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

Part of the Selected Works in Probability and Statistics book series (SWPS)


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.


Brownian Motion Harmonic Function Maximal Function Local Convergence Area Function 


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  1. 1.
    J. M. Brelot and L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), 13, (1963), 395-415.MATHMathSciNetGoogle Scholar
  2. 2.
    D. L. Burkholder and R. F. Gunpy, Distribution function inequalities for the area integral, Studio, Math., 44, (1972), 527-544MATHGoogle 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.MATHMathSciNetGoogle Scholar
  4. 4.
    A. P. CalderónOn the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc., 68, (1950), 47-54.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. P. CalderónOn a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc, 68, (1950), 55-61.MATHCrossRefMathSciNetGoogle 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.MATHMathSciNetGoogle Scholar
  8. 8.
    J. L. DoobConditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, (1957), 431-458.MATHMathSciNetGoogle 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.MATHCrossRefMathSciNetGoogle 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.MATHGoogle 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.MATHGoogle 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.MATHMathSciNetGoogle 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.MATHCrossRefMathSciNetGoogle 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.MATHCrossRefMathSciNetGoogle 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.MATHMathSciNetGoogle 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.MATHCrossRefMathSciNetGoogle Scholar

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© 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

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