Electronic Journal of Differential Equations, Vol. 1997(1997), No. 25, pp. 1-4. Title: A note on very weak p--harmonic mappings Authors: Juha Kinnunen (Univ. of Helsinki, Finland) Shulin Zhou (Peking Univ., Beijing, China) Abstract: We prove a new a priori estimate for very weak p-harmonic mappings when p is close to two. This sheds some light on a conjecture posed by Iwaniec and Sbordone. Submitted November 27, 1997. Published December 19, 1997. Math Subject Classification: 35J60, 35B45. Key Words: Very weak p--harmonic mapping.