Bi-parameter paraproducts
Let \(m\) be a bounded symbol on \(\mathbb R^4\), smooth away from the origin and such that \[ | \partial^{\alpha_1}_{\xi_1}\partial^{\alpha_2}_{\xi_2}\partial^{\beta_1}_{\eta_1}\partial^{\beta_2}_{\eta_2} m(\xi,\eta)\leq C| (\xi_1,\eta_1)| ^{-\alpha_1-\beta_1}| (\xi_2,\eta_2)| ^{-\alpha_2-\beta_2}, \] and denote \[ T(f,g)(x):= \int m(\xi,\eta)\widehat f(\xi)g(\eta)e^{2\pi i x(\xi+\eta)}\, d\xi\, d\eta. \] The main result of the paper states that \(T:L^p\times L^q\to L^r\) as long as \(1<p,q\leq \infty\), \(0<r<\infty\) and \(1/r=1/p+1/q\). To prove this fact, first \(T\) is discretized to a bi-parameter general paraproduct, and then a new technique in introduced to present a proof of the classical one-parameter case. This technique turns out to be very helpfull to handle an error term in the proof of the main theorem. The paper also contains two counterexamples concerning the double bilinear Hilbert transform \[ B(f,g):= \int f(x-t,y-s)g(x+t,y+s)(dt/t)(ds/s) \] and the trilinear operator \[ V(f,g,h)(x):= \int_{\xi_1<\xi_2<\xi_3} \widehat f(\xi_1)\widehat g(\xi_2)\widehat h(\xi_3) e^{2\pi i x(\xi_1-\xi_2+xi_3)} \, d\xi_1\, d\xi_2\, d\xi_3. \] The last part of the article contains some further comments and open questions.
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