On bilinear Littlewood-Paley square functions (Q5891144)

Let \(\omega \) be a cube in \(\mathbb{R}^{d},\) \(d\geq 1.\) For \(f,g\in S(\mathbb{R}^{d}) \), we can define a bilinear operator \(S_{\omega}\) associated with the symbol \(\chi _{\omega }\left( \xi -\eta \right) \) as follows: NEWLINE\[NEWLINE S_{\omega}\left( f,g\right) \left( x\right) =\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}\left( \xi \right) \hat{g}\left( \eta \right) \chi _{\omega }\left( \xi -\eta \right) e^{2\pi ix \cdot\left( \xi +\eta \right) }\,d\xi d\eta NEWLINE\]NEWLINE where \(S(\mathbb{R}^{d}) \) is the Schwartz space. Assume that \(\left\{ \omega _{l}\right\} _{l\in\mathbb{Z}^{d}}\) is a sequence of disjoint cubes in \(\mathbb{R}^{d}.\) Let \(S_{\omega _{l}}\) be the bilinear operator associated with the symbol \(\chi _{\omega }\left( \xi -\eta \right) \) as defined above. Then for \(f,g\in S(\mathbb{R}^{d}),\) the bilinear Littlewood-Paley square function associated with the sequence \(\left\{ \omega _{l}\right\} _{l\in\mathbb{Z}^{d}}\) is defined by NEWLINE\[NEWLINE S\left( f,g\right) \left( x\right) =\left( \sum_{l\in\mathbb{Z}^{d}}\left| S_{\omega_l }\left( f,g\right) \left( x\right) \right| ^{2}\right) ^{\frac{1}{2}}. NEWLINE\]NEWLINE \textit{P. Mohanty} and \textit{S. Shrivastava} proved in [Proc. Am. Math. Soc. 138, No. 6, 2095--2098 (2010; Zbl 1196.42006); Theorem 1.6] that if \(\phi \in C^{\infty }\left(\mathbb{R}\right) \) hase support contained in the unit interval of \(\mathbb{R}\), and if for \(l\in\mathbb{Z}\), \(\phi _{l}\left( \xi \right) =\phi \left( \xi -l\right) ,\) then for \( 2\leq p,q\leq \infty \) and \(1<r\leq \infty \) satisfying \(\frac{1}{p}+\frac{1 }{q}=\frac{1}{r},\) there exists a constant \(C=C\left( \phi ,p,q\right) >0\) such that for all \(f,g\in S(\mathbb{R}^{d}) \), NEWLINE\[NEWLINE \left\| \left( \sum_{l\in\mathbb{Z}}\left| S_{\phi _{l}}\left( f,g\right) \right| ^{2}\right) ^{\frac{1 }{2}}\right\| _{L^{r}(\mathbb{R}) }\leq C\left\| f\right\| _{L^{p}(\mathbb{R}) }\left\| g\right\| _{L^{q}( \mathbb{R}) }. NEWLINE\]NEWLINE In Section 2, the authors study smooth bilinear square functions on \(\mathbb{R}^{d}\) and give an easy proof of above theorem. They also prove the following Theorem 2.1.NEWLINENEWLINETheorem 2.1. Let \(K\) be a measurable function on \(\mathbb{R}^{d}\) such that NEWLINE\[NEWLINE \sum_{m\in\mathbb{Z}^{n}}\left( \int_{Q_{m}}\left| K\left( y\right) \right| ^{\rho }dy\right) ^{\frac{1}{\rho }}<\infty , NEWLINE\]NEWLINE where \(\rho =\max \{ 2,r'\} \) and \(Q_{m}=\Pi _{j=1}^{d}[m_{j},m_{j}+1).\) For \(l\in\mathbb{Z}^{n},\) define \(\widehat{K}_{l}\left( \xi \right) =\widehat{K}\left( \xi -l\right) \) and let \(T_{l}\) be the bilinear operator associated with \( \widehat{K}_{l}\left( \xi -l\right) .\) Then for exponents \(2\leq p,q\leq \infty \) and \(1\leq r\leq \infty \) satisfying \(\frac{1}{p}+\frac{1}{q}=\frac{ 1}{r},\) there exists a constant \(C=C\left( K,r\right) \) such that NEWLINE\[NEWLINE \left\| \left( \sum_{l\in\mathbb{Z}^{d}}\left| T_{l}\left( f,g\right) \right| ^{2}\right) ^{\frac{1}{2} }\right\| _{L^{r}(\mathbb{R}^{d}) }\leq C\left\| f\right\|_{L^{p}(\mathbb{R}^{d}) }\left\| g\right\| _{L^{q}(\mathbb{R}) }. NEWLINE\]NEWLINE In Section 3, the authors prove the following Theorem 3.1.NEWLINENEWLINETheorem 3.1. Let \(K\) be a measurable function on \(\mathbb T^{d}\) such that \(\left\| K\right\| _{L^{t}( \mathbb T^{d}) }<\infty ,\) where \(t=\max \{2,r'\}\). For \(l\in\mathbb{Z}^{n},\) define \(\widehat{K}_{l}\left( n\right) =\widehat{K}\left( n-l\right) \) and let \(T_l\) be the bilinear multiplier operator associated with the symbol \(\widehat{K}_{l}\left( n-m\right) .\) Then for exponents \( 2\leq p,q\leq \infty \) and \(1\leq r\leq \infty \) satisfying \(\frac{1}{p}+ \frac{1}{q}=\frac{1}{r},\) there exists a constant \(C=C\left( K,r\right) \) such that for all trigonometric polynomials \(f,g\) on \(\mathbb T^{d},\) we have NEWLINE\[NEWLINE\left\| \left( \sum_{l\in\mathbb{Z}}\left| T_{l}\left( f,g\right) \right| ^{2}\right) ^{\frac{1}{2} }\right\| _{L^{r}( \mathbb T^{d}) }\leq C\left\| f\right\| _{L^{p}(\mathbb T^{d}) }\left\| g\right\| _{L^{q}( \mathbb T^{d}) }. NEWLINE\]