## Uniform estimates on multi-linear operators with modulation symmetry.(English)Zbl 1041.42013

In earlier work [J. Am. Math. Soc. 15, 469–496 (2002; Zbl 0994.42015)] the authors investigated boundedness properties of multilinear singular integrals associated to an $$n$$-linear form $\Lambda_m (f_1,\dots, f_n)=\int \delta (\xi_1+\cdots +\xi_n) m(\xi) \widehat f_1(\xi)\cdots \widehat f_n(\xi) \, d\xi.$ They proved that if $$m$$ and its derivatives decay at appropriate rates away from a suitably nondegenerate singular subspace $$\Gamma'$$ of $$\Gamma = \{\xi_1+\cdots +\xi_n=0\}$$, then $$\Lambda$$ is bounded from $$\prod_{k=1}^n L^{p_k}$$ to $$\mathbb R$$ in the Hölder range $$\sum 1/p_k =1$$.
In the present paper, the authors restrict to the case in which $$\Gamma'$$ is the line in direction $$V=(v_1,\dots,v_n)$$ with $$\sum v_k=0$$, all $$v_k\neq 0$$, and $$n\geq 3$$. Bounds on $$\Lambda$$ are then obtained with weaker decay conditions on $$m$$ described as follows: for $$x,y\in \Gamma$$ one sets $$d_v (x,y) = \max_{1\leq k\leq n} | x_k-y_k| /| v_k|$$ and $$d_v (x,\Gamma')=\inf_{y\in \Gamma'} d_v(x,y)$$. One then assumes that, for all partial derivatives up to some suitable finite order, one has $$| \partial_\xi^\alpha m(\xi)| \leq C\prod_{k=1}^n | v_k d_v(\xi,\Gamma')| ^{-\alpha_k}$$. Then $$| \Lambda_m (f_1,\dots, f_n)| \leq C\prod_{k=1}^n \| f_k\| _{L^{p_k}}$$ whenever each $$p_k>2$$ and $$\sum_k 1/p_k=1$$.
The proof relies on the same intricate phase space machinery used in the authors’ earlier work. The sharper bounds attained here are only known to apply to one-dimensional singularities because of difficulty in finding a suitable analogue of $$d_v$$ for higher dimensional singularities.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47B38 Linear operators on function spaces (general) 47G10 Integral operators

Zbl 0994.42015
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### References:

 [1] C. Calderon,On commutators of singular integrals, Studia Math.53 (1975), 139–174. · Zbl 0315.44006 [2] R. R. Coifman and Y. Meyer,On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc.212 (1973), 315–331. · Zbl 0324.44005 [3] R. R. Coifman and Y. Meyer,Au delà des opérateurs pseudo-différentiels, Astérisque57, Société Mathématique de France, Paris, 1978. [4] R. R. Coifman and Y. Meyer,Commutateurs d’integrales singulières et opérateurs muhilinéaires, Ann. Inst Fourier (Grenoble)28 (1978), 177–202. · Zbl 0368.47031 [5] R. R. Coifman and Y. Meyer,Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves, inEuclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, Md.), Lecture Notes in Math.779, Springer-Verlag, Berlin, 1979, pp. 104–122. [6] R. R. Coifman and Y. Meyer,Nonlinear harmonic analysis, operator theory and P.D.E., Beijing Lectures in Analysis, Ann. of Math. Stud.112 (1986), 3–46. [7] R. R. Coifman and Y. Meyer,Ondelettes et opérateurs III, Opérateurs multilinéaires, Actualités Mathématiques, Hermann, Paris, 1991. [8] C. Fefferman,Pointwise convergence of Fourier series, Ann. of Math. (2)98 (1973), 551–571. · Zbl 0268.42009 [9] J. Gilbert and A. Nahmod,Hardy spaces and a Walsh model for bilinear cone operators, Trans. Amer. Math. Soc.351 (1999), 3267–3300. · Zbl 0919.42009 [10] J. Gilbert and A. Nahmod,Bilinear operators with non-smooth symbol, J. Fourier Anal. Appl.7 (2001), 435–467. · Zbl 0994.42014 [11] J. Gilbert and A. Nahmod,L p-boundedness for time-frequency paraproducts, J. Fourier Anal. Appl.8 (2002), 109–171. · Zbl 1028.42013 [12] L. Grafakos and X. Li,Uniform bounds for the bilinear Hilbert transforms, I., preprint. · Zbl 1071.44004 [13] L. Grafakos and R. Torres, R.On multilinear singular integrals, preprint. · Zbl 1016.42009 [14] S. Janson,On interpolation of multilinear operators, inFunction Spaces and Applications (Lund 1986), Lecture Notes in Math.1302, Springer, Berlin-New York, 1988. [15] C. Kenig and E. Stein,Multilinear estimates and fractional interpolation, Math. Res. Lett.6 (1999), 1–15. · Zbl 0952.42005 [16] M. Lacey and C. Thiele,L p estimates on the bilinear Hilbert transform for 2p Ann. of Math. (2)146 (1997), 693–724. · Zbl 0914.46034 [17] M. Lacey and C. Thiele,On Calderon’s conjecture. Ann. of Math. (2)149 (1999), 475–496. · Zbl 0934.42012 [18] M. Lacey and C. Thiele,A proof of boundedness of the Carleson operator, Math. Res. Lett.7 (2000), 361–370. · Zbl 0966.42009 [19] X. Li,Uniform bounds for the bilinear Hilbert transforms, II., preprint. [20] C. Muscalu, T. Tao and C. Thiele,Multi-linear operators given by singular multipliers. J. Amer. Math. Soc.15 (2002), 469–496. · Zbl 0994.42015 [21] C. Muscalu, T. Tao and C. Thiele,Uniform estimates on paraproducts, J. Analyse Math.87 (2002), 369–384. · Zbl 1043.42012 [22] C. Muscalu, T. Tao and C. Thiele, Lpestimates for the biest I. The Walsh case, preprint. · Zbl 1073.42009 [23] C. Muscalu, T. Tao and C. Thiele, Lpestimates for the biest II. The Fourier case, preprint. · Zbl 1073.42010 [24] E. Stein,Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, 1993. · Zbl 0821.42001 [25] T. Tao,Multilinear weighted convolution of L 2 functions, and applications to non-linear dispersive equations, Amer. J. Math.123 (2001), 839–908. · Zbl 0998.42005 [26] C. Thiele,On the bilinear Hilbert transform, Habilitationsschrift, UniversitÄt Kiel, 1998. [27] C. Thiele,The quartile operator and almost everywhere convergence of Walsh-Fourier series, Trans. Amer. Math. Soc.352 (2000), 5745–5766. · Zbl 0976.42017 [28] C. Thiele,A uniform estimate for the quartile operator, Rev. Mat. Iberoamericana18 (2002), 115–134. · Zbl 1023.42004 [29] C. Thiele,A uniform estimate, Ann. of Math. (2)157 (2002), 1–45. · Zbl 1038.42019
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