Abstract
In this chapter we gather together some background material such as “Basics of Fourier Transformation”, some aspects of the “Theory of Fourier Multipliers”, some “Function Spaces”, “Some tools from distribution theory” and “Useful Inequalities”. There is no attempt mode to present these sections in a self-contained form. Readers are encouraged to study these topics more in detail by utilizing the related literature.
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
References
H. Bahouri, J.Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343 (Springer, Berlin-Heidelberg, 2011)
J. Bergh, J. Löfström, Interpolation Spaces. An Introduction. Grundlagen der Mathematischen Wissenschaften, vol. 223 (Springer, Berlin-New York, 1976)
R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960)
P. Brenner, On \(L_p-L_{p^{\prime }}\) estimates for the wave equation. Math. Z. 145, 251–254 (1975)
L. Debnath, D. Bhatta, Integral Transforms and Their Applications, 2nd edn. (Chapman & Hall, Boca Raton-London-New York, 2007)
A. Friedman, Partial Differential Equations. Corrected reprint of the original edition. (Robert E. Krieger Publishing Co., Huntington, New York, 1976)
K. Gröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001)
H. Hajaiej, L. Molinet, T. Ozawa, B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, in Harmonic Analysis and Nonlinear Partial Differential Equations. RIMS Kokyuroku Bessatsu, B26 (Research Institute for Mathematical Sciences, Kyoto, 2011), pp. 159–175
L. Hörmander, Estimates for translation invariant operators in L p spaces. Acta Math. 104, 93–140 (1960)
A. Palmieri, M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II, to appear in Math. Nachr. 33pp.
J. Petree, New Thoughts on Besov Spaces. Duke University Mathematical Series I (Duke University Press, Durham, 1976)
R. Racke, Lectures on Nonlinear Evolution Equations. Initial Value Problems. Aspects of Mathematics, vol. E19 (Vieweg, Braunschweig/Wiesbaden, 1992)
T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications (de Gruyter, Berlin, 1996)
W. Sickel, H. Triebel, Hölder inequalities and sharp embeddings in function spaces of \(B^s_{p,q}\) and \(F^s_{p,q}\) type. Z. Anal. Anwendungen 14(1), 105–140 (1995)
E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, NJ, 1970)
G.O. Thorin, Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Comm. Sem. Math. Univ. Lund. 9, 1–58 (1948)
H. Triebel, Theory of Function Spaces. Monographs in Mathematics, vol. 78 (Birkhäuser, Basel, 1983)
B. Wang, H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ. 232, 36–73 (2007)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Ebert, M.R., Reissig, M. (2018). Background Material. In: Methods for Partial Differential Equations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66456-9_24
Download citation
DOI: https://doi.org/10.1007/978-3-319-66456-9_24
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-66455-2
Online ISBN: 978-3-319-66456-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)