Advertisement

Preliminaries

  • Dagmar Medková
Chapter

Abstract

Chapter 1 gathers fundamental facts from functional analysis.

References

  1. 1.
    R.A. Adams, J.L. Fournier, Sobolev Spaces, 2nd edn. (Elsevier, Oxford, 2003)Google Scholar
  2. 2.
    D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314 (Springer, Berlin, Heidelberg, 1996)Google Scholar
  3. 4.
    G.P. Akilov, L.V. Kantorovich, Funkcionalnyi Analiz (Nauka, Moskva, 1977)Google Scholar
  4. 5.
    H. Amann, J. Escher, Analysis III (Birkhäuser, Basel, Boston, Berlin, 2009)Google Scholar
  5. 16.
    C. Bennett, R. Sharpley, Interpolation of Operators (Academic, Orlando, FL, 1988)Google Scholar
  6. 17.
    J. Berg, J. Löström, Interpolation Spaces. An Introduction (Springer, Berlin, Heidelberg, New York, 1976)Google Scholar
  7. 19.
    N. Bourbaki, Integrirovanie. Mery, integrirovanie mer (Nauka, Moskva, 1967)Google Scholar
  8. 24.
    G. Chen, J. Zhou, Boundary Element Methods (Academic, London, 1992)Google Scholar
  9. 25.
    M. Chipot, Elliptic Equations: An Introductory Course (Birkhäuser, Basel, Boston, Berlin, 2009)Google Scholar
  10. 38.
    R.A. Devore, R.C. Sharpley, Besov spaces on domains in R d. Trans. Math. Soc. 335(2), 843–864 (1993)Google Scholar
  11. 40.
    M. Dobrowolski, Angewandte Functionanalysis. Functionanalysis, Sobolev-Räume und elliptische Differentialgleichungen (Springer, Berlin, Heidelberg, 2006)Google Scholar
  12. 41.
    R.G. Douglas, Banach Algebra Techniques in Operator Theory (Academic, New York and London, 1972)Google Scholar
  13. 42.
    J.J. Duistermaat, J.A.C. Kolk, Distributions. Theory and Applications (Birkhäuser, New York, Dordrecht, Heidelberg, London, 2010)Google Scholar
  14. 43.
    N. Dunford, J.T. Schwartz, Linear Operators I, General Theory (Interscience Publishers, New York, London, 1958)Google Scholar
  15. 49.
    H. Federer, Geometric Measure Theory (Springer, Berlin, Heidelberg, New York, 1969)Google Scholar
  16. 55.
    M. Giaquinta, G. Modica, Mathematical Analysis. Linear and Metric Structures and Continuity (Birkhäuser, Boston, 2007)Google Scholar
  17. 57.
    S. Goldberg, Unbounded Linear Operators. Theory and Applications (McGraw-Hill Book Company, New York, 1966)Google Scholar
  18. 58.
    P. Grisvard, Elliptic Problems in Nonsmooth Domains (SIAM, Philadelphia, 2011)Google Scholar
  19. 60.
    W. Hackbush, Integral Equations. Theory and Numerical Treatment (Bikhäuser, Bassel, 1995)Google Scholar
  20. 62.
    P.R. Halmos, Measure Theory (D. van Nostrand, New York, 1950)Google Scholar
  21. 67.
    L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis (Springer, Berlin, Heidelberg, 2003)Google Scholar
  22. 81.
    C.E. Kenig, Weighted H p spaces on Lipschitz domains. Am. J. Math. 102, 129–163 (1980)Google Scholar
  23. 86.
    S.G. Krantz, H.R. Parks, Geometric Integration Theory (Birkhäuser, Boston, 2008)Google Scholar
  24. 87.
    M.A. Krasnosel’skii, P.P. Zabreiko, E.I. Pustyl’nik, P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Izdat. Nauka, Moskva, 1966) (Russian)Google Scholar
  25. 94.
    A. Kufner, O. John, S. Fučík, Function Spaces (Academia, Prague, 1977)Google Scholar
  26. 98.
    E.H. Lieb, M. Loss, Analysis (American Mathematical Society, Providence, RI, 1997)Google Scholar
  27. 101.
    J. Malý, W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations (American Mathematical Society, Providence, RI, 1997)Google Scholar
  28. 105.
    V.G. Maz’ya, S.V. Poborchi, Differentiable Functions on Bad Domains (World Scientific, Singapore, 1997)Google Scholar
  29. 112.
    Y. Meyer, R. Coifman, Wavelets. Calderón–Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997)Google Scholar
  30. 118.
    M. Mitrea, M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque 344, Paris (2012)Google Scholar
  31. 121.
    V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory Advances and Applications, vol. 139 (Birkhäuser, Basel, 2007)Google Scholar
  32. 122.
    J. Nečas, Les méthodes directes en théorie des équations élliptiques (Academia, Prague, 1967)zbMATHGoogle Scholar
  33. 123.
    J. Peetre, New Thoughts on Besov Spaces. Duke University Mathematics Series (Duke University, Durham, 1976)zbMATHGoogle Scholar
  34. 125.
    W. Rudin, Real and Complex Analysis, 2nd edn. (McGraw-Hill, New York, 1974)zbMATHGoogle Scholar
  35. 127.
    M. Schechter, Principles of Functional Analysis (Academic, London, 1971)zbMATHGoogle Scholar
  36. 128.
    M. Schechter, Principles of Functional Analysis (American Mathematical Society, Providence, RI, 2002)zbMATHGoogle Scholar
  37. 130.
    G.E. Shilov, Mathematical Analysis. Second special course (Nauka, Moskva, 1965) (Russian)Google Scholar
  38. 136.
    E.M. Stein, Singular Integrals and Differentiability of Functions (Princeton University Press, Princeton, NJ, 1970)zbMATHGoogle Scholar
  39. 137.
    E.M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscilatory Integrals (Princeton University Press, Princeton, NJ, 1993)Google Scholar
  40. 140.
    L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces (Springer, Berlin, Heidelberg, 2007)zbMATHGoogle Scholar
  41. 141.
    A.E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1967)Google Scholar
  42. 142.
    W.F. Trench, Introduction to Real Analysis (Prentice Hall, Princeton, NJ, 2002)zbMATHGoogle Scholar
  43. 143.
    H. Triebel, Höhere Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972)zbMATHGoogle Scholar
  44. 144.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978)zbMATHGoogle Scholar
  45. 145.
    H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, Boston, Stuttgart, 1983)CrossRefzbMATHGoogle Scholar
  46. 146.
    H. Triebel, Theory of Function Spaces III. Monographs in Mathematics, vol. 100 (Birkhäuser, Basel, 2006)Google Scholar
  47. 147.
    G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 149.
    K. Yosida, Functional Analysis (Springer, Berlin, 1965)CrossRefzbMATHGoogle Scholar
  49. 150.
    W.P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Dagmar Medková
    • 1
  1. 1.Institute of Mathematics of the Czech, Academy of SciencesPraha 1Czech Republic

Personalised recommendations