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Let $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n be a domain that supports the $$p$$ p -Poincaré inequality. Given a homeomorphism $$\varphi \in L^1_p(\Omega )$$ φ ∈ L p 1 ( Ω ) , for $$p>n$$ p > n we show that the domain $$\varphi (\Omega )$$ φ ( Ω ) has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain $$\Omega ' \subset {\mathbb {C}}$$ Ω ′ ⊂ C with non-empty boundary and for any conformal homeomorphism $$\varphi $$ φ from the unit disc $${\mathbb {D}}$$ D onto $$\Omega '$$ Ω ′ the complex derivative $$\varphi '$$ φ ′ is integrable in the degree $$s, -2<s<2/3$$ s , - 2 < s < 2 / 3 . If $$\Omega '$$ Ω ′ is bounded then $$-2<s\le 2$$ - 2 < s ≤ 2 . We prove that integrability in the degree $$s> 2$$ s > 2 is not possible for domains $$\Omega '$$ Ω ′ with infinite geodesic diameter.
Computational Methods and Function Theory – Springer Journals
Published: Apr 5, 2014
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