By R. Beals

As soon as upon a time scholars of arithmetic and scholars of technological know-how or engineering took an analogous classes in mathematical research past calculus. Now it's common to split" complex arithmetic for technological know-how and engi neering" from what should be known as "advanced mathematical research for mathematicians." it sort of feels to me either worthwhile and well timed to aim a reconciliation. The separation among types of classes has bad results. Mathe matics scholars opposite the ancient improvement of study, studying the unifying abstractions first and the examples later (if ever). technology scholars research the examples as taught generations in the past, lacking sleek insights. a decision among encountering Fourier sequence as a minor example of the repre sentation idea of Banach algebras, and encountering Fourier sequence in isolation and built in an advert hoc demeanour, isn't any selection in any respect. you can realize those difficulties, yet much less effortless to counter the legiti mate pressures that have resulted in a separation. glossy arithmetic has broadened our views through abstraction and impressive generalization, whereas constructing thoughts that could deal with classical theories in a definitive method. nevertheless, the applier of arithmetic has persisted to wish a number of convinced instruments and has no longer had the time to obtain the broadest and so much definitive grasp-to study valuable and enough stipulations whilst uncomplicated enough stipulations will serve, or to profit the overall framework surround ing diverse examples.

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**Example text**

Clearly, she knew what was important. Mike Wilson 44 Remembering Cora Sadosky From left to right: Weissman, Cora, Manuel Sadosky, Rome 1948 Photo Courtesy of Cora Sol Goldstein, photographer unknown Graduation Photo: Cora Sadosky, PhD University of Chicago, 1965 Photo Courtesy of Cora Sol Goldstein, photographer unknown Remembering Cora Sadosky From left to right: [unknown], Zygmund, Cora Sadosky, Chicago, 1981 Photo Courtesy of Cora Sol Goldstein, photographer unknown 45 46 Remembering Cora Sadosky Group photo, taken in Caracas in January 1994, on the occasion of Mischa Cotlar’s conference celebrating his 80th birthday.

T2 /, which as usual is defined as in (4) but with averages on cubes (squares) in T2 . Multiparameter Analysis Other BMO spaces surfaced in Cotlar–Sadosky’s work on the product Hilbert transform and the study of Hankel operators, as well as in further collaborations of Cora with Ferguson [12] and with Pott [20]. All such spaces coincide in 1-d but they become different substitutes for BMO in product dimensions. Moreover different characterizations in the one parameter case produce different spaces in the multiparameter setting.

Ferguson and Sadosky characterized the spaces bmo and BMOrec in terms of (big) Hankel and little Hankel operators [12]. We shall only state the characterization of bmo. Let P W L2 ! H2 be the projection operator and P? D I P. f / D P? ˆf / and let Mˆ f D ˆf . Theorem (Ferguson–Sadosky). The following are equivalent. • • • • The function ˆ 2 bmo. ˆ and ˆN are both bounded on H2 . The commutators ŒMˆ ; Hj D Mˆ Hj Hj Mˆ are bounded on L2 . The commutator ŒMˆ ; H1 H2 D Mˆ H1 H2 H1 H2 Mˆ is bounded on L2 .