Skip to main content
SpringerLink
Log in

Smoothness and Function Spaces

  • Chapter
  • First Online:
Modern Fourier Analysis

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 250))

  • 8981 Accesses

Abstract

We embark on the study of smoothness with a quick examination of differentiability properties of functions. There are several ways to measure differentiability and numerous ways to quantify smoothness. In this chapter we measure smoothness using the Laplacian, which is easily related to the Fourier transform. This relation becomes the foundation of a very crucial and deep connection between smoothness and Littlewood–Paley theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    If v i ≠ 1, then v i is irrational. Let q k  ∈ Q such that \(q_{k} \rightarrow v_{i}\) as \(k \rightarrow \infty \). Then \(f(q_{k}) = q_{k} \rightarrow v_{i}\) as \(k \rightarrow \infty \) but \(f(v_{i}) = -1\neq v_{i}\); thus, f is discontinuous at v i and by linearity everywhere else.

  2. 2.

    We used that \(g(b) - g(a) =\int_{ 0}^{1}\nabla g\big((1 - t)a + tb\big) \cdot (b - a)\,dt = \nabla g\big((1 - t^{{\ast}})a + t^{{\ast}}b\big) \cdot (b - a)\) for all \(a,b \in \mathbf{R}^{n}\), for a \(\mathcal{C}^{1}\) function g on \(\mathbf{R}^{n}\) and some t ∗∈ (0, 1), depending on g,a,b.

References

  1. Adams, D. R., A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778.

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, R. A., Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York–London, 1975.

    Google Scholar 

  3. Aronszajn, N., Smith, K. T., Theory of Bessel potentials, I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475.

    Article  MATH  MathSciNet  Google Scholar 

  4. Calderón, A. P., Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math. 4 (1961), 33–49, Amer. Math. Soc., Providence, RI.

    Google Scholar 

  5. Campanato, S., Proprietà di hölderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 175–188.

    Google Scholar 

  6. Campanato, S., Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137–160.

    Google Scholar 

  7. Christ, M., Estimates for the k -plane transform, Indiana Univ. Math. J. 33 (1984), no. 6, 891–910.

    Article  MATH  MathSciNet  Google Scholar 

  8. Duren, P. L., Romberg, B. W., Shields, A. L., Linear functionals on H p spaces with \(0 < p < 1\), J. Reine Angew. Math. 238 (1969), 32–60.

    MATH  MathSciNet  Google Scholar 

  9. Fefferman, C., Stein, E. M., H p spaces of several variables, Acta Math. 129 (1972), no. 3–4, 137–193.

    Article  MATH  MathSciNet  Google Scholar 

  10. Folland, G. B., Stein, E. M., Estimates for the \(\overline{\partial }_{b}\) complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522.

    Article  MATH  MathSciNet  Google Scholar 

  11. Gagliardo, E., Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 7 (1958), 102–137.

    MATH  MathSciNet  Google Scholar 

  12. García-Cuerva, J., Rubio de Francia, J.-L., Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116, North-Holland Publishing Co., Amsterdam, 1985.

    Google Scholar 

  13. Grafakos, L., Classical Fourier Analysis, Third edition, Graduate Texts in Math. 249, Springer, New York, 2014.

    Google Scholar 

  14. Hardy, G. H., Littlewood, J. E., Some properties of fractional integrals I, Math. Z. 27 (1927), no. 1, 565–606.

    Article  MathSciNet  Google Scholar 

  15. Hardy, G. H., Littlewood, J. E., Some properties of fractional integrals II, Math. Z. 34 (1932), no. 1, 403–439.

    Article  MathSciNet  Google Scholar 

  16. Hardy, G. H., Littlewood, J. E., Generalizations of a theorem of Paley, Quarterly Jour. 8 (1937), 161–171.

    Google Scholar 

  17. Hedberg, L., On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.

    Article  MathSciNet  Google Scholar 

  18. Krantz, S. G., Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math. 1 (1983), no. 3, 193–260.

    MATH  MathSciNet  Google Scholar 

  19. Lieb, E. H., Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2nd Ser.) 118 (1983), no. 2, 349–374.

    Google Scholar 

  20. Lieb, E. H., Loss, M., Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 1997.

    Google Scholar 

  21. Maz’ya, V. G., Sobolev Spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  22. Meyers, N. G., Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc. 15 (1964), 717–721.

    MATH  MathSciNet  Google Scholar 

  23. Nirenberg, L., On elliptic partial differential equations, Ann. di Pisa 13 (1959), 116–162.

    MathSciNet  Google Scholar 

  24. Riesz, M., L’ intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), no. 1, 1–223.

    Article  MATH  MathSciNet  Google Scholar 

  25. Sobolev, S. L., On a theorem in functional analysis [in Russian], Mat. Sob. 46 (1938), 471–497.

    Google Scholar 

  26. Taibleson, M. H., On the theory of Lipschitz spaces of distributions on Euclidean n-space, I, J. Math. Mech. 13 (1964), 407–479.

    MathSciNet  Google Scholar 

  27. Taibleson, M. H., On the theory of Lipschitz spaces of distributions on Euclidean n-space, II, J. Math. Mech. 14 (1965), 821–839.

    MathSciNet  Google Scholar 

  28. Taibleson, M. H., On the theory of Lipschitz spaces of distributions on Euclidean n-space, III, J. Math. Mech. 15 (1966), 973–981.

    MATH  MathSciNet  Google Scholar 

  29. Walsh, T., The dual of \(H^{p}(\mathbb{R}_{+}^{n+1})\) for \(p < 1\), Canad. J. Math. 25 (1973), 567–577.

    Article  MATH  MathSciNet  Google Scholar 

  30. Welland, G. V., Weighted norm inequalities for fractional integrals, Proc. Amer. Math. Soc. 51 (1975), 143–148.

    Article  MATH  MathSciNet  Google Scholar 

  31. Weyl, H., Bemerkungen zum Begriff der Differentialquotienten gebrochener Ordnung, Viertel Natur. Gesellschaft Zürich 62 (1917), 296–302.

    Google Scholar 

  32. Zygmund, A., On a theorem of Marcinkiewicz concerning interpolation of operators, J. Math. Pures Appl. (9) 35 (1956), 223–248.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Loukas Grafakos

Authors
  1. Loukas Grafakos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Grafakos, L. (2014). Smoothness and Function Spaces. In: Modern Fourier Analysis. Graduate Texts in Mathematics, vol 250. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1230-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation