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January  2009, 11(1): 145-152. doi: 10.3934/dcdsb.2009.11.145

$L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$

Tadeusz Iwaniec 1, , Gaven Martin 2, and  Carlo Sbordone 3,

1. 

Syracuse University, Syracuse, NY 10065, United States
2. 

Massey University, Albany, Auckland, New Zealand
3. 

Università di Napoli, Napoli

Received  October 2007 Published  November 2008

We give an elementary proof for the integrability properties of the gradient of the harmonic extension of a self homeomorphism of the circle giving explicit bounds for the $p$-norms, $p<2$, estimates in Orlicz classes and also an $L^2(\mathbb D)$-weak type estimate.
Keywords: weak type estimates, mean distortion, Harmonic mapping.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145
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