The 2d Case of the Bourgain-demeter-guth Argument

and so geomxi is the geometric mean of xi. The goal of this note is to sketch the proof of the following theorem, concentrating on the iteration scheme from [4] in the two dimensional case. This justifies (19) of [4] without an appeal to [1]. We also include some routine applications of Hölder, Minkowski, etc. that are omitted from [4] for completeness. We will prove Theorem 1. Let 0 < δ ≤ 1. For each ball B ⊂ R of radius δ−2 and each g : [0, 1] → C, we have ∥E[0,1]g∥L6(wB) .ε δ−ε( ∑