# American Institute of Mathematical Sciences

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 $\mathbb D$

 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.
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 and Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145
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