Paraproducts and Hankel operators of Schatten class via p-John–Nirenberg Theorem

Abstract

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten–von Neumann class S p , if and only if its symbol is in the dyadic Besov space B p d . Our main tools are a product formula for paraproducts and a “p-John–Nirenberg-Theorem” due to Rochberg and Semmes. We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols. Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten–von Neumann class S p for 1< p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten–von Neumann class, answering a question by Bonami and Peloso. Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.