Bilinear square functions and vector-valued Calderón-Zygmund operators
The bilinear square function corresponding to a sequence of kernels \(\theta_{k}:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{C}\) is defined by \[ \Bigg(\sum_{k\in\mathbb{Z}}\mid\Theta_{k}(f,g)\mid^{2}\Bigg)^{1/2}, \] where \[ \Theta_{k}(f,g)=\int\int\theta_{k}(x,y,z)f(y)g(z)dydz. \] Optimal conditions on the kernels \(\theta_{k}\) are found to provide boundedness of bilinear square functions on products of Lebesgue, Sobolev and some other spaces of smooth functions. Bilinear vector-valued Calderón-Zygmund operators are introduced and used to obtain boundedness result in the case of products of Lebesgue spaces for the full range of exponents: \(1<p,q<\infty\) and \(1/2<r<\infty\).
scientific article; zbMATH DE number 6132226