Oberwolfach Reports

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Volume 1, Issue 3, 2004, pp. 1883–1970
DOI: 10.4171/OWR/2004/36

Published online: 2005-06-30

Spectral Theory in Banach Spaces and Harmonic Analysis

Nigel Kalton[1], Alan G.R. McIntosh[2] and Lutz Weis[3]

(1) University of Missouri, Columbia, USA
(2) Australian National University, Canberra, Australia
(3) Universit├Ąt Karlsruhe, Germany

The conference was motivated by the recent solutions of two longstanding questions. The first one is Kato's square root problem, i.e.~whether for an elliptic operator in divergence form on $L_2(\R^n)$ we have $\norm{L^{1/2}u}\leq C \norm{\nabla u} + C\norm{u}$ for $u\in W^1_2(\R^n)$. After a long development over 40 years this was shown in a joint effort by P.~Auscher, S.~Hofmann, M.~Lacey, A.~McIntosh and Ph.~Tchamitchian. For a new approach to this important result via Dirac operators and extensions of it, see the abstracts of A.~McIntosh and S.~Keith. The second problem, attributed to Br{\'e}zis in the eighties, asks whether the Cauchy problem for every generator of an analytic semigroup $A$ in an $L_q(\Omega)$-space with $1

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Kalton Nigel, McIntosh Alan, Weis Lutz: Spectral Theory in Banach Spaces and Harmonic Analysis. Oberwolfach Rep. 1 (2004), 1883-1970. doi: 10.4171/OWR/2004/36