Skip to main content
SpringerLink
Log in

Boundedness of Pseudodifferential Operators on Banach Function Spaces

  • Conference paper
  • First Online:
Operator Theory, Operator Algebras and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 242))

  • 1034 Accesses

Abstract

We show that if the Hardy–Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^{n})\) and on its associate space \(X^{\prime}(\mathbb{R}^{n})\), then a pseudodifferential operator \( {\rm Op}(a)\) is bounded on \(X(\mathbb{R}^{n})\) whenever the symbol a belongs to the Hörmander class \({S}_{\rho,\delta}^{{n}(\rho-1)} \mathrm{with}\,0<\rho\leq1\), \( 0 \leq \delta < 1 \) or to the Miyachi class \({S}_{\rho,\delta}^{{n} (\rho-1)} (\aleph, n) \ \mathrm{with} \, 0 \leq \delta \leq \rho \leq 1\) with \( 0 \leq \delta < 1\) and \( \aleph < 0 \) This result is applied to the case of variable Lebesgue spaces \( L^{p(.)} {({\mathbb{R}}^{n})} \).

Mathematics Subject Classification (2010). Primary 47G30; Secondary 35S05, 42B25, 46E30.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Abels, Pseudodifferential and Singular Integral Operators. An Introduction with Applications. De Gruyter, Berlin, 2012.

    Google Scholar 

  2. J. Álvarez and J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators. Ark. Mat. 28 (1990), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Álvarez, J. Hounie, and C. Pérez, A pointwise estimate for the kernel of a pseudodifferential operator, with applications. Rev. Union Mat. Argent. 37 (1991), 184–199.

    MATH  Google Scholar 

  4. J. Álvarez and C. Pérez, Estimates with A ∞ weights for various singular integral operators. Boll. Un. Mat. Ital. A (7) 8 (1994), 123–133.

    Google Scholar 

  5. R. Ashino, M. Nagase, and R. Vaillancourt, Pseudodifferential operators in L p(ℝn) spaces. Cubo 6 (2004), 91–129.

    MATH  MathSciNet  Google Scholar 

  6. C. Bennett and R. Sharpley, Interpolation of Operators. Academic Press, New York, 1988.

    Google Scholar 

  7. C. Capone, D. Cruz-Uribe, and A. Fiorenza, The fractional maximal operator and fractional integrals on variable L p spaces. Rev. Mat. Iberoamericana 23 (2007), 743–770.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Coifman and Y. Meyer. Au délà des Opérateurs Pseudo-Différentiels. Astérisque 57, Soc. Math. France, Paris, 1978 (in French).

    Google Scholar 

  9. D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Birkhäuser, Basel, 2013.

    Google Scholar 

  10. D. Cruz-Uribe, A. Fiorenza, and C.J. Neugebauer, The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223–238.

    MATH  MathSciNet  Google Scholar 

  11. D. Cruz-Uribe, A. Fiorenza, and C.J. Neugebauer, Corrections to: “The maximal function on variable L p spaces”. Ann. Acad. Sci. Fenn. Math. 29 (2004), 247–249.

    MATH  MathSciNet  Google Scholar 

  12. L. Diening, Maximal function on generalized Lebesgue spaces L p(·) . Math. Inequal. Appl. 7 (2004), 245–253.

    MATH  MathSciNet  Google Scholar 

  13. L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), 657–700.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin, 2011.

    Google Scholar 

  15. Ch. Fefferman and E.M. Stein, H p spaces of several variables. Acta Math. 129 (1972), 137–193.

    Google Scholar 

  16. L. Hörmander, Pseudo-differential operators and hypoelliptic equations. In: “Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966)”, pp. 138–183. Amer. Math. Soc., Providence, R.I., 1967.

    Google Scholar 

  17. L. Hörmander, The Analysis of Linear Partial Differential Operators. III: Pseudo- Differential Operators. Reprint of the 1994 edition. Classics in Mathematics. Springer, Berlin, 2007.

    Google Scholar 

  18. J.-L. Journe, Calderόn–Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderόn. Lecture Notes in Mathematics 994. Springer, Berlin, 1983.

    Google Scholar 

  19. A.Yu. Karlovich and I.M. Spitkovsky, Pseudodifferential operators on variable Lebesgue spaces. In: “Operator Theory, Pseudodifferential Equations and Mathematical Physics”. Operator Theory: Advances and Applications 228 (2013), 173–184.

    Google Scholar 

  20. A.Yu. Karlovich and I.M. Spitkovsky, The Cauchy singular integral operator on weighted variable Lebesgue spaces. In: “Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation”. Operator Theory: Advances and Applications 236 (2014), 275–291.

    Google Scholar 

  21. H. Kumano-go, Pseudo-Differential Operators. The MIT Press. Cambridge, MA, 1982.

    Google Scholar 

  22. A.K. Lerner, Some remarks on the Hardy–Littlewood maximal function on variable L p spaces. Math. Z. 251 (2005), 509–521.

    Article  MATH  MathSciNet  Google Scholar 

  23. A.K. Lerner, Some remarks on the Fefferman–Stein inequality. J. Anal. Math. 112 (2010), 329–349.

    Article  MATH  MathSciNet  Google Scholar 

  24. A.K. Lerner and S. Ombrosi, A boundedness criterion for general maximal operators. Publ. Mat. 54 (2010), 53–71.

    Article  MATH  MathSciNet  Google Scholar 

  25. A.K. Lerner and C. Pérez, A new characterization of the Muckenhoupt A p weights through an extension of the Lorentz–Shimogaki theorem. Indiana Univ. Math. J. 56 (2007), 2697–2722.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Michalowski, D. Rule, and W. Staubach, Weighted L p boundedness of pseudodifferential operators and applications. Canad. Math. Bull. 55 (2012), 555–570.

    Article  MATH  MathSciNet  Google Scholar 

  27. N. Miller, Weighted Sobolev spaces and pseudodifferential operators with smooth symbols. Trans. Amer. Math. Soc. 269 (1982), 91–109.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Miyachi, Estimates for pseudo-differential operators of class S 0,0 . Math. Nachr. 133 (1987), 135–154.

    Article  MATH  MathSciNet  Google Scholar 

  29. A. Miyachi, Estimates for pseudo-differential operators of class \( S_{{p,\delta }}^{m} \) in L p , hp, and bmo. In: “Analysis, Proc. Conf., Singapore 1986”, North-Holland Math. Stud. 150 (1988), 177–187.

    Google Scholar 

  30. A. Miyachi and K. Yabuta, Sharp function estimates for pseudo-differential operators of class \( S_{{p,\delta }}^{m} \). Bull. Fac. Sci., Ibaraki Univ., Ser. A 19 (1987), 15–30.

    Google Scholar 

  31. A. Nekvinda, Hardy–Littlewood maximal operator on L p(x)(ℝn). Math. Inequal. Appl. 7 (2004), 255–265.

    MATH  MathSciNet  Google Scholar 

  32. A. Nekvinda, Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. Appl. 337 (2008), 1345–1365.

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Nishigaki, Weighted norm inequalities for certain pseudo-differential operators. Tokyo J. Math. 7 (1984), 129–140.

    Article  MATH  MathSciNet  Google Scholar 

  34. L. Pick and M. Růžička, An example of a space of L p (x) on which the Hardy– Littlewood maximal operator is not bounded. Expo. Math. 19 (2001), 369–371.

    Article  MATH  MathSciNet  Google Scholar 

  35. V.S. Rabinovich and S.G. Samko, Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integral Equations Operator Theory 60 (2008), 507–537.

    Article  MATH  MathSciNet  Google Scholar 

  36. E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ, 1993.

    Google Scholar 

  37. M. Taylor, Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics 100. Birkhäuser Boston, Boston, MA, 1991.

    Google Scholar 

  38. K. Yabuta, Generalizations of Calderόn–Zygmund operators. Stud. Math. 82 (1985), 17–31.

    MATH  MathSciNet  Google Scholar 

  39. K. Yabuta, Calderόn–Zygmund operators and pseudo-differential operators. Commun. Partial Differ. Equations 10 (1985), 1005–1022.

    Article  MATH  MathSciNet  Google Scholar 

  40. K. Yabuta, Weighted norm inequalities for pseudodifferential operators. Osaka J. Math. 23 (1986), 703–723.

    MATH  MathSciNet  Google Scholar 

  41. K. Yabuta, Sharp function estimates for a class of pseudodifferential operators. Bull. Fac. Sci., Ibaraki Univ., Ser. A 21 (1989), 1–7.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829–516, Caparica, Portugal

    Alexei Yu. Karlovich

Authors
  1. Alexei Yu. Karlovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexei Yu. Karlovich .

Editor information

Editors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal

    Amarino Lebre

  3. Departamento de Matemática, Universidade do Algarve, Faro, Portugal

    Stefan Samko

  4. Department of Mathematics, College of William & Mary, Williamsburg, Virginia, USA

    Ilya M. Spitkovsky

Additional information

To Professor António Ferreira dos Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this paper

Cite this paper

Karlovich, A.Y. (2014). Boundedness of Pseudodifferential Operators on Banach Function Spaces. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_10

Download citation

Publish with us

Policies and ethics

Search

Navigation