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Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators. (English) Zbl 1242.47040
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 $$\biggl\|\biggl(\sum_{\omega\in\Omega} |T_{\chi\omega}(f,g)|^2\biggr)^{1/2}\biggr\|_{L^{p-3}(\mathbb R)}\leq C\|f\|_{L^{p_1}(\mathbb R)} \|g\|_{L^{p_2}(\mathbb R)}.$$ 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.

47G30Pseudodifferential operators
42B15Multipliers, several variables
42C10Fourier series in special orthogonal functions
35S99Pseudodifferential operators
47A07Forms on topological linear spaces
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