## Analysis & PDE

• Anal. PDE
• Volume 11, Number 6 (2018), 1535-1586.

### Square function estimates for the Bochner–Riesz means

Sanghyuk Lee

#### Abstract

We consider the square-function (known as Stein’s square function) estimate associated with the Bochner–Riesz means. The previously known range of the sharp estimate is improved. Our results are based on vector-valued extensions of Bennett, Carbery and Tao’s multilinear (adjoint) restriction estimate and an adaptation of an induction argument due to Bourgain and Guth. Unlike the previous work by Bourgain and Guth on $L p$ boundedness of the Bochner–Riesz means in which oscillatory operators associated to the kernel were studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing, which is essential for obtaining the sharp estimates for the square functions.

#### Article information

Source
Anal. PDE, Volume 11, Number 6 (2018), 1535-1586.

Dates
Accepted: 12 January 2018
First available in Project Euclid: 23 May 2018

https://projecteuclid.org/euclid.apde/1527040819

Digital Object Identifier
doi:10.2140/apde.2018.11.1535

Mathematical Reviews number (MathSciNet)
MR3803718

Zentralblatt MATH identifier
06881250

Subjects
Primary: 42B15: Multipliers
Secondary: 35B65: Smoothness and regularity of solutions

#### Citation

Lee, Sanghyuk. Square function estimates for the Bochner–Riesz means. Anal. PDE 11 (2018), no. 6, 1535--1586. doi:10.2140/apde.2018.11.1535. https://projecteuclid.org/euclid.apde/1527040819

#### References

• M. Annoni, “Almost everywhere convergence for modified Bochner–Riesz means at the critical index for $p\geq 2$”, Pacific J. Math. 286:2 (2017), 257–275.
• I. Bejenaru, “Optimal multilinear restriction estimates for a class of surfaces with curvature”, preprint, 2016.
• I. Bejenaru, “The optimal trilinear restriction estimate for a class of hypersurfaces with curvature”, Adv. Math. 307 (2017), 1151–1183.
• J. Bennett, “Aspects of multilinear harmonic analysis related to transversality”, pp. 1–28 in Harmonic analysis and partial differential equations, edited by P. Cifuentes et al., Contemp. Math. 612, Amer. Math. Soc., Providence, RI, 2014.
• J. Bennett, A. Carbery, and T. Tao, “On the multilinear restriction and Kakeya conjectures”, Acta Math. 196:2 (2006), 261–302.
• J. Bennett, N. Bez, T. C. Flock, and S. Lee, “Stability of the Brascamp–Lieb constant and applications”, preprint, 2015.
• J. Bourgain, “Besicovitch type maximal operators and applications to Fourier analysis”, Geom. Funct. Anal. 1:2 (1991), 147–187.
• J. Bourgain, “On the restriction and multiplier problems in $\mathbb{R}^3$”, pp. 179–191 in Geometric aspects of functional analysis (1989–90), edited by V. D. Milman and J. Lindenstrauss, Lecture Notes in Math. 1469, Springer, 1991.
• J. Bourgain, “$L^p$-estimates for oscillatory integrals in several variables”, Geom. Funct. Anal. 1:4 (1991), 321–374.
• J. Bourgain, “On the Schrödinger maximal function in higher dimension”, Tr. Mat. Inst. Steklova 280 (2013), 53–66. In Russian; translated in Proc. Steklov Inst. Math. 280:1 (2013), 46–60.
• J. Bourgain and C. Demeter, “The proof of the $l^2$ decoupling conjecture”, Ann. of Math. $(2)$ 182:1 (2015), 351–389.
• J. Bourgain and L. Guth, “Bounds on oscillatory integral operators based on multilinear estimates”, Geom. Funct. Anal. 21:6 (2011), 1239–1295.
• A. Carbery, “The boundedness of the maximal Bochner–Riesz operator on $L\sp{4}(\mathbb{R}\sp{2})$”, Duke Math. J. 50:2 (1983), 409–416.
• A. Carbery, “Radial Fourier multipliers and associated maximal functions”, pp. 49–56 in Recent progress in Fourier analysis (El Escorial, 1983), edited by I. Peral and J. L. Rubio de Francia, North-Holland Math. Stud. 111, North-Holland, Amsterdam, 1985.
• A. Carbery and S. I. Valdimarsson, “The endpoint multilinear Kakeya theorem via the Borsuk–Ulam theorem”, J. Funct. Anal. 264:7 (2013), 1643–1663.
• A. Carbery, G. Gasper, and W. Trebels, “Radial Fourier multipliers of $L\sp{p}(\mathbb{R}\sp{2})$”, Proc. Nat. Acad. Sci. U.S.A. 81:10 (1984), 3254–3255.
• A. Carbery, J. L. Rubio de Francia, and L. Vega, “Almost everywhere summability of Fourier integrals”, J. London Math. Soc. $(2)$ 38:3 (1988), 513–524.
• L. Carleson and P. Sjölin, “Oscillatory integrals and a multiplier problem for the disc”, Studia Math. 44 (1972), 287–299.
• M. Christ, “On almost everywhere convergence of Bochner–Riesz means in higher dimensions”, Proc. Amer. Math. Soc. 95:1 (1985), 16–20.
• M. Christ, “Weak type endpoint bounds for Bochner–Riesz multipliers”, Rev. Mat. Iberoamericana 3:1 (1987), 25–31.
• M. Christ, “Weak type $(1,1)$ bounds for rough operators”, Ann. of Math. $(2)$ 128:1 (1988), 19–42.
• X. Du, L. Guth, and X. Li, “A sharp Schrödinger maximal estimate in $\mathbb{R}^2$”, Ann. of Math. $(2)$ 186:2 (2017), 607–640.
• C. Fefferman, “Inequalities for strongly singular convolution operators”, Acta Math. 124 (1970), 9–36.
• C. Fefferman, “The multiplier problem for the ball”, Ann. of Math. $(2)$ 94 (1971), 330–336.
• C. Fefferman, “A note on spherical summation multipliers”, Israel J. Math. 15 (1973), 44–52.
• J. L. Rubio de Francia, “A Littlewood–Paley inequality for arbitrary intervals”, Rev. Mat. Iberoamericana 1:2 (1985), 1–14.
• L. Guth, “The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture”, Acta Math. 205:2 (2010), 263–286.
• L. Guth, “A restriction estimate using polynomial partitioning”, J. Amer. Math. Soc. 29:2 (2016), 371–413.
• L. Guth, “Restriction estimates using polynomial partitioning, II”, preprint, 2016.
• L. Guth, J. Hickman, and M. Iliopoulou, “Sharp estimates for oscillatory integral operators via polynomial partitioning”, preprint, 2017.
• L. Hörmander, “Oscillatory integrals and multipliers on $F\!L\sp{p}$”, Ark. Mat. 11 (1973), 1–11.
• S. Lee, “Improved bounds for Bochner–Riesz and maximal Bochner–Riesz operators”, Duke Math. J. 122:1 (2004), 205–232.
• S. Lee, “Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces”, J. Funct. Anal. 241:1 (2006), 56–98.
• S. Lee, “On pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}^2$”, Int. Math. Res. Not. 2006 (2006), art. id. 32597.
• S. Lee and A. Seeger, “On radial Fourier multipliers and almost everywhere convergence”, J. Lond. Math. Soc. $(2)$ 91:1 (2015), 105–126.
• S. Lee and A. Vargas, “Sharp null form estimates for the wave equation”, Amer. J. Math. 130:5 (2008), 1279–1326.
• S. Lee and A. Vargas, “Restriction estimates for some surfaces with vanishing curvatures”, J. Funct. Anal. 258:9 (2010), 2884–2909.
• S. Lee and A. Vargas, “On the cone multiplier in $\mathbb{R}^3$”, J. Funct. Anal. 263:4 (2012), 925–940.
• S. Lee, K. M. Rogers, and A. Seeger, “Improved bounds for Stein's square functions”, Proc. Lond. Math. Soc. $(3)$ 104:6 (2012), 1198–1234.
• S. Lee, K. M. Rogers, and A. Seeger, “On space-time estimates for the Schrödinger operator”, J. Math. Pures Appl. $(9)$ 99:1 (2013), 62–85.
• S. Lee, K. M. Rogers, and A. Seeger, “Square functions and maximal operators associated with radial Fourier multipliers”, pp. 273–302 in Advances in analysis: the legacy of Elias M. Stein, edited by C. Fefferman et al., Princeton Math. Ser. 50, Princeton Univ. Press, 2014.
• Y. Ou and H. Wang, “A cone restriction estimate using polynomial partitioning”, preprint, 2017.
• J. Ramos, “The trilinear restriction estimate with sharp dependence on the transversality”, preprint, 2016.
• A. Seeger, “On quasiradial Fourier multipliers and their maximal functions”, J. Reine Angew. Math. 370 (1986), 61–73.
• A. Seeger, “Endpoint inequalities for Bochner–Riesz multipliers in the plane”, Pacific J. Math. 174:2 (1996), 543–553.
• B. Shayya, “Weighted restriction estimates using polynomial partitioning”, Proc. Lond. Math. Soc. $(3)$ 115:3 (2017), 545–598.
• E. M. Stein, “Localization and summability of multiple Fourier series”, Acta Math. 100 (1958), 93–147.
• E. M. Stein, “Oscillatory integrals in Fourier analysis”, pp. 307–355 in Beijing lectures in harmonic analysis (Beijing, 1984), edited by E. M. Stein, Ann. of Math. Stud. 112, Princeton Univ. Press, 1986.
• E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
• G. Sunouchi, “On the Littlewood–Paley function $g\sp*$ of multiple Fourier integrals and Hankel multiplier transformations”, Tôhoku Math. J. $(2)$ 19 (1967), 496–511.
• T. Tao, “Weak-type endpoint bounds for Riesz means”, Proc. Amer. Math. Soc. 124:9 (1996), 2797–2805.
• T. Tao, “The weak-type endpoint Bochner–Riesz conjecture and related topics”, Indiana Univ. Math. J. 47:3 (1998), 1097–1124.
• T. Tao, “The Bochner–Riesz conjecture implies the restriction conjecture”, Duke Math. J. 96:2 (1999), 363–375.
• T. Tao, “On the maximal Bochner–Riesz conjecture in the plane for $p<2$”, Trans. Amer. Math. Soc. 354:5 (2002), 1947–1959.
• T. Tao, “A sharp bilinear restrictions estimate for paraboloids”, Geom. Funct. Anal. 13:6 (2003), 1359–1384.
• T. Tao and A. Vargas, “A bilinear approach to cone multipliers, I: Restriction estimates”, Geom. Funct. Anal. 10:1 (2000), 185–215.
• T. Tao and A. Vargas, “A bilinear approach to cone multipliers, II: Applications”, Geom. Funct. Anal. 10:1 (2000), 216–258.
• T. Tao, A. Vargas, and L. Vega, “A bilinear approach to the restriction and Kakeya conjectures”, J. Amer. Math. Soc. 11:4 (1998), 967–1000.
• F. Temur, “A Fourier restriction estimate for surfaces of positive curvature in $\mathbb{R}^6$”, Rev. Mat. Iberoam. 30:3 (2014), 1015–1036.
• P. A. Tomas, “A restriction theorem for the Fourier transform”, Bull. Amer. Math. Soc. 81 (1975), 477–478.
• L. Wisewell, “Kakeya sets of curves”, Geom. Funct. Anal. 15:6 (2005), 1319–1362.
• T. Wolff, “A sharp bilinear cone restriction estimate”, Ann. of Math. $(2)$ 153:3 (2001), 661–698.
• R. Zhang, “The endpoint perturbed Brascamp–Lieb inequality with examples”, preprint, 2015.