe show that, if we replace the optimal constantwith a smaller one, 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.

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In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.

We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by [3] in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.