The theory of weights and the Dirichlet problem for elliptic equations. (English) Zbl 0770.35014

The authors study the Dirichlet problem for second order elliptic operators in divergence form with bounded measurable coefficients. The analysis involves the interplay between partial differential equations and harmonic analysis. The general idea is to consider two operators \(L_ 0\) and \(L_ 1\) for which “good” estimates for the Dirichlet problem for \(L_ 0\) are known. The question is to determine optimal conditions on the difference between the coefficients which guarantee that the Dirichlet problem for \(L_ 1\) has “good” estimates. “Good” can have several different interpretations. One of them is that there exists \(p:1<p<\infty\) for which the Dirichlet problem may be solved in an appropriate setting with \(L_ p\) data.
The approach employs work of R. Fefferman [J. Am. Math. Soc. 2, No. 1, 127-135 (1989; Zbl 0694.35050)] in a critical way. The main results of Fefferman and of B. E. J. Dahlberg [Am. J. Math. 108, 1119-1138 (1986; Zbl 0644.35032); Arch. Ration. Mech. Anal. 65, 275-288 (1977; Zbl 0406.28009)] are proved by using a “differential inequality” for a family of harmonic measures introduced by Dahlberg. The authors also give new direct proofs of the results in these papers without requiring the use of the differential inequality.
Examples are presented for the theorems cited by “pulling back” the Laplacian via a quasiconformal mapping of the plane into itself. Specializing the main theorem in this case indicates a new characterization of the class of \(A^ \infty\) weights introduced in relation to problems in harmonic analysis by B. Muckenhoupt [Trans. Amer. Math. Soc. 165, 207-226 (1972; Zbl 0236.26016)] and R. Coifman and C. Fefferman [Studia Math. 51, 241-250 (1974; Zbl 0291.44007)]. A criterion is stated in terms of Carleson measures that is very close to \(\log w\in BMO\) and is necessary and sufficient for \(w\in A^ \infty\). This can be viewed as an \(A^ \infty\) version of the Helson-Szego characterization of \(A^ 2\) in \(\mathbb{R}\). However, the results obtained in this paper are valid for all dimensions. The authors conclude by using the characterization of \(A^ \infty\) to establish that their theorems on the Dirichlet problem are optimal.


35J25 Boundary value problems for second-order elliptic equations
30C62 Quasiconformal mappings in the complex plane
42B30 \(H^p\)-spaces
30C85 Capacity and harmonic measure in the complex plane
31B25 Boundary behavior of harmonic functions in higher dimensions
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