Abstract
In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that
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Bernicot, F. Lp estimates for non-smooth bilinear Littlewood–Paley square functions on \({\mathbb{R}}\). Math. Ann. 351, 1–49 (2011). https://doi.org/10.1007/s00208-010-0588-1
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DOI: https://doi.org/10.1007/s00208-010-0588-1