Approximation of harmonic functions

Björn E. J. Dahlberg

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 2, page 97-107
  • ISSN: 0373-0956

Abstract

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Let u be harmonic in a bounded domain D with smooth boundary. We prove that if the boundary values of u belong to L p ( σ ) , where p 2 and σ denotes the surface measure of D , then it is possible to approximate u uniformly by function of bounded variation. An example is given that shows that this result does not extend to p < 2 .

How to cite

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Dahlberg, Björn E. J.. "Approximation of harmonic functions." Annales de l'institut Fourier 30.2 (1980): 97-107. <http://eudml.org/doc/74453>.

@article{Dahlberg1980,
abstract = {Let $u$ be harmonic in a bounded domain $D$ with smooth boundary. We prove that if the boundary values of $u$ belong to $L^p(\sigma )$, where $p\ge 2$ and $\sigma $ denotes the surface measure of $\partial D$, then it is possible to approximate $u$ uniformly by function of bounded variation. An example is given that shows that this result does not extend to $p&lt; 2$.},
author = {Dahlberg, Björn E. J.},
journal = {Annales de l'institut Fourier},
keywords = {approximation of harmonic functions; boundary values; functions of bounded variation},
language = {eng},
number = {2},
pages = {97-107},
publisher = {Association des Annales de l'Institut Fourier},
title = {Approximation of harmonic functions},
url = {http://eudml.org/doc/74453},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Dahlberg, Björn E. J.
TI - Approximation of harmonic functions
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 2
SP - 97
EP - 107
AB - Let $u$ be harmonic in a bounded domain $D$ with smooth boundary. We prove that if the boundary values of $u$ belong to $L^p(\sigma )$, where $p\ge 2$ and $\sigma $ denotes the surface measure of $\partial D$, then it is possible to approximate $u$ uniformly by function of bounded variation. An example is given that shows that this result does not extend to $p&lt; 2$.
LA - eng
KW - approximation of harmonic functions; boundary values; functions of bounded variation
UR - http://eudml.org/doc/74453
ER -

References

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  1. [1] L. CARLESON, Interpolation by bounded analytic functions and the Corona problem, Ann. Math., 76 (1962), 547-559. Zbl0112.29702MR25 #5186
  2. [2] L. CARLESON, The Corona Problem, in Lecture Notes in Mathematics, vol 118, Springer Verlag, Berlin, 1969. Zbl0192.16801
  3. [3] B. E. J. DAHLBERG, Weighted norm inequalities for the Lusin area integral and the non tangential maximal functions for functions harmonic in a Lipschitz domain, to appear in Studia Math. Zbl0449.31002
  4. [4] C. FEFFERMAN and E. M. STEIN, Hp-spaces of several variables, Acta Math., 129 (1972), 137-193. Zbl0257.46078MR56 #6263
  5. [5] J. GARNETT, to appear. 
  6. [6] N. G. MEYERS and W. P. ZIEMER, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. of Math., 99 (1977), 1345-1360. Zbl0416.46025MR58 #22443
  7. [7] W. RUDIN, The radial variation of analytic functions, Duke Math. J., 22 (1955), 235-242. Zbl0064.31105MR18,27g
  8. [8] E. M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, New Jersey, 1970. Zbl0207.13501MR44 #7280
  9. [9] N. Th. VAROPOULOS, BMO functions and the A T T -equation, Pacific J. Math., 71 (1977), 221-273. Zbl0371.35035

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