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Fractional Differentiation: Leibniz Meets Hölder

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Excursions in Harmonic Analysis, Volume 5

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

We discuss how to estimate the fractional derivative of the product of two functions, not in the pointwise sense, but on Lebesgue spaces whose indices satisfy Hölder’s inequality

Grafakos acknowledges the support of the Simons Foundation.

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References

  1. F. Bernicot, D. Maldonado, K. Moen, V. Naibo, Bilinear Sobolev-Poincaré inequalities and Leibniz-type rules. J. Geom. Anal. 24, 1144–1180 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bourgain, D. Li, On an endpoint Kato-Ponce inequality. Differ. Integr. Equ. 27(11/12), 1037–1072 (2014)

    MathSciNet  MATH  Google Scholar 

  3. F. Christ, M. Weinstein, Dispersion of small-amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal. 100, 87–109 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. R.R. Coifman, Y. Meyer, Commutateurs d’ intégrales singulières et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. R.R. Coifman, Y. Meyer, Au délà des opérateurs pseudo-différentiels. Astérisque No. 57 (Societé Mathematique de France, Paris, 1979)

    Google Scholar 

  6. E. Cordero, D. Zucco, Strichartz estimates for the vibrating plate equation. J. Evol. Equ. 11(4), 827–845 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Grafakos, Classical Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, no 249 (Springer, New York, 2008)

    Google Scholar 

  8. L. Grafakos, Multilinear operators in harmonic analysis and partial differential equations. Harmonic analysis and nonlinear partial differential equations, Research Institute of Mathematical Sciences (Kyoto), 2012

    MATH  Google Scholar 

  9. L. Grafakos, S. Oh, The Kato-Ponce inequality. Commun. Partial Differ. Equ. 39, 1128–1157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Grafakos, R.H. Torres, Multilinear Calderón-Zygmund theory. Adv. Math. 165, 124–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Grafakos, D. Maldonado, V. Naibo, A remark on an endpoint Kato-Ponce inequality. Differ. Integr. Equ. 27, 415–424 (2014)

    MathSciNet  MATH  Google Scholar 

  12. A. Gulisashvili, M. Kon, Exact smoothing properties of Schröndinger semigroups. Am. J. Math. 118, 1215–1248 (1996)

    Article  MATH  Google Scholar 

  13. T. Kato, Remarks on the Euler and Navier-Stokes equations in R 2, in Nonlinear Functional Analysis and Its Applications, Part 2 (Berkeley, California, 1983). Proceedings of Symposia in Pure Mathematics, vol. 45, Part 2 (American Mathematical Society, Providence, RI, 1986), pp. 1–7

    Google Scholar 

  14. T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. C.E. Kenig, E.M. Stein, Multilinear estimates and fractional integration. Math. Res. Lett. 6, 1–15 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de-Vries equation via the contraction principle. Commun. Pure Appl. Math. 46(4), 527–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis, Volume 2. Cambridge Studies in Advanced Mathematics, 138 (Cambridge University Press, Cambridge, 2013)

    Google Scholar 

  18. C. Muscalu, J. Pipher, T. Tao, C. Thiele, Bi-parameter paraproducts. Acta Math. 193, 269–296 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

Authors
  1. Loukas Grafakos
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Correspondence to Loukas Grafakos .

Editor information

Editors and Affiliations

  1. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Radu Balan

  2. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    John J. Benedetto

  3. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Wojciech Czaja

  4. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Matthew Dellatorre

  5. Norbert Wiener Center, University of Maryland, College Park, Maryland, USA

    Kasso A. Okoudjou

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Grafakos, L. (2017). Fractional Differentiation: Leibniz Meets Hölder. In: Balan, R., Benedetto, J., Czaja, W., Dellatorre, M., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 5. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-54711-4_2

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