We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.
On considère le problème de l’approximation qualitative par des solutions d’une équation elliptique, homogène, à coefficients constants, dans les normes de Lipschitz et BMO. Notre méthode est bien connue : on trouve une condition suffisante pour l’approximation en se réduisant à un problème de synthèse spectrale dans un certain espace de Lizorkin-Triebel doté de sa topologie faible . Deux exemples, dont l’origine est dans une construction de Hedberg, montrent que nos conditions sont fines.
@article{AIF_1996__46_4_1057_0, author = {Mateu, Joan and Netrusov, Yuri and Orobitg, Joan and Verdera, Joan}, title = {BMO and {Lipschitz} approximation by solutions of elliptic equations}, journal = {Annales de l'Institut Fourier}, pages = {1057--1081}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {46}, number = {4}, year = {1996}, doi = {10.5802/aif.1540}, mrnumber = {98c:41029}, zbl = {0853.31007}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1540/} }
TY - JOUR AU - Mateu, Joan AU - Netrusov, Yuri AU - Orobitg, Joan AU - Verdera, Joan TI - BMO and Lipschitz approximation by solutions of elliptic equations JO - Annales de l'Institut Fourier PY - 1996 SP - 1057 EP - 1081 VL - 46 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1540/ DO - 10.5802/aif.1540 LA - en ID - AIF_1996__46_4_1057_0 ER -
%0 Journal Article %A Mateu, Joan %A Netrusov, Yuri %A Orobitg, Joan %A Verdera, Joan %T BMO and Lipschitz approximation by solutions of elliptic equations %J Annales de l'Institut Fourier %D 1996 %P 1057-1081 %V 46 %N 4 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1540/ %R 10.5802/aif.1540 %G en %F AIF_1996__46_4_1057_0
Mateu, Joan; Netrusov, Yuri; Orobitg, Joan; Verdera, Joan. BMO and Lipschitz approximation by solutions of elliptic equations. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 1057-1081. doi : 10.5802/aif.1540. http://archive.numdam.org/articles/10.5802/aif.1540/
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