Applications of the Bellman Function Technique in Multilinear and Nonlinear Harmonic Analysis.

摘要

Large part of this work is motivated by a question raised by Demeter and Thiele in [14] on establishing Lp estimates for a two-dimensional bilinear operator of paraproduct type, called the twisted paraproduct. It is given by TF,Gx,y :=k∈Z 22kR Fx-s,y4 2ksds R Gx,y-ty 2ktdt, where ϕ, psi are Schwartz functions and y&d4; (xi) is supported "near" |xi| = 1. We confirm this conjecture by proving TF,G Lr R2 ≤Cp,q&dvbm0;F&dvbm0;Lp R2 &dvbm0;G&dvbm0;Lq R2 whenever 1 < p, q 12 . As a byproduct of the approach we develop a rather general technique for verifying multilinear estimates. This method is subsequently further applied to show Lp bounds for a class of two-dimensional until ilinear forms that generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct.;The remaining material is related to the one-dimensional Dirac scattering transform f ainfinity,x binfinity,x binfinity,x ainfinity,x , defined by the initial value problem 66x ax,x bx,x bx,x ax,x = ax,x bx,x bx,x ax,x 0fx e-2pixx fxe2p ixx0 , a-infinity,x=1, b-infinity,x=0. Muscalu, Tao, and Thiele asked in [33] if the analogues of Hausdorff-Young inequalities are valid with constants independent of p, ∥&parl0;ln&vbm0;a&parl0;infinity,x &parr0;&vbm0;&parr0;1/2∥ Lqx R≤C fLp R ,for1≤p≤2, 1p+1 q=1. We provide positive answer to this question in the case where the exponentials are replaced by the character function of the "d -adic model" of the real line.;Our main tool for all of the attempted problems, both multilinear and nonlinear in nature, is the Bellman function technique, briefly described as "systematic induction over scales".