## Analysis & PDE

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

### Multiple vector-valued inequalities via the helicoidal method

#### Abstract

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 $BHT⊗Π$ 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

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

Dates
Revised: 12 June 2016
Accepted: 12 July 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843379

Digital Object Identifier
doi:10.2140/apde.2016.9.1931

Mathematical Reviews number (MathSciNet)
MR3599522

Zentralblatt MATH identifier
1361.42005

#### Citation

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. https://projecteuclid.org/euclid.apde/1510843379

#### References

• 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.