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Beltrami operators in the plane. (English) Zbl 1009.30015

The paper deals with optimal \(L^p(\mathbb{C})\)-properties of solutions of the general nonlinear elliptic system \(f_{\overline z}= H(z, f_z)\), where \(H\) is a measurable function satisfying the Lipschitz condition \[ |H(z, w_1)- H(z, w_2)|\leq k|w_1- w_2|,\quad k< 1 \] and \(H(z,0)= 0\). The best known and most important example of such an equation is the Beltrami equation where \(H(z,w)= \mu(z)w\) and \(|\mu|_\infty= \text{ess sup}|\mu|< 1\). The regularity theory of this equation leads to the concept of weakly \(k\)-quasiregular mappings \(f\) in a domain \(D\subset\mathbb{C}\). These are solutions \(f\) of the Beltrami equation in the Sobolev space \(W^{1,q}_{\text{loc}}(D)\) where \(1\leq q\leq 2\) and \(|\mu|_\infty\leq k\). For \(q= 2\) these mappings are called \(k\)-quasiregular and are continuous. The regularity problem is to determine the range of \(q\) such that \(f\) is continuous. The authors show that this happens iff the operators \(I-\mu T\) are injective on \(L^q(\mathbb{C})\) for all \(\mu\) with \(|\mu|_\infty\leq k< 1\). Here \(T\) denotes the Beurling transform. They also show that \(|\mu|_\infty< q-1\) guarantees this and that \(|\mu|_\infty> q-1\) is not enough. The behavior of \(\mu\) at \(\infty\) is the essential feature. In the general case the nonlinear singular integral operator \(Bg= g- H(z,Tg)\) is invertible in \(L^p(\mathbb{C})\) whenever \(p\in (1+ k,1+ 1/k)\). The proof relies on a priori bounds for the operator \(I- \mu T\) and this is studied in \(L^p\)-spaces with weights constructed from the Jacobians of plane quasiconformal mappings.

MSC:

30C62 Quasiconformal mappings in the complex plane
35J60 Nonlinear elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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