Ad Honorem Yves Meyer For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1756 1378 Notices of the AMS Volume 65, Number 11 Ronald Coifman, Guest Editor of papers in areas of application ranging from signal processing to medical diagnostics. Modern engineering depends on his methods. Numerical analysis uses his Introduction tools for efficient numerical computation of linear and Yves Meyer was awarded the 2017 Abel Prize. His work nonlinear maps. has impacted mathematics in a broad and profound More recently Meyer has introduced new tools for way. Perhaps even more importantly, he has led a broad, analysis of the Navier-Stokes equations for fluid flow, multifaceted, worldwide network of research collabora- discovering remarkable profound mechanisms relating tions of mathematics with music, chemistry, physics, and oscillation to stability and blowups. signal processing. He has made seminal contributions Some of Meyer’s close collaborators have kindly pro- in a number of fields, from number theory to applied vided some descriptions of his work, with a goal of mathematics. Meyer started his career on the interface between covering a broad panorama of analysis. Stéphane Mallat, Fourier analysis and number theory. Early in his career who formalized with Meyer the orthogonal multiresolu- he introduced the theory of model sets [1], which have tion framework, describes Meyer’s celebrated work on become an important tool in the mathematical study wavelets. His student Stéphane Jaffard (1989), who wrote of aperiodic order two years before the discovery of the account [J] of his Abel Prize for the Société Mathé- Penrose pavings by Roger Penrose and ten years before matique de France Gazette des Mathématiciens, focuses the discovery of quasi-crystals by Dan Shechtman. on time-frequency analysis. Alexander Olevskii describes Around 1975 he initiated the field of nonlinear Fourier Meyer’s Sets, which modeled quasi-crystals before they analysis as a tool for organizing and analyzing nonlinear were discovered. My own contribution describes Meyer’s functional transformations of mathematics. He devel- work on nonlinear Fourier analysis. His student Albert oped all the fundamental tools and concepts necessary Cohen (1990) describes Meyer’s impact on sparse analysis. to understand the nonlinear dependence of solutions of Meyer’s work is characterized by an extraordinary boundary value problems on the shape of the bound- depth, solving longstanding problems, and starting new ary. In particular, together with Coifman and McIntosh fields of mathematics and applications. These seminal, [4], he solved in 1982 the last outstanding problem of broad contributions have had a profound impact on classical harmonic analysis by proving the continuity of the Cauchy integral operator on Lipschitz curves. It had different areas of science and establish him as a major been the key obstacle to the solvability of boundary figure in mathematics. value problems for Lipshitz domains (e.g. domains with corners). His methodologies prepared the way for Bony’s References para-differential calculus, Wu’s proof of the existence [DS] Bjørn Ian Dundas and Christian Skau, Inter- of water waves in three dimensions, and the proof of view with Abel Laureate Yves Meyer, Newsletter Kato’s conjecture, essentially changing the landscape of Eur. Math. Soc., Sept. 2017, 14–22, www.ems-ph.org analysis. /journals/newsletter/pdf/2017-09-105.pdf#page=16, pp. 14–22; reprinted in the May 2018 Notices https: Around 1984 Meyer [6]–[8] discovered the relation //www.ams.org/journals/notices/201805/rnoti-p520 between the analytic tools used in harmonic analysis .pdf. MR3726777 and various signal processing algorithms used in seismic [J] S. Jaffard, Yves Meyer, Prix Abel 2017 (French) exploration. In his Abel Prize interview [DS], Meyers said, Gaz. Math. No. 153 (2017), 20–26. smf4.emath.fr “Morlet, Grossmann, and Daubechies were in a sense ahead /Publications/Gazette/2017/153/. MR3701591 of me in their work on wavelets. So I was the ‘Quatrieme Mousquetaire.’ They were Les Trois Mousquetaires.” He A Few Representative Works by Meyer recognized their work as a rediscovery of Calderón’s [1] Algebraic Numbers and Harmonic Analysis, North Holland, formulas in harmonic analysis, thereby bridging fifty New York (1972). MR0485769 years of multiscale harmonic analysis with “wavelets.” [2] Wavelets and Operators, Vol. 1 and 2, revised version of 3, This discovery led later to the construction of the Meyer Cambridge Univ. Press (1992, 1997). MR1719426 wavelet basis, an orthonormal basis of functions localized [3] Wavelets, paraproducts and Navier–Stokes equations, in in space and frequency. His work inspired Daubechies to Current Developments in Mathematics 1996, MIT Press discover the compactly supported orthonormal wavelet (1997). bases, which profoundly affected the field of engineering, [4] (with R. R. Coifman and A. McIntosh) L’intégrale de Cauchy leading in subsequent work to nonlinear adapted Fourier sur les courbes Lipschitziennes, Annals of Math. 116 (1982) 361–387. MR672839 analysis and signal processing [3]–[5]. [5] (with Marco Cannone) Littlewood–Paley decompositions and The technological impact has been remarkable. For Navier-Stokes equations, J. Methods Appl. Anal. (1995) 307– example, the current JPEG 2000 standard for image 319. MR1362019 compression has evolved from the wavelet tools invented [6] (with A. Grossmann, and I. Daubechies) Painless nonorthog- by Meyer. The field of wavelet analysis has thousands onal expansions. Journal of Mathematical Physics 27, 1271 (1986). MR836025 Ronald Coifman is professor of mathematics at Yale University. [7] Wavelets and Applications, Proc. Int. Congr. Math. Kyoto His email address is [email protected]. (1990) 1619–1626. Math. Soc. Japan. December 2018 Notices of the AMS 1379 [8] (with M. Farge, E. Goirand, F. Pascal, and V. Wickerhauser) Lorenzo Brandolese 2001 Improved predictability of two-dimensional turbulent flows Diego Chamorro 2006 using wavelet packet compression, Fluid Dynamics Research Jérôme Gilles 2006 (1992) 229–250. Xiaolong Li 2006 Photo Credit Ronald Coifman Opening photo courtesy of Stéphane Jaffard. Yves Meyer’s Work on Nonlinear Fourier Analysis: Meyer’s PhD students Beyond Calderón-Zygmund Antoine Ayache It is a privilege to be able to relate ideas, explorations, Jean-Louis Clerc and visions that Yves, his collaborators, and his students Michel Bruneau (codir. P. A. Meyer) developed over the last forty years, for some of which I Sylvia Dobyinski was an active participant and observer. François Gramain We have had a lot of fun and excitement in this adven- Jean-Lin Journé ture, exploring and discovering beauty and structure. Pierre-Gilles Lemarié-Rieusset I will focus my narrative on the simplest illustrations Noel Lohoue and examples of Yves Meyer’s foundational contributions Martin Meyer to nonlinear harmonic analysis, and try to illuminate and Freddy Paiva motivate some key programmatic issues that continue Fabrice Planchon and build beyond the Calderón-Zygmund vision and Philippe Tchamitchian program. It was Zygmund’s view that harmonic analysis Chantale Tran-Oberlé provides the infrastructure linking all areas of analysis, Mohamed El Hodaibi from complex analysis to partial differential equations to Fatma Trigui probability and geometry. Taoufik El Bouayachi In particular he pushed forward the idea that the Oscar Barraza remarkable tools of complex analysis, such as contour Henri Oppenheim integration, conformal mappings, and factorization, used Patrick Andersson to provide miraculous proofs in real analysis should be Guillaume Bernuau deciphered and converted to real variable tools. Together Mehdi Abouda with Calderón, he bucked the trend for abstraction, preva- Frederic Oru lent at the time, and formed a school pushing forward this Ramzi Labidi interplay between real and complex analysis. A principal Soulaymane Korry (codir. Bernard Maurey) bridge was provided by real variable methods, multiscale Ali Haddad analysis, Littlewood Paley theory, and related Calderón Aline Bonami 1970 representation formulas, later rediscovered by Morlet and Jean-Pierre Schreiber 1972 others. They will be discussed here in relation to wavelets. Marc Frisch 1977 Jean-Paul Allouche 1978 Bilinear Convolutions, the Calderón Commutator, Salifou Tembely Complex Analysis, and Paraproducts Qixiang Yang In order to understand some of the basic ideas and Guillaume Bernuau methods introduced by Meyer and to illustrate the scope François Lust-Piquard 1978 (codir. N.Varopoulos) of the program, we start with the basic example of “para- Guy David 1981 calculus” introduced by Calderón as a bilinear operator Michel Zinsmeister 1981 needed to extend smooth pseudo-differential calculus to Alain Yger 1982 rough environments. He managed by an analytical tour Gérard Bourdaud 1983 de force using complex function theory to prove that Michel Emsalem 1987 the Calderón commutator defined below is a bounded Miguel Escobedo Martinez 1988 (codir. H.Brezis) operator on 퐿2. Pascal Auscher 1989 Given a Lipschitz function 퐴 on the real line (so ′ ∞ Stéphane Jaffard 1989 퐴 ∶= 푎 ∈ 퐿 (ℝ)) one formally defines the linear operator Albert Cohen 1990 퐶1(푓) by the formula Sylvain Durand 1993 퐴(푥) − 퐴(푦) (1) 퐶 (푓) = ∫ 푓(푦)푑푦 1 2 Marco Cannone 1994 ℝ (푥 − 푦) Khalid Daoudi 1996 (codir. J.Levy-Véhel, INRIA) = (퐴|푑/푑푥| − |푑/푑푥|퐴)푓 = [퐴, |푑/푑푥|](푓), Abderrafiaa El-Kalay 1996 Hong Xu 1996 where the meaning of the absolute value of the derivative Patrik Andersson 1997 operator is given as (2) |푑/푑푥|푓(푥) = 1/2휋 ∫ exp(푖푥휉)|휉|푓(휉)푑휉̂ ℝ 1380 Notices of the AMS Volume 65, Number 11 and 푓̂ is the Fourier transform of 푓. Applications to Analytic Dependence This is the first commutator of Calderón. The simplest One class of problems in nonlinear Fourier analysis particular case is obtained when 퐴(푥) = 푥 and 퐶1(푓) concerns the nonlinear analytic dependence of linear becomes the classical Hilbert transform. He introduced operators on functional arguments. As we will see, such an auxiliary related operator, linking complex function problems are deeply connected to all aspects of harmonic theory and Fourier analysis as we now describe.