Abstract
This paper surveys some recent results on vector-valued Fourier multiplier theorems and pseudo differential operators, which have found important application in the theory of evolution equations. The approach used combines methods from Fourier analysis and the geometry of Banach spaces, such as R-boundedness.
Girardi is supported in part by the Alexander von Humboldt Foundation.
Weis is supported in part by Landesforschungsschwerpunkt Evolutionsgleichungen des Landes Baden-Württenberg.
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References
Amann,H: Linear and quasilinear parabolic problems. I: Abstract Linear theory. Birkhauser Boston Inc., Boston,M.A, 1995
Amann,H: Operator-valued fourier multipliers, vertor-valued Besov spaces and applications. Math Nachr. 186 (1997), 5–56
Amann,H: Elliptic operators with infinite-dimensional state spaces. J. Evol. Equ. 1 (2001), no 2, 143–188
Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. (to appear).
Blunck, S.: Maximal regularity of discrete and continuous time evolution equations. Studia Math. 146 (2001), no 2, 157–176.
Bourgain, J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21 (1983), no 2, 163–168.
Bourgain, J.: Vector-valued singular integrals and the Hl-B M O duality. Probability theory and harmonic analysis (Cleveland, Ohio, 1983), Dekker, New York, 1986, 1–19.
Clément, P., de Pagter, B., Sukochev, F.A., Witvliet, H.: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no 2, 135–163.
Clément, P., Prüss, J.: An operator-valued transference principle and maximal regularity on vector-valued Lu-spaces. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Dekker, New York, 2001, 67–87.
Denk,R., Hieber, M., Prüß, J.: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. (to appear).
Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge 1995.
Girardi, M., Weis, L.: Criteria for R-boundedness of operator families. Recent Contributions to Evolution Equations. Marcel Dekker Lecture Notes in Math. (to appear).
Girardi, M., Weis, L.: Integral operators with operator-valued kernels. In preparation.
Girardi, M., Weis, L.: Operator-valued Fourier multiplier theorems on Besov spaces. Mathematische Nachrichten (to appear).
Girardi, M., Weis, L.: Operator-valued Fourier multiplier theorems on L P (X) and geometry of Banach spaces (submitted).
Haller, R., Heck, H., Noll, A.: Mikhlin’s theorem for operator-valued Fourier multipliers on n-dimensional domains. Math. Nachr. (to appear).
Kaiton, N.J., Lancien, G.: A solution to the problem of LP-maximal regularity. Math. Z. 235 (2000), 559–568.
Kunstmann, P.C.: Maximal LP-regularity for second order elliptic operators with uniformly continuous coefficients on domains (submitted).
Kunstmann, P.C., Weis, L.: Perturbation theorems for maximal Lu-regularity. Ann. Sc. Norm. Sup. Pisa 30 (2001), 415–435.
Kwapieri, S.: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. Studia Math. 44 (1972), 583–595.
Latushkin, Y., Räbiger, F.: Fourier multipliers in stability and control theory. Preprint.
Peetre, J.: Sur la transformation de Fourier des fonctions à valeurs vectorielles. Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26.
Vector-valued extentions of some classical theorems in harmonic analysis 185
Schulze, B.-W.: Boundary value problems and singular pseudo-differential operators. John Wiley & Sons Ltd., Chichester, 1998.
Strkalj, 2.: R.-Beschränktheit, Summensätze abgeschlossener Operatoren and operatorwertige Pseudodifferentialoperatoren. Ph.D. thesis, Universität Karlsruhe, Karlsruhe, Germany, 2000.
Strkalj, 2., Weis, L.: On operator-valued Fourier multiplier theorems (submitted).
Weis, L.: Stability theorems for semi-groups via multiplier theorems. Differential equations, asymptotic analysis, and mathematical physics (Potsdam 1996), Akademie Verlag, Berlin 1997, 407–411.
Weis, L.: A new approach to maximal Lu-regularity. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Dekker, New York 2001, 195–214.
Weis, L.: Operator-valued Fourier multiplier theorems and maximal Lu-regularity. Math. Ann. 319 (2001), 735–758.
Zimmermann, F.: On vector-valued Fourier multiplier theorems. Studia Math. 93 (1989), 201–222.
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Girardi, M., Weis, L. (2003). Vector-Valued Extentions of Some Classical Theorems in Harmonic Analysis. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_13
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