Skip to main content
Log in

Interpolation of Bilinear Operators Between Quasi-Banach Spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We study interpolation, generated by an abstract method of means, of bilinear operators between quasi-Banach spaces. It is shown that under suitable conditions on the type of these spaces and the boundedness of the classical convolution operator between the corresponding quasi-Banach sequence spaces, bilinear interpolation is possible. Applications to the classical real method spaces, Calderón-Lozanovsky spaces, and Lorentz-Zygmund spaces are presented.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.V. Astashkin, On Interpolation of Bilinear Operators by the Real Method of Interpolation, Mat. Zametki, 52 (1992), 15–24 (Russian).

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

  3. A.Yu. Brudnyi, N.Ya. Krugljak, Interpolation Functors and Interpolation Spaces I, North-Holland, Amsterdam, (1991).

  4. A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math., 24 (1964), 113–190.

  5. M. Cwikel, J. Peetre, Abstract K and J spaces, J. Math. Pures. Appl., 60 (1981), 1–50.

  6. R.R. Coifman, Y. Meyer, Commutateurs d'intègrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble), 28 (1978), 177–202.

    Google Scholar 

  7. J. Gilbert, A. Nahmod, Bilinear operators with non-smooth symbols, J. Fourier Anal. Appl. 7 (2001), 437–469.

    Google Scholar 

  8. L. Grafakos, R. Torres, Multilinear Calderón Zygmund theory, Adv. in Math. 40 (1996), 344–351.

  9. L. Grafakos, N.J. Kalton, The Marcinkiewicz multiplier condition for bilinear operators, Studia Math. 146 (2001), 115–156.

    Google Scholar 

  10. L. Grafakos, N.J. Kalton, Multilinear Calderón-Zygmund operators on Hardy spaces, Collect. Math., 52 (2001), 169–179.

  11. J. Gustavsson, A function parameter in connetion with interpolation of Banach spaces, Math. Scand., 42 (1978), 289–305.

    Google Scholar 

  12. J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces, Studia Math., 60 (1977), 33–59.

  13. N.J. Kalton, Convexity conditions for non-locally convex lattices, Glasgow Math. J., 25 (1984), 141–152.

    Google Scholar 

  14. N.J. Kalton, N.T. Peck, J.W. Roberts, An F-space Sampler, London Math. Soc. Lecture Notes 89, Cambridge University Press, (1985).

  15. N.J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math. 84 (1986), 297–324.

    Google Scholar 

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

    Google Scholar 

  17. S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English transl.: Amer. Math. Soc., Providence, (1982).

  18. M.T. Lacey, C.M. Thiele, Lp bounds for the bilinear Hilbert transform, p>2, Ann. of Math., 146 (1997), 693–724.

  19. M.T. Lacey, C.M. Thiele, On Calderón's conjecture, Ann. of Math., 149 (1999), 475–496.

    Google Scholar 

  20. J.L. Lions, J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), 5–68.

  21. G.Ya. Lozanovsky, On some Banach lattices IV, Sibirsk. Mat. Z. 14 (1973), 140–155 (in Russian); English transl.: Siberian. Math. J. 14 (1973), 97–108.

  22. M. Mastyło, On interpolation of some quasi-Banach spaces, J. Math. Anal. Appl. 147 (1990), 403–419.

    Google Scholar 

  23. M. Mastyło, On interpolation of bilinear operators, J. Funct. Anal., 214 (2004), 260–283.

    Google Scholar 

  24. M. Mastyło, Interpolation methods of means and orbits, Studia Math., 17 (2005), 53–175.

  25. C. Merucci, Application of interpolation with function parameter to Lorentz, Sobolev and Besov spaces, In: Interpolation Spaces and Allied Topics in Analysis, Lecture Notes in Math. 1070 (1984), 183–201.

  26. P. Nilsson, Interpolation of Banach lattices, Studia Math., 82 (1985), 135–154.

  27. Y. Sagher, Interpolation of r-Banach spaces, Studia Math., 26 (1966), 45–70.

  28. M. Zafran, A multilinear interpolation theorem, Studia Math., 62 (1978), 107–124.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Loukas Grafakos.

Additional information

The author is supported by the National Science Foundation under grant DMS 0099881.

The author is supported by KBN Grant 1 P03A 013 26.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grafakos, L., Mastyło, M. Interpolation of Bilinear Operators Between Quasi-Banach Spaces. Positivity 10, 409–429 (2006). https://doi.org/10.1007/s11117-005-0034-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-005-0034-x

Mathematics Subject Classification (2000)

Keywords

Navigation