Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators
The authors prove the following \(L^p\) estimates for smooth bilinear square functions. Let \(\Omega := (\omega)_{\omega\in\Omega}\) be a well-distributed collection of intervals of the same length and equidistant. Then, for exponents \(p_1, p_2, p_3\in[2,\infty]\) satisfying \(0 <1/p_3=(1/p_1)+(1/p_2)\), there exists a constant \(C\), independent of the collection \(\Omega\), such that for all \(f, g \in {\mathcal S}(\mathbb R)\), one has NEWLINE\[NEWLINE\biggl\|\biggl(\sum_{\omega\in\Omega} |T_{\chi\omega}(f,g)|^2\biggr)^{1/2}\biggr\|_{L^{p-3}(\mathbb R)}\leq C\|f\|_{L^{p_1}(\mathbb R)} \|g\|_{L^{p_2}(\mathbb R)}.NEWLINE\]NEWLINE Here \(T_\sigma (f, g)(x) :=\int_{\mathbb{R}^2}e^{ix(\xi+\eta}\hat f(\xi)\hat g(\eta) \sigma(x,\xi,\eta)\,d\xi\,d\eta\), for symbols in the exotic ``class'' \(B_{0,0}^0\). The boundedness of some bilinear pseudo-differential operators associated with symbols belonging to the subclass \(BS_{0,0}^0\) is deduced.
scientific article; zbMATH DE number 6039334