Multiple Hilbert transforms associated with polynomials

Kim, Joonil

doi:10.1090/memo/1119

Multiple Hilbert transforms associated with polynomials

About this Title

Joonil Kim

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1119
ISBNs: 978-1-4704-1435-1 (print); 978-1-4704-2505-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1119
Published electronically: January 21, 2015
Keywords:Multiple Hilbert transform, Newton polyhedron, face, cone

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Chapters

Chapter 1. Introduction
Chapter 2. Definitions of Polyhedra, Their Faces and Cones
Chapter 3. Main Theorem and Background
Chapter 4. Combinatorial Lemmas
Chapter 5. Descending Faces vs. Ascending Cones
Chapter 6. Preliminary Results of Analysis
Chapter 7. Proof of Sufficiency
Chapter 8. Necessity Theorem
Chapter 9. Preliminary Lemmas for Necessity Proof
Chapter 10. Proof of Necessity
Chapter 11. Proofs of Corollary 3.1 and Main Theorem 3.1
Chapter 12. Appendix

Abstract

Let $\Lambda =(\Lambda _1,\cdots ,\Lambda _d)$ with $\,\Lambda _\nu \subset \mathbb {Z}_+^n\,$ , and set $\mathcal {P}_{\Lambda }$ the family of all vector polynomials,

$\mathcal {P}_{\Lambda }=\{P_{\Lambda }: P_{\Lambda }(t)=(\sum _{\mathfrak {m}\,\in \Lambda _{1}}\,c_{\mathfrak {m}}^{1}\,t^{\mathfrak {m}},\cdots ,\sum _{\mathfrak {m}\,\in \Lambda _{d}}\,c_{\mathfrak {m}}^{d}\,t^{\mathfrak {m}}) \quad \text {with}\quad t\in \mathbb {R}^n\}.$

Given $P_\Lambda \in \mathcal {P}_{\Lambda }$ , we consider a class of multi-parameter oscillatory singular integrals,

$\mathcal {I}(P_\Lambda ,\xi ,r)=\text {p.v.}\int _{\prod [-r_j,r_j]}e^{i\langle \xi ,P_{\Lambda }(t)\rangle } \frac {dt_1}{t_1}\cdots \frac {dt_n}{t_n} \quad \text {where} \quad \xi \in \mathbb {R}^d, r\in \mathbb {R}_+^n.$

When $n=1$ , the integral $\mathcal {I}(P_\Lambda ,\xi ,r)$ for any $P_\Lambda \in \mathcal {P}_\Lambda$ is bounded uniformly in $\xi$ and $r$ . However, when $n\ge 2$ , the uniform boundedness depends on each individual polynomial $P_\Lambda$ . In this paper, we fix $\Lambda$ and find a necessary and sufficient condition on $\Lambda$ that

$\text {for all} \quad P_\Lambda \in \mathcal {P}_\Lambda , \sup _{\xi , \, r} |\mathcal {I}(P_\Lambda ,\xi ,r)|\le C_{P_\Lambda }<\infty .$

[1] Anthony Carbery, Stephen Wainger, and James Wright, Triple Hilbert transforms along polynomial surfaces in $\Bbb R^4$, Rev. Mat. Iberoam. 25 (2009), no. 2, 471–519. MR 2554163, 10.4171/RMI/573
[2] Anthony Carbery, Stephen Wainger, and James Wright, Singular integrals and the Newton diagram, Collect. Math. Vol. Extra (2006), 171–194. MR 2264209
[3] Anthony Carbery, Stephen Wainger, and James Wright, Double Hilbert transforms along polynomial surfaces in $\bold R^3$, Duke Math. J. 101 (2000), no. 3, 499–513. MR 1740686, 10.1215/S0012-7094-00-10135-4
[4] Yong-Kum Cho, Sunggeum Hong, Joonil Kim, and Chan Woo Yang, Triple Hilbert transforms along polynomial surfaces, Integral Equations Operator Theory 65 (2009), no. 4, 485–528. MR 2576306, 10.1007/s00020-009-1731-9
[5] Yong-Kum Cho, Sunggeum Hong, Joonil Kim, and Chan Woo Yang, Multiparameter singular integrals and maximal operators along flat surfaces, Rev. Mat. Iberoam. 24 (2008), no. 3, 1047–1073. MR 2490209, 10.4171/RMI/566
[6] Michael Christ, Alexander Nagel, Elias M. Stein, and Stephen Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann. of Math. (2) 150 (1999), no. 2, 489–577. MR 1726701, 10.2307/121088
[7] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037
[8] Magali Folch-Gabayet and James Wright, Singular integral operators associated to curves with rational components, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1661–1679 (electronic). MR 2357709, 10.1090/S0002-9947-07-04349-8
[9] Alexander Nagel, Fulvio Ricci, and Elias M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal. 181 (2001), no. 1, 29–118. MR 1818111, 10.1006/jfan.2000.3714
[10] Alexander Nagel, Fulvio Ricci, Elias Stein, and Stephen Wainger, Singular integrals with flag kernels on homogeneous groups, I, Rev. Mat. Iberoam. 28 (2012), no. 3, 631–722. MR 2949616, 10.4171/RMI/688
[11] Alexander Nagel and Malabika Pramanik, Maximal averages over linear and monomial polyhedra, Duke Math. J. 149 (2009), no. 2, 209–277. MR 2541704, 10.1215/00127094-2009-039
[12] Alexander Nagel and Stephen Wainger, $L^{2}$ boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), no. 4, 761–785. MR 0450901
[13] Sanjay Patel, $L^p$ estimates for a double Hilbert transform, Acta Sci. Math. (Szeged) 75 (2009), no. 1-2, 241–264. MR 2533413
[14] Sanjay Patel, Double Hilbert transforms along polynomial surfaces in $\bold R^3$, Glasg. Math. J. 50 (2008), no. 3, 395–428. MR 2451738, 10.1017/S0017089508004291
[15] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152. MR 1484770, 10.1007/BF02392721
[16] Malabika Pramanik and Chan Woo Yang, Double Hilbert transform along real-analytic surfaces in $\Bbb R^{d+2}$, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 363–386. MR 2400397, 10.1112/jlms/jdm113
[17] F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 637–670 (English, with English and French summaries). MR 1182643
[18] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
[19] Elias M. Stein and Brian Street, Multi-parameter singular Radon transforms, Math. Res. Lett. 18 (2011), no. 2, 257–277. MR 2784671, 10.4310/MRL.2011.v18.n2.a6
[20] Elias M. Stein and Brian Street, Multi-parameter singular Radon transforms III: Real analytic surfaces, Adv. Math. 229 (2012), no. 4, 2210–2238. MR 2880220, 10.1016/j.aim.2011.11.016
[21] Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, 10.1090/S0002-9904-1978-14554-6
[22] Brian Street, Multi-parameter singular Radon transforms I: The $L^2$ theory, J. Anal. Math. 116 (2012), 83–162. MR 2892618, 10.1007/s11854-012-0004-8
[23] A. Varchenko, Newton polyhedra and estimations of oscillatory integrals, Functional Anal. Appl. 18 (1976), 175–196.

References

[1] Anthony Carbery, Stephen Wainger, and James Wright, Triple Hilbert transforms along polynomial surfaces in $\mathbb {R}^4$ , Rev. Mat. Iberoam. 25 (2009), no. 2, 471–519. MR 2554163 (2010k:42021), 10.4171/RMI/573
[2] Anthony Carbery, Stephen Wainger, and James Wright, Singular integrals and the Newton diagram, Collect. Math. Vol. Extra (2006), 171–194. MR 2264209 (2008c:42011)
[3] Anthony Carbery, Stephen Wainger, and James Wright, Double Hilbert transforms along polynomial surfaces in $\mathbf {R}^3$ , Duke Math. J. 101 (2000), no. 3, 499–513. MR 1740686 (2001g:42025), 10.1215/S0012-7094-00-10135-4
[4] Yong-Kum Cho, Sunggeum Hong, Joonil Kim, and Chan Woo Yang, Triple Hilbert transforms along polynomial surfaces, Integral Equations Operator Theory 65 (2009), no. 4, 485–528. MR 2576306 (2011b:42037), 10.1007/s00020-009-1731-9
[5] Yong-Kum Cho, Sunggeum Hong, Joonil Kim, and Chan Woo Yang, Multiparameter singular integrals and maximal operators along flat surfaces, Rev. Mat. Iberoam. 24 (2008), no. 3, 1047–1073. MR 2490209 (2010a:42042), 10.4171/RMI/566
[6] Michael Christ, Alexander Nagel, Elias M. Stein, and Stephen Wainger, Singular and maximal Radon transforms: analysis and geometry, Ann. of Math. (2) 150 (1999), no. 2, 489–577. MR 1726701 (2000j:42023), 10.2307/121088
[7] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
[8] Magali Folch-Gabayet and James Wright, Singular integral operators associated to curves with rational components, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1661–1679 (electronic). MR 2357709 (2008i:42024), 10.1090/S0002-9947-07-04349-8
[9] Alexander Nagel, Fulvio Ricci, and Elias M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal. 181 (2001), no. 1, 29–118. MR 1818111 (2001m:22018), 10.1006/jfan.2000.3714
[10] Alexander Nagel, Fulvio Ricci, Elias Stein, and Stephen Wainger, Singular integrals with flag kernels on homogeneous groups, I, Rev. Mat. Iberoam. 28 (2012), no. 3, 631–722. MR 2949616, 10.4171/RMI/688
[11] Alexander Nagel and Malabika Pramanik, Maximal averages over linear and monomial polyhedra, Duke Math. J. 149 (2009), no. 2, 209–277. MR 2541704 (2010j:42043), 10.1215/00127094-2009-039
[12] Alexander Nagel and Stephen Wainger, $L^{2}$ boundedness of Hilbert transforms along surfaces and convolution operators homogeneous with respect to a multiple parameter group, Amer. J. Math. 99 (1977), no. 4, 761–785. MR 0450901 (56 \#9192)
[13] Sanjay Patel, $L^p$ estimates for a double Hilbert transform, Acta Sci. Math. (Szeged) 75 (2009), no. 1-2, 241–264. MR 2533413 (2011a:42018)
[14] Sanjay Patel, Double Hilbert transforms along polynomial surfaces in $\mathbf {R}^3$ , Glasg. Math. J. 50 (2008), no. 3, 395–428. MR 2451738 (2009k:42035), 10.1017/S0017089508004291
[15] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152. MR 1484770 (98j:42009), 10.1007/BF02392721
[16] Malabika Pramanik and Chan Woo Yang, Double Hilbert transform along real-analytic surfaces in $\mathbb {R}^{d+2}$ , J. Lond. Math. Soc. (2) 77 (2008), no. 2, 363–386. MR 2400397 (2009e:42032), 10.1112/jlms/jdm113
[17] F. Ricci and E. M. Stein, Multiparameter singular integrals and maximal functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 637–670 (English, with English and French summaries). MR 1182643 (94d:42020)
[18] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
[19] Elias M. Stein and Brian Street, Multi-parameter singular Radon transforms, Math. Res. Lett. 18 (2011), no. 2, 257–277. MR 2784671 (2012b:44007), 10.4310/MRL.2011.v18.n2.a6
[20] Elias M. Stein and Brian Street, Multi-parameter singular Radon transforms III: Real analytic surfaces, Adv. Math. 229 (2012), no. 4, 2210–2238. MR 2880220, 10.1016/j.aim.2011.11.016
[21] Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453 (80k:42023), 10.1090/S0002-9904-1978-14554-6
[22] Brian Street, Multi-parameter singular radon transforms I: The $L^2$ theory, J. Anal. Math. 116 (2012), 83–162. MR 2892618, 10.1007/s11854-012-0004-8
[23] A. Varchenko, Newton polyhedra and estimations of oscillatory integrals, Functional Anal. Appl. 18 (1976), 175–196.

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