Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 8 (2016), 1931-1988.

Multiple vector-valued inequalities via the helicoidal method

Cristina Benea and Camil Muscalu

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We develop a new method of proving vector-valued estimates in harmonic analysis, which we call “the helicoidal method”. As a consequence of it, we are able to give affirmative answers to several questions that have been circulating for some time. In particular, we show that the tensor product BHTΠ between the bilinear Hilbert transform BHT and a paraproduct Π satisfies the same Lp estimates as the BHT itself, solving completely a problem introduced by Muscalu et al. (Acta Math. 193:2 (2004), 269–296). Then, we prove that for “locally L2 exponents” the corresponding vector-valued BHT satisfies (again) the same Lp estimates as the BHT itself. Before the present work there was not even a single example of such exponents.

Finally, we prove a biparameter Leibniz rule in mixed norm Lp spaces, answering a question of Kenig in nonlinear dispersive PDE.

Article information

Anal. PDE, Volume 9, Number 8 (2016), 1931-1988.

Received: 20 January 2016
Revised: 12 June 2016
Accepted: 12 July 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42A45: Multipliers 42B15: Multipliers 42B25: Maximal functions, Littlewood-Paley theory 42B37: Harmonic analysis and PDE [See also 35-XX]

vector-valued estimates for singular and multilinear operators tensor products in mixed norms Leibniz rule AKNS systems


Benea, Cristina; Muscalu, Camil. Multiple vector-valued inequalities via the helicoidal method. Anal. PDE 9 (2016), no. 8, 1931--1988. doi:10.2140/apde.2016.9.1931.

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  • M. Bateman, “Single annulus $L\sp p$ estimates for Hilbert transforms along vector fields”, Rev. Mat. Iberoam. 29:3 (2013), 1021–1069.
  • M. Bateman and C. Thiele, “$L\sp p$ estimates for the Hilbert transforms along a one-variable vector field”, Anal. PDE 6:7 (2013), 1577–1600.
  • C. Benea and C. Muscalu, “Quasi-Banach valued inequalities via the helicoidal method”, preprint, 2016.
  • A. Benedek, A.-P. Calderón, and R. Panzone, “Convolution operators on Banach space valued functions”, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365.
  • J. Bourgain, “Vector-valued singular integrals and the $H\sp 1$-BMO duality”, pp. 1–19 in Probability theory and harmonic analysis (Cleveland, Ohio, 1983), Monogr. Textbooks Pure Appl. Math. 98, Dekker, New York, 1986.
  • D. L. Burkholder, “A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions”, pp. 270–286 in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I (Chicago, IL, 1981), Wadsworth, Belmont, CA, 1983.
  • L. Carleson, “On convergence and growth of partial sums of Fourier series”, Acta Math. 116 (1966), 135–157.
  • M. Christ and A. Kiselev, “Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results”, J. Amer. Math. Soc. 11:4 (1998), 771–797.
  • M. Christ and A. Kiselev, “Maximal functions associated to filtrations”, J. Funct. Anal. 179:2 (2001), 409–425.
  • M. Christ and A. Kiselev, “WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials”, J. Funct. Anal. 179:2 (2001), 426–447.
  • A. Culiuc, F. Di Plinio, and Y. Ou, “Domination of multilinear singular integrals by positive sparse forms”, 2016.
  • C. Demeter and P. Silva, “Some new light on a few classical results”, Colloq. Math. 140:1 (2015), 129–147.
  • F. Di Plinio and Y. Ou, “Banach-valued multilinear singular integrals”, preprint, 2015.
  • D. L. Fernandez, “Vector-valued singular integral operators on $L\sp p$-spaces with mixed norms and applications”, Pacific J. Math. 129:2 (1987), 257–275.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies 116, North-Holland, Amsterdam, 1985.
  • L. Grafakos and X. Li, “The disc as a bilinear multiplier”, Amer. J. Math. 128:1 (2006), 91–119.
  • T. P. Hyt önen and M. T. Lacey, “Pointwise convergence of vector-valued Fourier series”, Math. Ann. 357:4 (2013), 1329–1361.
  • T. P. Hyt önen, M. T. Lacey, and I. Parissis, “The vector valued quartile operator”, Collect. Math. 64:3 (2013), 427–454.
  • J.-L. Journé, “Calderón–Zygmund operators on product spaces”, Rev. Mat. Iberoamericana 1:3 (1985), 55–91.
  • C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle”, Comm. Pure Appl. Math. 46:4 (1993), 527–620.
  • R. Kesler, “Mixed estimates for degenerate multi-linear operators associated to simplexes”, J. Math. Anal. Appl. 424:1 (2015), 344–360.
  • M. Lacey and C. Thiele, “On Calderón's conjecture”, Ann. of Math. $(2)$ 149:2 (1999), 475–496.
  • A. K. Lerner, “A simple proof of the $A_2$ conjecture”, Int. Math. Res. Not. 2013:14 (2013), 3159–3170.
  • Y. Meyer and R. Coifman, Wavelets: Calderón–Zygmund and multilinear operators, Cambridge Studies in Advanced Mathematics 48, Cambridge University Press, 1997.
  • C. Muscalu and W. Schlag, Classical and multilinear harmonic analysis. Vol. II, Cambridge Studies in Advanced Mathematics 138, Cambridge University Press, 2013.
  • C. Muscalu, J. Pipher, T. Tao, and C. Thiele, “Bi-parameter paraproducts”, Acta Math. 193:2 (2004), 269–296.
  • C. Muscalu, T. Tao, and C. Thiele, “$L\sp p$ estimates for the biest, II: The Fourier case”, Math. Ann. 329:3 (2004), 427–461.
  • C. Muscalu, J. Pipher, T. Tao, and C. Thiele, “Multi-parameter paraproducts”, Rev. Mat. Iberoam. 22:3 (2006), 963–976.
  • C. Muscalu, T. Tao, and C. Thiele, “The bi-Carleson operator”, Geom. Funct. Anal. 16:1 (2006), 230–277.
  • R. Oberlin, A. Seeger, T. Tao, C. Thiele, and J. Wright, “A variation norm Carleson theorem”, J. Eur. Math. Soc. $($JEMS$)$ 14:2 (2012), 421–464.
  • R. E. A. C. Paley, “Some theorems on orthonormal functions”, Studia Math. 3 (1931), 226–238.
  • Z. Ruan, “Multi-parameter Hardy spaces via discrete Littlewood–Paley theory”, Anal. Theory Appl. 26:2 (2010), 122–139.
  • J. L. Rubio de Francia, “A Littlewood–Paley inequality for arbitrary intervals”, Rev. Mat. Iberoamericana 1:2 (1985), 1–14.
  • J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torrea, “Calderón–Zygmund theory for operator-valued kernels”, Adv. in Math. 62:1 (1986), 7–48.
  • P. Silva, “Vector-valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems”, J. Lond. Math. Soc. $(2)$ 90:3 (2014), 695–724.
  • C. Thiele, Wave packet analysis, CBMS Regional Conference Series in Mathematics 105, American Mathematical Society, Providence, RI, 2006.
  • R. H. Torres and E. L. Ward, “Leibniz's rule, sampling and wavelets on mixed Lebesgue spaces”, J. Fourier Anal. Appl. 21:5 (2015), 1053–1076.