# Approximation of harmonic functions

Annales de l'institut Fourier (1980)

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

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topDahlberg, 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 $p< 2$.},

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 $p< 2$.

LA - eng

KW - approximation of harmonic functions; boundary values; functions of bounded variation

UR - http://eudml.org/doc/74453

ER -

## References

top- [1] L. CARLESON, Interpolation by bounded analytic functions and the Corona problem, Ann. Math., 76 (1962), 547-559. Zbl0112.29702MR25 #5186
- [2] L. CARLESON, The Corona Problem, in Lecture Notes in Mathematics, vol 118, Springer Verlag, Berlin, 1969. Zbl0192.16801
- [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] C. FEFFERMAN and E. M. STEIN, Hp-spaces of several variables, Acta Math., 129 (1972), 137-193. Zbl0257.46078MR56 #6263
- [5] J. GARNETT, to appear.
- [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] W. RUDIN, The radial variation of analytic functions, Duke Math. J., 22 (1955), 235-242. Zbl0064.31105MR18,27g
- [8] E. M. STEIN, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, New Jersey, 1970. Zbl0207.13501MR44 #7280
- [9] N. Th. VAROPOULOS, BMO functions and the $ATT$-equation, Pacific J. Math., 71 (1977), 221-273. Zbl0371.35035

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