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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.

MSC:

47G30 Pseudodifferential operators
42B15 Multipliers for harmonic analysis in several variables
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
35S99 Pseudodifferential operators and other generalizations of partial differential operators
47A07 Forms (bilinear, sesquilinear, multilinear)
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