Approximation of harmonic functions
Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 97-107.

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 p2 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.

Soit D un domaine borné à frontière régulière, et u une fonction harmonique dans D. On montre que si les valeurs de u à la frontière appartiennent à L p (σ) avec p2 (σ étant la mesure de surface à la frontière), u est approchable uniformément par des fonctions à variation bornée, et on montre que le résultat ne s’étend pas au cas p<2.

     author = {Dahlberg, Bj\"orn E. J.},
     title = {Approximation of harmonic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {97--107},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {2},
     year = {1980},
     doi = {10.5802/aif.787},
     mrnumber = {82i:31010},
     zbl = {0417.31005},
     language = {en},
     url = {}
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PB  - Institut Fourier
PP  - Grenoble
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DO  - 10.5802/aif.787
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%T Approximation of harmonic functions
%J Annales de l'Institut Fourier
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Dahlberg, Björn E. J. Approximation of harmonic functions. Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 97-107. doi : 10.5802/aif.787.

[1] L. Carleson, Interpolation by bounded analytic functions and the Corona problem, Ann. Math., 76 (1962), 547-559. | MR | Zbl

[2] L. Carleson, The Corona Problem, in Lecture Notes in Mathematics, vol 118, Springer Verlag, Berlin, 1969.

[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. | EuDML | MR | Zbl

[4] C. Fefferman and E. M. Stein, Hp-spaces of several variables, Acta Math., 129 (1972), 137-193. | MR | Zbl

[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. | MR | Zbl

[7] W. Rudin, The radial variation of analytic functions, Duke Math. J., 22 (1955), 235-242. | MR | Zbl

[8] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, New Jersey, 1970. | MR | Zbl

[9] N. Th. Varopoulos, BMO functions and the ATT-equation, Pacific J. Math., 71 (1977), 221-273. | MR | Zbl

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