By Juan M. Delgado Sanchez, Tomas Dominguez Benavides

ISBN-10: 9812818448

ISBN-13: 9789812818447

This quantity contains a set of articles through prime researchers in mathematical research. It offers the reader with an in depth evaluate of the present-day examine in several components of mathematical research (complex variable, harmonic research, genuine research and sensible research) that holds nice promise for present and destiny advancements. those evaluation articles are hugely valuable should you are looking to know about those issues, as many effects scattered within the literature are mirrored in the course of the many separate papers featured herein.

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Additional info for Advanced Course Of Mathematical Analysis III: Proceedings of the Third International School La Rabida, Spain, 3 - 7 September 2007

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Blasco and P. Villarroya [1,5]. The second one is the bilinear formulation of the Coifman– Weiss transference method [7] that has been extensively studied in [2–4] by O. Blasco, E. Berkson, M. J. T. Gillespie. We shall only mention one theorem and its application of each of the procedures considered in the just mentioned papers. 1 can be transferred to both situations periodic and discrete. We will also present a detailed proof for the reader to see the tools used in our approaches. The interested reader can consult the references in the bibliography for a further study of the topic.

Indeed, using the expression Bm (f, g)(x) = R f (x − t)g(x + t)dµ(t) one gets Bm (f, g) p3 ≤ R f (· − t)g(· + t) ≤ R f (· − t) = f p1 g p1 p3 d|µ|(t) g(· + t) p2 d|µ|(t) d|µ|(t) = µ p2 1 f p1 g p2 . R However, the case where the symbol m is not smooth has a much shorter story. A very non trivial example is given by m (x) = −i sign (x) which leads to the bilinear Hilbert transform and it was first considered by Lacey and Thiele in [16–18] and then extended to other cases in [10,11]. The solution took many years to be achieved after the formulation of the question by A.

Anal. Appl. 15 (1981), 1–17. 37. -C. R. Acad. Sci. Paris 306 (1988), 55–58. 38. -C. Yoccoz, Th´eor`eme de Siegel, nombres de Bryuno et polynˆ omes quadratiques, Ast´erisque 231 (1995), 3–88. es Keywords: Bilinear Hilbert transform, bilinear maximal functions, transference methods, discretization. 1. Notation and preliminaries These notes contain an extended version of the talk given in the III International Course of Mathematical Analysis held in La R´ abida (Huelva, Spain) in September 2007 and they are based on results appeared in [1–4].

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