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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Kristina Ana Škreb (University of Zagreb)

Bellman functions and $L^p$ estimates for paraproducts

We regard dyadic paraproducts as trilinear forms. Even though they are well-known to satisfy $L^p$ estimates in the whole Banach range of exponents, one might want to give a direct proof or study the behavior of the constants. We find an explicit formula for one possible Bellman function associated with the $L^p$ boundedness of dyadic paraproducts in the spirit of the Bellman function by Nazarov, Treil, and Volberg. 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. This is a joint work with Vjekoslav Kovač (University of Zagreb).