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Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts (1806.10085v1)
Published 25 Jun 2018 in math.CA
Abstract: We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $|b|{\operatorname{bmo}(\mathbb{R}{n+m})} = 1$ and $1 < p, q \le \infty$ and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ |[b, T]_1(f_1, f_2)|{Lr(\mathbb{R}{n+m})} \lesssim |f_1|{Lp(\mathbb{R}{n+m})} |f_2|{Lq(\mathbb{R}{n+m})}. $$ Previously the range $r \le 1$ was proved only in the paraproduct free situation. The main novelty lies in the treatment of the so called partial paraproducts.