Abstract
We consider Fourier series of functions in L p(0, 2π; X) where X is a Banach space. In particular, we show that the Fourier series of each function in L p(0, 2π; X) converges unconditionally if and only if p=2 and X is a Hilbert space. For operator-valued multipliers we present the Marcinkiewicz theorem and give applications to differential equations. In particular, we characterize maximal regularity (in a slightly different version than the usual one) by R-sectoriality. Applications to non-autonomous problems are indicated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
[Am04] H. Amann: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4 (2004), 417–430.
[ABHN01] W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel, 2001.
[ACFP07] W. Arendt, R. Chill, S. Fornaro, C. Poupaud: L p-maximal regularity for nonautonomous evolution equations. J. Diff. Equ. 237 (2007), 1–26.
[AB02] W. Arendt, S. Bu: The operator-valued Marcinkiewicz multiplier theorems and maximal regularity. Math. Z. 240 (2002), 311–343.
[AR09] W. Arendt, P. Rabier: Linear evolution operators on spaces of periodic functions. Comm. Pure and Applied Analysis 8 (2009), 5–36.
[Bur01] D. Burkholder: Martingales and singular integrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I (W.B. Johnson and J. Lindenstrauss Eds.), Elsevier, 2001, 233–269.
[CPSW00] Ph. Clément, B. de Pagter, F. A. Sukochev, H. Witvliet: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), 135–163.
[CP00] Ph. Clément, J. Prüss: An operator-valued transference principle and maximal regularity on vector-valued L p-spaces. In: Evolution Equations and Their Applications in Physical and Life Sciences (G. Lumer and L. Weis Eds.), Marcel Dekker, 2000, 67–78.
[Do93] G. Dore: L p-regularity of abstract differential equations. In: Functional Analysis and Related Topics (H. Komatsu Ed.), Springer LNM 1540 (1993), 25–38.
[KW04] P. Kunstmann, L. Weis: Maximal L p -regularity for parabolic equations, Fourier multipliers theorems and H∞-functional calculus. In: Functional Analytic Methods for Evolution Equations. Springer LNM 1855, 2004, 65–311.
[Kw72] S. Kwapien: Isomorphic characterization of inner product spaces by orthogonal series with vector-valued coefficients. Studia Math. 44 (1972), 583–595
[LeTa91] M. Ledoux, M. Talagrand: Probability in Banach Spaces, Isoperimetry and Processes. Springer-Verlag, Berlin, 1991.
[LT79] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces II, Springer, Berlin, 1979.
[Lun95] A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995.
[Pie07] A. Pietsch: History of Banach Spaces and Linear Operators. Birkhäuser, Basel, 2007.
[Sch61] J. Schwartz: A remark on inequalities of Calderon-Zygmund type for vectorvalued functions. Comm. Pure Appl. Math. 14 (1961), 785–799.
[SW07] Z. Strkalj, L. Weis: On operator-valued Fourier multiplier theorem. Trans. Amer. Math. Soc. 359 (2007), 3529–3547.
[W01] L. Weis: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann. 319 (2001), 735–758.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Arendt, W., Bu, S. (2009). Fourier Series in Banach spaces and Maximal Regularity. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol 201. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0211-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0211-2_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0210-5
Online ISBN: 978-3-0346-0211-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)