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.


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


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