Paraproducts and Products of functions in BMO(\mathbb Rⁿ) and H¹(\mathbb Rⁿ) through wavelets

2011/03/09 by Aline Bonami, Bonami, Aline, Sandrine Grellier +3
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.1103.1822

openalex publication_date 2011/03/09 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in \BMO(\bRⁿ) and \H¹(\bRⁿ), may be written as the sum of two continuous bilinear operators, one from \H¹(\bRⁿ)× \BMO(\bRⁿ) into L¹(\bRⁿ), the other one from \H¹(\bRⁿ)× \BMO(\bRⁿ) into a new kind of Hardy-Orlicz space denoted by \H^log(\bRⁿ). More precisely, the space \H^log(\bRⁿ) is the set of distributions f whose grand maximal function \mathcal Mf satisfies ∫_{\mathbb Rⁿ} \frac |\mathcal M f(x)|log(e+|x|) +log (e+ |\mathcal Mf(x)|)dx

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