Existence of maximizers for scaling invariant Lp inequalities

We summarize methods to prove that best constants are attained for certain inequalities which are most useful in PDEs. The earliest systematic results have been obtained by Lieb in 1983. They apply e.g. to the Hardy-Littlewood-Sobolev inequality and related estimates (like the classical Sobolev inequality). Later a further approach has been developed by P.-L. Lions which allowed to deal with inequalities having no favourable properties under rearrangement. In a final section we present a recent new method to show that the best constant is attained in the standard Strichartz estimate for the Schrödinger equation.

Markus Kunze

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