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A Carleson type estimate provides control of the bound of positive harmonic func- tions vanishing on a portion of the boundary. Such an estimate is well-known for harmonic functions in certain Euclidean domains. In this note we prove a Carleson type estimate for p-harmonic functions on bounded John domains in a complete met- ric space equipped with an Ahlfors Q-regular measure supporting a (1;p)-Poincar´e inequality for some 1 < p • Q. This result is new even in the Euclidean setting when p 6= 2. We then use the Carleson estimate to study the conformal Martin boundary of bounded John domains in metric measure spaces of Q-bounded geometry.

Carleson type estimates for p-harmonic functions and the conformal Martin boundary of John domains in metric measure spaces