## Multi-linear operators given by singular multipliers.(English)Zbl 0994.42015

The authors study $$L^{p}$$-boundedness for a class of multilinear operators $$T_{m}$$ defined in terms of Fourier multipliers $$m$$ by $(T_{m}(f_{1},\dots ,f_{n-1}))^{\wedge }(-\xi _{n})=\int \delta (\xi _{1}+\dots +\xi _{n})m(\xi)\hat{f}_{1}(\xi _{1})\cdots \hat{f}_{n-1}(\xi _{n-1}) d\xi _{1}\cdots d\xi _{n-1}.$ where $$\xi =(\xi _{1},\dots ,\xi _{n})$$. Expressing $$T_{m}$$ this way allows one to obtain bounds for $$T_{m}$$ by interpolating from restricted weak-type estimates for the $$n$$-linear form $\Lambda _{m}(f_{1},\dots ,f_{n})=\int T_{m}(f_{1},\dots ,f_{n-1})(x)f_{n}(x) dx.$ One says that $$\Lambda$$ is of restricted weak-type $$\alpha =(\alpha _{1},\dots ,\alpha _{n})$$ provided $|\Lambda (f_{1},\dots ,f_{n})|\leq C|E_{1}|^{\alpha _{1}}\cdots |E_{n}|^{\alpha _{n}}$ where $$|f_{i}|\leq 1$$ on its support $$E_{i}$$. Here one thinks of $$\alpha _{i}=\frac{1}{p_{i}}$$. In order to obtain operator bounds $$T_{m}:L^{p_{1}}\times \cdots \times L^{p_{n-1}}\rightarrow L^{p_{n}'}$$ that allow $$p_{n}<0$$ one must allow $$\alpha _{n}<0$$ . Moreover, one only needs $$|f_{n}|\leq 1$$ on the complement of an exceptional subset of $$E_{n}$$ where the maximal function of $$\chi _{E_{n}}$$ is large. Dilation structure imposes the added condition $$\sum \alpha _{i}=1$$. Once a restricted weak-type estimate is established, multilinear interpolation is used to pass to $$L^{p}$$-estimates. The restricted weak-type estimates in the current work are based on certain Carleson type tile-norms similar to ones used by the authors in previous work on bilinear singular integrals. Here those Carleson-type estimates are extended to multi-tiles. These are products of tiles – essentially dyadic time-frequency rectangles of unit area – in which each tile in the product shares same time interval. The authors consider symbols $$m$$ that can have singularities along a subspace $$\Gamma '$$ in $$\Gamma =\{\xi :\xi _{1}+\dots +\xi _{n}=0\}$$ having dimension $$k<n/2$$ while satisfying symbol estimates $$|\partial _{\xi }^{\alpha }m(\xi)|\leq C_{\alpha }\text{dist} (\xi ,\Gamma ')^{-|\alpha |}$$. Suppose for convenience that $$\Gamma '$$ is parameterized in terms of $$\xi _{1},\dots ,\xi _{k}$$. It is shown that under these conditions $$T_{m}$$ continuously maps $$L^{p_{1}}\times \dots \times L^{p_{n-1}}$$ into $$L^{p_{n}}$$ provided (i) $$1<p_{i}\leq \infty (i=1,\dots ,n-1)$$, (ii) $$1/(n-1)<p_{n}'<\infty$$, and (iii) $$\frac{1}{p_{1}}+\dots +\frac{1}{p_{n}}=1$$. In addition, (iv) $$\frac{1}{p_{i_{1}}} +\dots +\frac{1}{p_{i_{r}}}<\frac{n-2k+r}{2}$$ for all choices of $$1\leq i_{1}<\dots <i_{r}\leq n$$ and $$1\leq r\leq n$$. The range of exponents (i)-(iii) is familiar from work of C. E. Kenig and E. M. Stein [Math. Res. Lett. 6, No. 1, 1-15; 6, No. 3-4, 467 (1999; Zbl 0952.42005)]. The role of the singularity was clarified by J. E. Gilbert and A. R. Nahmod [J. Fourier Anal. Appl. 7, No. 5, 435-467 (2001; preceding review)] who proved the result for the case $$n=3$$ and $$k=1$$. The present work involves an induction on $$k$$. The endpoint case $$k=n/2$$ is still open when $$n$$ is even, although the authors prove a substitute in which $$L^{p_{i}}$$ is replaced by the Wiener algebra in enough coordinates.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47H60 Multilinear and polynomial operators 47G10 Integral operators 47B25 Linear symmetric and selfadjoint operators (unbounded)

### Citations:

Zbl 0994.42014; Zbl 0952.42005
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### References:

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