The Hilbert transform is an mathematical object intimately related to physical situations such as charge distribution. It and related objects such as the Cauchy transform, occur in a range of analytical questions, from orthogonal polynomials to dynamics, to analytic function spaces and partial differential equations. The two-weight problem for the Hilbert transform concerns a characterization of those pairs of measures so that the transform maps Hilbert space of one measure into Hilbert space of the other. This problem has been solved by the proposer, Eric T Sawyer and Ignacio Uriate-Tuero, in work sponsored by the National Science Foundation. This has immediate application to for instance a long-standing conjecture of Sarason in operator theory. The goal of this proposal, is to build upon this success, turning to extensions of this characterization for other natural questions, such as that for the Cauchy transform, which is fundamental for the theory of analytic function spaces.
This work seeks to detail subtle properties of transforms which closely model physical situations, such as electrical charge distributions. As such, the techniques will impact the range of analytical tools that can be brought to bear on these questions that entail fine knowledge about these operators. In addition, the proposer carries out a variety of roles in mentoring and training a next generation of scientific workforce. This includes: (1) The work of the investigator on an MCTP grant to recruit and train talented undergraduate majors at the Georgia Institute of Technology. (2) Directing two graduate students in their thesis work. (3) Mentoring a number of postdoctoral fellows. (4) Conference organization at research centers in the US, Canada and Europe. (5) Editorial work for the Proceedings of the American Mathematical Society and the Journal of Geometric Analysis. (6) Dissemination of research accomplishments and goals, including lectures at venues around the world.
Throughout signal processing, and modeling of complex systems, one encounters transforms of signals that have a subtle non-local behavior. Accordingly, one wants to know the fullest circumstances in which these transforms are certain to yield another signal. This is the central question of weighted estimates for these transforms. This question has throughly investigated, with the support of this grant. One result of Lacey-Petermichl-Reguera, set out a simple conceptual approach to this question in a classical setting. This result, which has 72 citations the Google scholar database, set off a race to settle the so-called A2 conjecture. This led to the final form of a theory of the classical (Muckenhoupt) weighted theory, first started in the early 1970's. The finest question in the weighted theory is the complete characterization of the individual two weight inequality for the Hilbert transform. This is a core question, one whose history and applications reach across three decades, and several areas of mathematics. The main line of investigation that led to the solution started during this grant period. Its resolution gives immediate implication, and the full meaning of the solution is under intense investigation. The PI actively engages in recruiting and training a new generation of scientists. At the undergraduate level, I advise majors about placement into graduate programs, and those that I advise are placed in top programs throughout the country. Georgia Tech also trains a number of new US citizens, who are an important, and under utilized source of stem majors. I also advise these students. Graduate students that I had a role in advising include James Scurry, Maria Carmen Reguera, and Gagik Amirkhanyan. All three have graduated. Publications written during the grant period. Citation counts collected from Google Scholar. Lacey, Michael T. Two weight inequality for the Hilbert transform: A real variable characterization, II, Duke Math J, to appear. Lacey, Michael T., et al, Two weight inequality for the Hilbert transform: A real variable characterization, I Duke Math J, to appear. (17 citations) Lacey, Michael. On the A2 inequality for Calderón-Zygmund operators. Recent advances in harmonic analysis and applications, 235--242, Springer Proc. Math. Stat., 25, Springer, New York, 2013. Lacey, Michael, Konstantin Ilyich Oskolkov, Recent advances in harmonic analysis and applications, 35--37, Springer Proc. Math. Stat., 25, Springer, New York, 2013. Hytönen, Tuomas; Lacey, Michael. Pointwise convergence of vector-valued Fourier series. Math. Ann. 357 (2013), no. 4, 1329--1361. Hytönen, Tuomas; Lacey, Michael; Parissis, Ioannis. The vector valued quartile operator. Collect. Math. 64 (2013), no. 3, 427--454. Hytönen, Tuomas; Lacey, Michael; Pérez, Carlos. Sharp weighted bounds for the q-variation of singular integrals. Bull. Lond. Math. Soc. 45 (2013), no. 3, 529--540. (15 citations) Hytönen, Tuomas; Lacey, Michael. The Ap-A-infty inequality for general Calderón-Zygmund operators. Indiana Univ. Math. J. 61 (2012), no. 6, 2041--2092. (30 citations) Lacey, Michael; Petermichl, Stefanie; Pipher, Jill; Wick, Brett. Multi-parameter Div-Curl lemmas. Bull. Lond. Math. Soc. 44 (2012), no. 6, 1123--1131. Do, Yen; Lacey, Michael. Weighted bounds for variational Walsh-Fourier series. J. Fourier Anal. Appl. 18 (2012), no. 6, 1318--1339. Do, Yen; Lacey, Michael. Weighted bounds for variational Fourier series. Studia Math. 211 (2012), no. 2, 153--190. Hytönen, Tuomas; Lacey, Michael, et al, Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on Ap weighted spaces. J. Anal. Math. 118 (2012), no. 1, 177--220. (30 citations) Lacey, Michael. An Ap-Ainfty inequality for the Hilbert transform. Houston J. Math. 38 (2012), no. 3, 799--814. Lacey, Michael, et al, A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure. Anal. PDE 5 (2012), no. 1, 1--60. Lacey, Michael, et al, A two weight inequality for the Hilbert transform assuming an energy hypothesis. J. Funct. Anal. 263 (2012), no. 2, 305--363. (13 citations) Do, Yen Q.; Lacey, Michael. On the convergence of lacunacy Walsh-Fourier series. Bull. Lond. Math. Soc. 44 (2012), no. 2, 241--254. (6 citations) Lacey, Michael. The linear bound in A2 for Calderón-Zygmund operators: a survey. Marcinkiewicz centenary volume, 97--114, Banach Center Publ., 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011. Bilyk, Dmitriy; Lacey, Michael, et al, A three-dimensional signed small ball inequality. Dependence in probability, analysis and number theory, 73--87, Kendrick Press, Heber City, UT, 2010. Lacey, Michael; Petermichl, Stefanie; Reguera, Maria Carmen. Sharp A2 inequality for Haar shift operators. Math. Ann. 348 (2010), no. 1, 127--141. (72 citations) Lacey, Michael; Moen, Kabe; Pérez, Carlos; Torres, Rodolfo H. Sharp weighted bounds for fractional integral operators. J. Funct. Anal. 259 (2010), no. 5, 1073--1097. (37 citations) Lacey, Michael; Petermichl, Stefanie; Pipher, Jill; Wick, Brett. Iterated Riesz commutators: a simple proof of boundedness. Harmonic analysis and partial differential equations, 171--178, Contemp. Math., 505, Amer. Math. Soc., Providence, RI, 2010. Lacey, Michael; Sawyer, Eric; Uriarte-Tuero, Ignacio. Astala's conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane. Acta Math. 204 (2010), no. 2, 273--292. (23 citations)
