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.

Trilinear embedding for divergence-form operators with complex coefficients

“…On top of grossly extending [28,Section 3.3], the trilinear embedding of Theorem 1 also directly implies the above-said bilinear embeddings from [16,15]. See Section 1.6 for a more precise formulation of this statement and its proof.…”

“…Other works using the Bellman-heat approach include Domelevo-Petermichl [20], Mauceri-Spinelli [30], Wróbel [42], Betancor-Dalmasso-Fariña-Scotto [7], and the article [28] by the latter two authors of the present paper. Since we will be dealing with pairs and triples of matrices, it is useful to introduce further notation, as in [16,15]: ∆ p (A 1 , .…”

“…Thus the proof of Theorem 1 is based on combining the elements from [28] on one hand, and the technique from [16] (for Ω = R d ) and [15] (for general Ω) on the other. 1.6.…”

“…We use a three-variable Bellman function found by the latter two authors of the present paper [28]. It is modelled over a two-variable Bellman function due to Nazarov and Treil [36].…”

“…Compared to [28], the novelty in the convexity estimate formulated here is the presence of complex elliptic matrices, as specified below.…”