Geodesic
Pseudo-holomorphic functions at the critical exponent
Journal of the European Mathematical Society, Tome 18 (2016) no. 9, pp. 1919-1960

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We study Hardy classes on the disk associated to the equation ∂ˉw=αwˉ for α∈Lr with 2≤r∞. 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 Lp boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in W1,2. In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for W01,2​-functions.
DOI : 10.4171/jems/634
Classification : 30-XX, 35-XX, 46-XX
Keywords: Pseudo-holomorphic functions, Hardy spaces, conjugate Beltrami equation, nonstrictly elliptic equations, Dirichlet problem 
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Laurent Baratchart; Alexander Borichev; Slah Chaabi. Pseudo-holomorphic functions at the critical exponent. Journal of the European Mathematical Society, Tome 18 (2016) no. 9, pp. 1919-1960. doi: 10.4171/jems/634

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