FINE REGULARITY OF SUPERHARMONIC FUNCTIONS ON METRIC SPACES Juha Kinnunen and Visa Latvala 1. Introduction In this note we study capacity estimates related to p-superharmonic functions on metric spaces. In particular we want to point out that the suitable Caccioppoli inequality and the weak Harnack inequality established in [KM4] yield a type of estimate which is essential in the standard test function technique used to obtain certain fine regularity properties of p-superharmonic functions. We apply our esti- mate to give a new proof for the fact that the p-capacity of the infinity set of any p-superharmonic function is equal to zero (Theorem 3.6). We are also able to prove the fine continuity of the p-superharmonic functions (Theorem 4.2). Unfortunately, since our estimate is probably not sharp, the results in Section 4 are most likely not optimal. One of the main purposes of this paper is to motivate the following open question (see Section 2 for the assumptions and notation): Suppose that u> 0 is bounded and p-superharmonic in an open set Ω ⊂ X . Let η be a Lipschitz continuous function with the properties 0 ≤ η ≤ 1, η =0 in Ω \ B(z,R), and g η ≤ c/R, B(z, 10R) ⊂ Ω. Does there exist a constant c> 0 such that  B(z,R)∩{u≤λ} g p u η p dμ ≤ cλμ(B(z,R))R −p (ess inf B(z,R) u) p−1 for any λ> 0? Our key lemma (Lemma 3.4) states that the answer is positive if p − 1 on the right hand is replaced by p − 1 − ε for any small ε> 0. It is well-known ([MZ], Theorem 2.118) that the answer is positive in the Euclidean case. 2. Preliminaries Let X be a metric space and let μ be a Borel measure on X . Throughout the paper we assume that the measure of every nonempty open set is positive and that the measure of every bounded set is finite. Let u be a real-valued function on X . A nonnegative Borel measurable function g on X is said to be an upper gradient of u if (2.1) |u(x) − u(y )|≤  γ gds. 2000 Mathematics Subject Classification 31C45, 46E35. Typeset by A M S-T E X 1