## 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

### Citations:

Zbl 0994.42015; Zbl 0934.42012
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### References:

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