BMO and Lipschitz approximation by solutions of elliptic equations
Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 1057-1081.

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.

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.

@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://www.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://www.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://www.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, Tome 46 (1996) no. 4, pp. 1057-1081. doi : 10.5802/aif.1540. http://www.numdam.org/articles/10.5802/aif.1540/

[A] D R. Adams, A note on the Choquet integrals with respect to Hausdorff capacity, Lecture Notes in Math., 1302 (1988), 115-124. | MR | Zbl

[AH] D.R. Adams and L. I. Hedberg, Function spaces and Potential Theory, Springer, Berlin and Heidelberg, 1996. | MR | Zbl

[B] T. Bagby, Approximation in the mean by solutions of elliptic equations, Trans. Amer. Math. Soc., 281 (1984), 761-784. | MR | Zbl

[CA] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Sc. Norm. Sup. Pisa, 18 (1964), 137-160. | Numdam | MR | Zbl

[C] L. Carleson, Selected problems on exceptional sets, Van Nostrand Math. Studies, 13, Van Nostrand, Princeton, N. J., 1967. | MR | Zbl

[FJ] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Func. Anal., 93 (1990), 34-170. | MR | Zbl

[GR] J. García-Cuerva and J. L. Rubio De Francia, Weighted norm inequalities and related topics, North-Holland Mathematical Studies, 116, Amsterdam, 1985. | MR | Zbl

[GT] P.M. Gauthier and N. Tarkhanov, Degenerate cases of approximation by solutions of systems with injective symbols, Canad. J. Math., 20 (1993), 1-18. | Zbl

[H] L.I. Hedberg, Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81. | MR | Zbl

[Ho] P.J. Holden, Extension theorems for functions of vanishing mean oscillation, Pacific J. of Math., 142 (1990), 277-295. | MR | Zbl

[JW] A. Jonsson and H. Wallin, Function spaces on subsets of Rn, Harwood Academic Publishers, Math. Reports, 2, Part 1, 1984. | MR | Zbl

[M] J. Mateu, A counterexample in Lp approximation by harmonic functions, preprint, 1995. | Zbl

[MO] J. Mateu and J. Orobitg, Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J., 39 (1990), 703-736. | MR | Zbl

[MV] J. Mateu and J. Verdera, BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces, Rev. Mat. Iberoamericana, 4 (1988), 291-318. | MR | Zbl

[MP] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, Cambridge, 1995. | MR | Zbl

[MPO] P. Mattila and J. Orobitg, On some properties of Hausdorff content related to instability, Ann. Acad. Sci. Fenn. Ser. A I Math., 19 (1994), 393-398. | MR | Zbl

[ME] N.G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc., 15 (1964), 717-721. | MR | Zbl

[N1] Y. Netrusov, Spectral synthesis in spaces of smooth functions, Russian Acad. Sci. Dokl. Math., 46, 1993), 135-138. | MR | Zbl

[N2] Y. Netrusov, Sets of singularities of functions in spaces of Besov and Lizorkin-Triebel type, Proc. Steklov Inst. Math., 187 (1990), 185-203. | Zbl

[N3] Y. Netrusov, Imbedding theorems for Lizorkin-Triebel spaces, Zapiski Nauchn. Sem. LOMI, 159 (1987), 103-112, English trans.: J. Soviet Math., 47 (1989). | Zbl

[N4] Y. Netrusov, Metric estimates of capacities of sets in the Besov spaces, Trudy Mian USSR, 190 (1989), 159-185, English trans.: Proc. Steklov Ins. Math., 190 (1992), 167-192. | MR | Zbl

[OF1] A.G. O'Farrell, Rational approximation in Lipschitz norms II, Proc. R. Ir. Acad., 75 A (1975), 317-330. | MR | Zbl

[OF2] A.G. O'Farrell, Hausdorff content and rational approximation in fractional Lipschitz norms, Trans. Amer. Math. Soc., 288 (1977), 187-206. | MR | Zbl

[OF3] A.G. O'Farrell, Localness of certain Banach modules, Indiana Univ. Math. J., 24 (1975), 1135-1141. | MR | Zbl

[St] E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. | MR | Zbl

[TA] N. Tarkhanov, Approximation on compact sets by solutions of systems with surjective symbol, Russian Math. Surveys, 48-5 (1994), 103-145. | MR | Zbl

[T] H. Triebel, Theory of function spaces, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1983. | Zbl

[V1] J. Verdera, Cm approximation by solutions of elliptic equations, and Calderón Zygmund operators, Duke Math. J., 55 (1987), 157-187. | MR | Zbl

[V2] J. Verdera, Removability, capacity and approximation, Complex Potential Theory, NATO ASI Series C, 439, Kluwer Academic Publishers, Dordrecht (1993), 419-473. | MR | Zbl

[V3] J. Verdera, BMO rational approximation and one dimensional Hausdorff content, Trans. Amer. Math. Soc., 297 (1986), 283-304. | MR | Zbl

[W] B.M. Weinstock, Uniform approximation by solutions of elliptic equations, Proc. Amer. Math. Soc., 41 (1973), 267-290. | MR | Zbl

Cité par Sources :