2015/06/18 by Francesco Di Plinio, Yumeng Ou, Di Plinio, Francesco +1 · 3 citations
Mathematics · #42B20 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1506.05827
openalex publication_date 2015/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued Hörmander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed Lp-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rules of Muscalu et. al.. Our approach works just as well for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform, extending results of Silva. We also prove several operator-valued T (1)-type theorems both in one parameter, and of multi-para\-meter, mixed-norm type. A distinguishing feature of our T(1) theorems is that the usual explicit assumptions on the distributional kernel of T are replaced with testing-type conditions. Our proofs rely on a newly developed Banach-valued version of the outer Lp space theory of Do and Thiele.