# Spectral Theory in Banach Spaces and Harmonic Analysis

### Alan G.R. McIntosh

Australian National University, Canberra, Australia### Nigel Kalton

University of Missouri, Columbia, USA### Lutz Weis

Universität Karlsruhe, Germany

## Abstract

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 $1<q<\infty$ has maximal $L_p$-regularity, i.e.~does the solution $y$ of the Cauchy problem $y'=Ay+f$, $y(0)=0$, satisfy $\norm[W^1_p(\R_+,L_q)]{y} \leq C \norm[L_p(\R_+,L_q)]{f}$. Recently G.~Lancien and N.J.~Kalton gave counterexamples to this questions while L.~Weis gave a characterization of maximal $L_p$-regularity in terms of R-boundedness. This criterium has since been shown to be useful in establishing maximal regularity for large classes of Cauchy problems for elliptic differential operators with rather general coefficients and boundary values, e.g.~Schr\"odinger operators with singular potentials and Stokes operators (see the abstracts of M.~Hieber, P.~Kunstmann, J.~Pr\"uss and R.~Schnaubelt). Applications of maximal regularity to non-linear differential equations were presented in the talks of H.~Amann and G.~Simonett. The two problems of Kato and Br{\'e}zis share a common background in the mathematical methods employed in their solution, e.g.~the $H^\infty$-functional calculus for sectorial operators in Banach spaces, boundedness of singular integral operators and square function estimates. Obviously, further progress will depend on an indepth study of these methods and their interrelations. The workshop contributions featured some of the most recent progress in these directions. N.J.~Kalton used deep results from Banach space theory to crystallize the difficult perturbation theory of the $H^\infty$-functional calculus. Some of the randomization techniques he used can be seen as an extension of the classical square function estimates from $L_p$-spaces to general Banach spaces. This method also underlies boundedness results for Fourier multiplier operators, Calder\'on-Zygmund operators and wavelet transforms for functions with values in a Banach space with the UMD-property (see the abstracts of W.~Arendt, O.~Blasco, T.~Hytonen and C.~Kaiser) and applications of those to spectral theory (A.~Gillespie). Also in the scalar-valued case several extensions of basic methods in the theory of singular integral operators were presented, e.g.~new spaces of BMO- and $H^1$-type associated to a given operator (X.~Duong), Calder\'on-Zygmund operators associated to non-doubling measures (J.~Garcia-Cuerva), bilinear pseudo-differential operators (A.~Nahmod) and optimal domains for convolution operators (W.~Ricker). As so often, new tools in harmonic analysis lead to new results for the $H^\infty$-functional calculus of partial differential operators (M.~Cowling, X.~Duong). Some of the talks reached beyond the circle of questions raised by the problems of Kato and Br{\'e}zis. They were closely connected to the main topics by a shared methological background in spectral theory and harmonic analysis. Ch.~Thiele discussed an approach to non-linear differential equations via the non-linear Fourier transform. B.~Jefferies and J.~van Neerven described the connection between spectral theory, square function estimates and stochastic integration on infinite dimensional Banach spaces. S.~Grivaux and Y.~Latushkin gave applications of spectral theory to instability and stability theorems for evolutionary systems. The talks of of J.~Eschmeier, M.~Haase and F.~Sukochev were focused on the functional calculus and its applications in operator theory. The varied background of the participants lead to a number of new collaborations started during the workshop. Last but not least, the unique setting of Oberwolfach and a week of beautiful sunshine contributed to a memorable and successful meeting.

## Cite this article

Alan G.R. McIntosh, Nigel Kalton, Lutz Weis, Spectral Theory in Banach Spaces and Harmonic Analysis. Oberwolfach Rep. 1 (2004), no. 3, pp. 1883–1970

DOI 10.4171/OWR/2004/36