Baratchart, Laurent; Borichev, Alexander; Chaabi, Slah Pseudo-holomorphic functions at the critical exponent. (English) Zbl 1353.30044 J. Eur. Math. Soc. (JEMS) 18, No. 9, 1919-1960 (2016). Authors’ abstract: We study Hardy classes on the disk associated to the equation \(\bar \partial w= \alpha \bar w\) for \(\alpha \in L^r\) with \(2 \leq r < \infty\). The paper seems to be the first to deal with the case \(r=2\). We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted \(L^p\) boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in \(W^{1,2}\). In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for \(W^{1,2}_0\)-functions. Reviewer: Miloš Čanak (Beograd) Cited in 8 Documents MSC: 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) 30E25 Boundary value problems in the complex plane 35J56 Boundary value problems for first-order elliptic systems 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:pseudo-holomorphic functions; Hardy spaces; conjugate Beltrami equation; Dirichlet problem PDFBibTeX XMLCite \textit{L. Baratchart} et al., J. Eur. Math. Soc. (JEMS) 18, No. 9, 1919--1960 (2016; Zbl 1353.30044) Full Text: DOI arXiv References: [1] Adams, R., Fournier, J.: Sobolev Spaces. Academic Press (2003)Zbl 1098.46001 MR 2424078 · Zbl 1098.46001 [2] Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Univ. Press (2009)Zbl 1182.30001 MR 2472875 · Zbl 1182.30001 [3] Astala, K., P¨aiv¨arinta, L.: Calder´on’s inverse conductivity problem in the plane. 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