Improving trace Hardy inequalities and hardy inequalities for fractional laplacians on bounded domains
Βελτίωση Hardy ανισοτήτων ίχνους και ανισοτήτων Hardy για κλασματικές Λαπλασιανές σε φραγμένα χωρία
Τζιράκης, Κωνσταντίνος Δ.
This thesis is devoted to inequalities which interpolate weighted Hardy and trace Hardy inequalities.We first derive a sharp interpolation between the two Hardy inequalities. Then we proceed to improving these inequalities by adding in the least hand side correction terms, that amount to positive integrals of the functions under consideration. In particular, we concern with integrals, over the half space or its boundary, which involve either the critical Hardy potential or the critical Sobolev exponent. In all cases, it turns out that correction terms of such type can be added at the expense of a logarithmic corrective weight, which is optimal in the sense that the inequality fails for smaller powers of this weight. Furthermore, we show that the aforementioned inequalities can be repeatedly improved, obtaining an infinite correction series.
The results in the two borderline cases of these interpolation inequalities yield refinements of the weighted Hardy and the trace weighted Hardy inequality respectively, thus unify and extend some earlier results. In particular, it follows that the trace Hardy and the Hardy weighted inequalities admit the same infinite improvement.
Moreover, we apply the resulted improvements of the trace Hardy inequality with trace remainder terms, to derive refinements of Hardy inequalities associated with two different fractional Laplacians defined on bounded domains.