## 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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