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Bilinear operators with non-smooth symbol. I. (English) Zbl 0994.42014

A bilinear operator that maps pairs of Schwartz functions to Schwartz distributions and commutes with simultaneous translations can be written in terms of its multiplier \(m(\xi ,\eta)\) as \[ B(f,g)(x)=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }m(\xi ,\eta)\widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi ix(\xi +\eta)} d\xi d\eta. \] While boundedness of operators for which \(m\) is smooth away from the origin and satisfies symbol estimates can be proven by more or less standard methods of Calderón-Zygmund theory, multipliers with singularities along other sets require new techniques. The symbol \(m(\xi ,\eta)=\chi _{\{\xi <\eta \}}\) of the bilinear Hilbert transform is prototypical. Here, the authors prove the following extension to ‘cone operators’ of Lacey and Thiele’s theorem on boundedness of the bilinear Hilbert transform [M. Lacey and C. Thiele, Ann. Math. 149, No. 2, 475-496 (1999; Zbl 0934.42012)] : Let \(\Gamma \) be a one-sided cone with vertex at the origin and suppose that, inside \(\Gamma \), \(m\) satisfies the symbol estimates \(|D^{\alpha }m(\xi ,\eta)|\leq C_{\alpha }\text{dist}((\xi ,\eta),\Gamma)^{-|\alpha |}\) (\(|\alpha |\geq 0\)). Then the operator \(B\) defined as above with multiplier \(m(\xi ,\eta)\chi _{\Gamma }(\xi ,\eta)\) is bounded from \(L^{p}\times L^{q}\) to \(L^{r}\) where \(1/p+1/q=1/r<3/2\), provided no edge of \(\Gamma \) lies on the diagonal \(\xi +\eta =0\) or on a coordinate axis. The authors also show that the range of \(B\) lies in the corresponding complex Hardy subspace of \(L^{r}\) when \(r\geq 1\) and \(\Gamma \) strictly lies inside \(\xi +\eta >0\). These facts follow from a boundedness result for generalized paraproduct type operators built from modulated wavelets or wave-packets. The paraproduct bound is proved in a companion paper in the same journal. In the present paper the reduction of the cone operator to a discretized modulated paraproduct is proved. Furthermore, \( L^{p}\)-boundedness is proved for analogues of standard paraproducts having the form \[ (c,f,g)\mapsto \sum_{k,l=-\infty }^{\infty }c_{kl}2^{jk/2}\left\langle f,\psi _{jkl}^{(1)}\right\rangle \left\langle g,\psi _{jkl}^{(3)}\right\rangle \psi _{jkl}^{(3)} \] where \(\psi _{jkl}^{(i)}(x)=2^{jk/2}\psi ^{(i)}(2^{jk}x-la_{i})\). This operator is bounded – with a bound depending on the shifts \(a_{i}\) – from \( l^{\infty }\times L^{p}\times L^{q}\) to \(L^{r}\) provided \(1/r=1/p+1/q<2\). Here one assumes at least two of the \(\psi ^{(i)\text{ }}\) have vanishing moment. The full paraproduct result assumes that the \(\widehat{\psi}^{(i)}\) have disjoint supports. The techniques used are substantially different from those of C. Muscalu, T. Tao and C. Thiele [J. Am. Math. Soc. 15, No. 2, 469-496 (2002; following review)] who independently generalized the \(L^{p}\) result – though not the Hardy space result – to the case \(n\geq 2\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B15 Multipliers for harmonic analysis in several variables
42B25 Maximal functions, Littlewood-Paley theory
47G10 Integral operators
47H60 Multilinear and polynomial operators
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