Bellman functions and $L^p$ estimates for paraproducts

Abstract

We give an explicit formula for one possible Bellman function associated with the $L^p$ boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.

arXiv comment18 pages, 1 figure, 1 ancillary file (Mathematica notebook); v3: exposition is slightly improved, additional calculations and remarks included

Cited by in corpus (3)

• Variational estimates for martingale paraproducts
• Convergence of ergodic-martingale paraproducts
• Trilinear embedding for divergence-form operators with complex coefficients