18 . 118 Topics in Analysis Problem Set 3

j=1 ‖fj‖ 1 n Lavg(BR) p = 2n n− 1 where f = ∑n j=1 fj , supp f̂j ⊂ N1/RΣj and Σj are hypersurfaces with normal vector almost parallel to xj-axis. Take p-th power on both sides and induction on scales. Tile BR using balls of radius R1/2. The averaged integral over BR on the left hand side is approximately the average over all balls of radius R1/2 in the tiling of the averaged integral over one ball of radius R1/2. Suppose the statement is true at scale R1/2, use local orthogonality and local constancy at scale R1/2, and put everything back to integration over BR. Then use Multilinear Kakeya and the reverse direction of local orthogonality.