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

Tadeusz Iwaniec; Gaven Martin; Carlo Sbordone

doi:10.3934/dcdsb.2009.11.145

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2009, Volume 11, Issue 1: 145-152. Doi: 10.3934/dcdsb.2009.11.145

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

1.
Syracuse University, Syracuse, NY 10065
2.
Massey University, Albany, Auckland
3.
Università di Napoli, Napoli

Received: September 30, 2007

Published: June 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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