By M. V. Velasco, A. Rodriguez-Palacios
This quantity contains a suite of articles by way of top researchers in mathematical research. It offers the reader with an in depth review of recent instructions and advances in themes for present and destiny examine within the box.
Read Online or Download Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004 PDF
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Additional info for Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004
I) For p = 00 the result is trivial: llMflloo 5 I l f l l m ; (ii) but for p = 1 the result is false: indeed, M f ( x ) 1z1-l when IC is large if f is not identically zero (integrability can fail even locally as was shown in ["I). : M f ( x )> A}l L C ,Ilflll forA > 0. ) It is good enough because 2 can be obtained for p > 1 from this estimate together with the trivial L" result (see the interpolation theorem below), and 3 is also good because Lebesgue's differentiation theorem comes as a corollary.
The last three results are due to J . Bourgain, in a series of deep papers published in 1983-84 in Studia Math. and in Acta Math. He also gave a correct proof of a result announced by Grothendieck: the space C k ( U )of all complex-valued functions which are continuous with all their derivatives or order 5 k , on a d-dimensional compact manifold U , has the DPP. 2. The reciprocal Dunford Pettis and the Dieudonne‘ properties. Following the same idea, Grothendieck introduces two more properties. ) -P(D,W ) :the Dieudonne‘ property (DP for short; the name is due to a Dieudonnk’s result on weakly convergent sequences of Radon measures).
1. ” As we have seen, Grothendieck went to Paris to do his doctorate in 1949. After attending some courses there, he followed the advise of H. Cartan and went t o Nancy t o work with L a u r e n t Schwartz and Jean DieudonnB. Allow Schwartz himself describe his impressions: W e [Dieudonnk and Schwartz] received Grothendieclc in October, 1951. He showed a 50 pages paper o n “Integration with values in a topological group” to Dieudonne‘. I t was correct, but uninteresting. Dieudonne‘, with all the aggressiveness he could have, ([and he could a lot] gave him a severe tongue-lashing, arguing that he should not work in such a manner, just generalizing for the pleasure of doing so..
Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004 by M. V. Velasco, A. Rodriguez-Palacios