Advertisement

Boundary Behaviour of Harmonic Functions in a Half-Space and Brownian Motion

  • Burgess DavisEmail author
  • Renming Song
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

The behaviour of harmonic functions in the half-space \(R_ + ^{n + 1}\) has been discussed from two points of view: geometrical and probabilistic. In this paper, we compare these two view points with respect to (1) local convergence at the boundary and (2) the Hp-spaces. The results are as follows: (1) The existence of a nontangential limit for almost all points in a set E of positive Lebesgue measure in \({R^n}\left( { = \partial \,R_ + ^{n + 1}} \right)\) is more restrictive than the existence of a « fine » or probability limit almost everywhere in E when \(n \geqslant 2\). When \(n=1\), the existence of a nontangential limit almost everywhere in E implies the existence of a « fine » limit almost everywhere in E and conversely.

Keywords

Brownian Motion Harmonic Function Maximal Function Local Convergence Area Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. M. Brelot and L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), 13, (1963), 395-415.zbMATHMathSciNetGoogle Scholar
  2. 2.
    D. L. Burkholder and R. F. Gunpy, Distribution function inequalities for the area integral, Studio, Math., 44, (1972), 527-544zbMATHGoogle Scholar
  3. 3.
    D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal function characterization of the class HP, Trans. Amer. Math. Soc, 157(1971), 137-153.zbMATHMathSciNetGoogle Scholar
  4. 4.
    A. P. CalderónOn the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc., 68, (1950), 47-54.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. P. CalderónOn a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc, 68, (1950), 55-61.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. CarlesonOn the existence of boundary values for harmonic functions in several variables, Arkw för Mathematik, 4, (1961), 393-399.CrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Constantinescu and A. Cornea, Über das Verhalten der analytischen Abildungen Riemannscher Flachen auf dem idealen Rand von Martin, Nagoya Math. J., 17, (1960), 1-87.zbMATHMathSciNetGoogle Scholar
  8. 8.
    J. L. DoobConditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, (1957), 431-458.zbMATHMathSciNetGoogle Scholar
  9. 9.
    J. L. Doob, Boundary limit theorems for a half-space, J. Math. Pures Appl, (9) 37, (1958), 385-392.MathSciNetGoogle Scholar
  10. 10.
    C. Fefferman and E. M. Stein, HP-spaces in several variables, Acta Math. 129, (1972), 137-193.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. fur Mat., 167, (1931), 405-423.Google Scholar
  12. 12.
    J. LelongReview 4471, Math. Reviews, 40, (1970), 824-825.Google Scholar
  13. 13.
    J. LelongÉtude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sei. École Norm. Sup., 66, (1949), 125-159.zbMATHGoogle Scholar
  14. 14.
    J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J., 4, (1938), 473-485.CrossRefMathSciNetGoogle Scholar
  15. 15.
    H. P. McKean Jr., Stochastic integrals, Academic Press, New York, 1969.zbMATHGoogle Scholar
  16. 16.
    L. NaïmSur le rôle de la frontière de R. S. Martin dans la Théorie du potential, Ann. Inst. Fourier(Grenoble), 7, (1957), 183-285.zbMATHMathSciNetGoogle Scholar
  17. 17.
    I. I. Privalov, Integral Cauchy, Saratov, 1919.Google Scholar
  18. 18.
    D. SpencerA function-theoretic identity, Amer. J. Math., 65, (1943), 147-160.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    E. M. SteinOn the theory of harmonic functions of several variables IL Behaviour near the boundary, Acta Math., 106, (1961), 137-174.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    E. M. Stein and G. WeissOn the theory of harmonic functions of several variables I. The theory of Hp-spaces, Acta Math., 103, (1960), 25-62.zbMATHMathSciNetGoogle Scholar
  21. 21.
    J. L. WalshThe approximation of harmonic functions by harmonic polynomials and harmonic rational functions, Bull. Amer. Math. Soc., 35, (1929), 499-544.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations