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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols pp 1–67Cite as

Function spaces and distributions

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Part of the book series: Developments in Mathematics ((DEVM,volume 41))

Abstract

This chapter is devoted to function and distribution spaces. We first recall definitions of some well-known classical function and distribution spaces, simultaneously introducing the terminology and notations used in this book. Then we introduce (see Section 1.10) a new class of test functions and the corresponding space of distributions (generalized functions), which play an important role in the theory of pseudo-differential operators with singular symbols introduced in Chapter 2 By singular symbols we mean, if not otherwise assumed, symbols singular in dual variables.

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Notes

  1. 1.

    In fact, (L∞(Ω))∗ is isomorphic to the space of finite Borel measures with the total variation norm. The latter contains L 1 (Ω) as a linear subspace, see, e.g., [Tri77].

References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)

    MATH  Google Scholar 

  2. Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer (1976)

    Google Scholar 

  3. Besov, O.B., Il’in, V.P., Nikolskii, S.M.: Integral Representation of Functions and Embedding Theorems I, II. Willey, New York (1979)

    Google Scholar 

  4. Brezis, H., Lions, J.L.: Nonlinear Partial Differential Equations and Their Applications. Chapman & Hall (1981)

    Google Scholar 

  5. Dubinskii, Yu.A.: The algebra of pseudo-differential operators with analytic symbols and its applications to mathematical physics. Russ. Math. Surv., 37, 109–153 (1982)

    Article  Google Scholar 

  6. Fefferman, Ch.: The multiplier theorem for the ball. The Annals of Mathematics, 94 (2), 330–336 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gel’fand I.M., Shilov, G.E.: Fourier transforms of rapidly growing functions and questions of uniqueness of the solution of Cauchy’s problem. Usp. Mat. Nauk. 8 (6), 3–54 (1953)

    MATH  Google Scholar 

  8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1977)

    Book  MATH  Google Scholar 

  9. Graf, U.: Introduction to Hyperfunctions and Their Integral Transformations. Birkhäuser, Basel (2010)

    Book  Google Scholar 

  10. Hörmander, L.: The Analysis of Linear Partial Differential Operators, I - IV. Springer-Verlag, Berlin-Heidelberg-New-York (1983)

    Google Scholar 

  11. Kaneko, A.: Representation of hyperfunctions by measures and some of its applications. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 19, 321–352 (1972)

    Google Scholar 

  12. Lebesgue, H.: Sur une généralisation de l’intégrale définie. Comptes Rend. l’Acad. Sci., 132, 1025–1028 (1901)

    MATH  Google Scholar 

  13. Lizorkin P.I.: Generalized Liouville differentiation and functional spaces L p r(E n ). Embedding theorems. Mat. Sb. 60, 325–353 (1963) (in Russian)

    MathSciNet  Google Scholar 

  14. Lizorkin, P.I.: Multipliers of Fourier integrals in the spaces L p . Proc. Steklov Inst. Math. 89, 269–290 (1967)

    MATH  Google Scholar 

  15. Lizorkin, P.I.: Generalized Liouville differentiation and the method of multipliers in the theory of embeddings of classes of differentiable functions. Proc. Steklov Inst. Math. 105, 105–202 (1969)

    Google Scholar 

  16. Mikhlin, S.G.: On the multipliers of Fourier integrals, Dokl. Akad. Nauk. 109, 701–703 (1956)(in Russian)

    MATH  Google Scholar 

  17. Nikolskii, S.M.: Approximation of Functions of Several Variables and Embedding Theorems. Springer-Ferlag (1975)

    Google Scholar 

  18. Peetre, J.: New thoughts on Besov spaces. Duke Univ. Math. Series, Duke Univ., Durham (1976)

    Google Scholar 

  19. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press (1980)

    MATH  Google Scholar 

  20. Riesz, F.: Untersuchungen über Systeme integrierbarer Funktionen. Mathematische Annalen 69 (4), 449–497 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  21. Robertson, A.P., Robertson, W.: Topological Vector Spaces. Cambridge University Press (1964)

    Google Scholar 

  22. Samko, S.G.: Fundamental functions vanishing on a given set and division by functions. Mathematical notes, 21 (5), 379–386 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  23. Samko, S.G.: Denseness of Lizorkin-type spaces Φ V in L p . Mat. Notes, 31, 432–437 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Samko, S.G.: Denseness of the spaces Φ V of Lizorkin type in the mixed in L p (ℝ n)-spaces. Stud. Math. 113, 199–210 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Sato, M.: Theory of Hyperfunctions, I. Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, astronomy, physics, chemistry, 8 (1), 139–193 (1959)

    Google Scholar 

  26. Sato, M.: Theory of Hyperfunctions, II. Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, astronomy, physics, chemistry, 8 (2), 387–437 (1960)

    Google Scholar 

  27. Schaefer, H.H., Wolff, M.P.: Topological vector spaces, 2nd ed. Springer (1999)

    Google Scholar 

  28. Schwartz, L.: Théorie des distributions I, II. Hermann, Paris (1951)

    Google Scholar 

  29. Sobolev, S.L.: Le probléme de Cauchy dans l’espace des fonctionelles (Russian and French). Dokl. Akad. Nauk SSSR (Comptes Rend. l’Acad. Sci. URSS) 3 (8), 7 (67), 291–294 (1935)

    Google Scholar 

  30. Sobolev, S.L.: Méthode nouvelle á resoudre le probléme de Cauchy pour les équations linéaires hyperboliques normales. Mat. Sb. 1 (43), 39–72 (1936)

    MATH  Google Scholar 

  31. Sobolev, S.L.: On a theorem in functional analysis. Mat. Sb. 4 (46), 471–497 (1938)(Russian) (Engl. transl.: Amer. Math. Soc. Transl. 34 (2), 39–68 (1963))

    Google Scholar 

  32. Sobolev, S.L.: Some applications of functional analysis in mathematical physics (Russian). 1st ed.: Leningr. Goz. Univ., Leningrad (1950) 3rd enlarged ed.: Izd. Nauka, Moskva (1988) (Engl. transl.: Amer. Math. Soc., Providence, R. I. (1963))

    Google Scholar 

  33. Sobolev, S.L.: Introduction to the theory of cubature formulas. Nauka, Moscow (1974)(in Russian) (Engl. transl: Cubature formulas and modern analysis: An introduction. Gordon and Breach (1992))

    Google Scholar 

  34. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970)

    Google Scholar 

  35. Sutradhar, B.C.: On the characteristic function of multivariate Student t-distribution. The Canadian Journal of Statistics (La Revue Canadienne de Statistique), 14 (4), 329–337 (1986)

    Google Scholar 

  36. Taylor, M.: Pseudo differential operators. Prinston University Press (1981)

    Google Scholar 

  37. Treves, F.: Introduction to Pseudo-Differential and Fourier Integral Operators. Plenum Publishing Co., New York (1980)

    Google Scholar 

  38. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Birkhäuser, Basel (1977)

    Google Scholar 

  39. Triebel, H.: Theory of Function Spaces. Leipzig, Birkhäuser Verlag, Basel-Boston-Stuttgart (1983)

    Book  Google Scholar 

  40. Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations I. Differ. Equations, 33, 831–840 (1997)

    MathSciNet  MATH  Google Scholar 

  41. Umarov, S.R.: Nonlocal boundary value problems for pseudo-differential and differential operator equations II. Differ. Equations, 34, 374–381 (1988)

    MathSciNet  Google Scholar 

  42. Vladimirov, V.S.: Generalized Functions in Mathematical Physics. Mir Publishers, Moscow (1979)

    Google Scholar 

  43. Zygmund, A.: Smooth functions. Duke Math. J. 12 (1), 47–76 (1945)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of New Haven, West Haven, CT, USA

    Sabir Umarov

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Umarov, S. (2015). Function spaces and distributions. In: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols. Developments in Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-20771-1_1

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