BMO rational approximation and one-dimensional Hausdorff content

Verdera, Joan

doi:10.1090/S0002-9947-1986-0849480-5

BMO rational approximation and one-dimensional Hausdorff content
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by Joan Verdera PDF

Trans. Amer. Math. Soc. 297 (1986), 283-304 Request permission

Abstract:

Let $X \subset {\mathbf {C}}$ be compact and let $f \in \operatorname {VMO} ({\mathbf {C}})$. We give necessary and sufficient conditions on $f$ and $X$ for ${f_{|X}}$ to be the limit of a sequence of rational functions without poles on $X$ in the norm of $\operatorname {BMO} (X)$, the space of functions of bounded mean oscillation on $X$. We also characterize those compact $X \subset {\mathbf {C}}$ with the property that the restriction to $X$ of each function in $\operatorname {VMO} ({\mathbf {C}})$, which is holomorphic on $\mathring {X}$, is the limit of a sequence of rational functions with poles off $X$. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.

References

Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225986
A. M. Davie, Analytic capacity and approximation problems, Trans. Amer. Math. Soc. 171 (1972), 409–444. MR 350009, DOI 10.1090/S0002-9947-1972-0350009-9
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971

Positive length but zero analytic capacity

24

John Garnett, Metric conditions for rational approximation, Duke Math. J. 37 (1970), 689–699. MR 269851
Reese Harvey and John Polking, Removable singularities of solutions of linear partial differential equations, Acta Math. 125 (1970), 39–56. MR 279461, DOI 10.1007/BF02838327
Robert Kaufman, Hausdorff measure, BMO, and analytic functions, Pacific J. Math. 102 (1982), no. 2, 369–371. MR 686557
Per Lindberg, A constructive method for $L^{p}$-approximation by analytic functions, Ark. Mat. 20 (1982), no. 1, 61–68. MR 660125, DOI 10.1007/BF02390498

Metric properties of analytic

capacity and approximation of analytic functions with Holder condition by rational functions

8

Analytic capacity and rational approximation

Anthony G. O’Farrell, Hausdorff content and rational approximation in fractional Lipschitz norms, Trans. Amer. Math. Soc. 228 (1977), 187–206. MR 432887, DOI 10.1090/S0002-9947-1977-0432887-2
Anthony G. O’Farrell, Estimates for capacities, and approximation in Lipschitz norms, J. Reine Angew. Math. 311(312) (1979), 101–115. MR 549960, DOI 10.1515/crll.1979.311-312.101
A. G. O’Farrell, Qualitative rational approximation on plane compacta, Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981) Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 103–122. MR 717230, DOI 10.1007/BFb0061890
Cora Sadosky, Interpolation of operators and singular integrals, Monographs and Textbooks in Pure and Applied Mathematics, vol. 53, Marcel Dekker, Inc., New York, 1979. An introduction to harmonic analysis. MR 551747
Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3
Sven Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 593–608. MR 190729
A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838

Restrictions of

weights