Arkiv för Matematik

  • Ark. Mat.
  • Volume 51, Number 2 (2013), 293-313.

On the Carleson duality

Tuomas Hytönen and Andreas Rosén

Full-text: Open access

Abstract

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L2 Whitney averages belongs to L2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$, and characterize the pointwise multipliers from ${\mathcal{X}}$ to L2 on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to Lp generalizations of the space ${\mathcal{X}}$. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

Note

Andreas Rosén was earlier named Andreas Axelsson.

Article information

Source
Ark. Mat., Volume 51, Number 2 (2013), 293-313.

Dates
Received: 11 April 2011
First available in Project Euclid: 1 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907215

Digital Object Identifier
doi:10.1007/s11512-012-0167-7

Mathematical Reviews number (MathSciNet)
MR3090198

Zentralblatt MATH identifier
1294.42002

Rights
2012 © Institut Mittag-Leffler

Citation

Hytönen, Tuomas; Rosén, Andreas. On the Carleson duality. Ark. Mat. 51 (2013), no. 2, 293--313. doi:10.1007/s11512-012-0167-7. https://projecteuclid.org/euclid.afm/1485907215


Export citation

References

  • Auscher, P. and Axelsson [Rosén], A., Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math. 184 (2011), 47–115.
  • Auscher, P. and Rosén, A., Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II, to appear in Anal. PDE.
  • Carleson, L., An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930.
  • Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547–559.
  • Coifman, R. R., Meyer, Y. and Stein, E. M., Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.
  • Dahlberg, B., On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119–1138.
  • Fefferman, C. and Stein, E. M., Hp spaces of several variables, Acta Math. 129 (1972), 137–193.
  • Fefferman, R. A., Kenig, C. E. and Pipher, J., The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65–124.
  • Kenig, C. and Pipher, J., The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), 447–509.
  • Kenig, C. and Pipher, J., The Neumann problem for elliptic equations with nonsmooth coefficients: part II, Duke Math. J. 81 (1995), 227–250.
  • Stein, E. M., Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43. Princeton University Press, Princeton, NJ, 1993.