A Short Course on Rearrangement Inequalities

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Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, background): “, [12, 20, 26, 75] (and also the references in the Bibliographical Remarks at the end of this lecture).”
Sharp reversed Hardy–Littlewood–Sobolev inequality on Rn (2015, background): “implicity, we still denote q= r/(r− 1) <0. Clearly 1/p− λ/n= 1 + 1/q. Let fbe a function on Rn which vanishes at inﬁnity, its symmetric decreasing rearrangement is denoted by f⋆, see [LieLos01] or [Bur09] for the deﬁnitions. It is wellknown that if f∈ Lp(Rn), p>0 then f⋆ ∈ Lp(Rn) and kfk Lp(R ”
A probabilistic take on isoperimetric-type inequalities (2012, methodology): “We refer the reader to the book [22] or the introductory notes [5] for further background material on rearrangements of functions.”