Improving interpolated Hardy and trace Hardy inequalities on bounded domains

Abstract

In this work we study inequalities which interpolate weighted Hardy and trace Hardy inequalities on bounded domains. We establish improvements of these inequalities by adding correction terms of Hardy type with a singular logarithmic weight. We show that this weight is optimal in the sense that the inequality fails for more singular weights. Moreover, we determine the best values of the constants for the remaining terms. We also show that the aforementioned inequalities can be repeatedly improved, obtaining an infinite series of correction terms involving specific Hardy type potentials. In the two borderline cases of these interpolation inequalities we obtain refinements of the weighted Hardy and the trace weighted Hardy inequality respectively. It follows that the trace Hardy and the Hardy weighted inequalities share the same optimal constant for the specific remaining terms. These results constitute an extension of a well known result when the interpolation inequality reduces to the classical nonweighted Hardy inequality.