Approximation of harmonic functions

Dahlberg, Björn E. J.

Annales de l'Institut Fourier, Tome 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 $L^p\left(\sigma \right)$, 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 $p<2$.

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 $L^p\left(\sigma \right)$ avec $p\ge 2$ ($\sigma$ 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 $p<2$.

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

[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. | Zbl | MR | EuDML

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

[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. | Zbl | MR

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

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

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