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15) ω = 1. Sn−1 Let λ denote the differential form on Rn \{0} obtained by pulling ω back using πx. 16) |λ(u)| ≤ C |u − x|−n+1 for all u ∈ Rn \{x}, where C is a slightly different constant from before. In particular, λ is locally integrable across x (and smooth everywhere else). This permits one to take the exterior derivative of λ on all of Rn in the (distributional) sense of currents [Fed, Morg], and the result is that dλ is the current of degree n which is a Dirac mass at x. More precisely, dλ = 0 away from x because ω is automatically closed (being a form of top degree on Sn−1 ), and because the pull-back of a closed form is always closed.

The next two definitions give the conditions on M that we shall consider. These and similar notions have come up many times in various parts of geometry and analysis, as in [Ale, AleV2, AleV3, As1, As2, As3, CoiW1, CoiW2, Gro1, Gro2, HeiKo1, HeiKo2, HeiKo2, HeiY, Pet1, Pet2, V¨ai6]. 9 (The doubling condition) A metric space (M, d(x, y)) is said to be doubling (with constant L0 ) if each ball B in M with respect to d(x, y) can be covered by at most L0 balls of half the radius of B. Notice that Euclidean spaces are automatically doubling, with a constant L0 that depends only on the dimension.

20) for B centered at 0. If one is far enough away from the origin (compared to r), then Nr (x) simply vanishes, and there is nothing to do. In general one can have mixtures of the two types of phenomena. 23) dist(x, F ) = inf{|x − z| : z ∈ F }. It is a standard exercise that such a function f is always Lipschitz with norm at most 1. Depending on the behavior of the set F , this function can have 45 plenty of sharp corners, like |x| has at the origin, and plenty of oscillations roughly like the ones in the functions gρ .