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The following inequality is sometimes called a Sobolev–Poincar´e inequality. The above discussion shows that it holds with the exponent p in both sides. This is the most useful inequality for functions u in N01,p (Ω) when µ(X \ Ω) > 0 and Ω is bounded. Then u is = 0 in the large subset of X \ Ω. 9. Let X be a doubling metric space supporting a weak (t, q)– Poincar´e inequality for some 1 < q < p and t > 1. Suppose that u ∈ N 1,p (X) and let A = {y ∈ B(x, R) : |u(y)| > 0}. If µ(A) ≤ γµ(B(x, R)) for some γ with 0 < γ < 1, then there is a constant c > 0 so that − |u|t dµ 1/t ≤ cR guq dµ) − 1/q .

However, for many fractal sets X of Rn the space M 1,p (X) with a natural Hausdorﬀ measure is a natural space to study minimizers of variational integrals and it would be interesting to know if a potential theory similar to the theory based on the Newtonian space can be constructed on M 1,p (X). 3. Newtonian space N 1,p (X) This is the second choice to form a function space in X similar to the classical ﬁrst order Sobolev space W 1,p . In some sense it is closer to the classical Sobolev space W 1,p than M 1,p .

5). The function u is 2λ–lipschitz on Eλ since for x, y ∈ Eλ |u(x) − u(y)| ≤ d(x, y)(g(x) + g(y)) ≤ 2λd(x, y); April 7, 2008 17:13 World Scientific Review Volume - 9in x 6in 36 O. Martio we have assumed that the inequality |u(x) − u(y)| ≤ d(x, y)(g(x) + g(y)) holds everywhere in X. 3 u|Eλ can be extended to a 2λ–bilipschitz function vλ : X → R. Now vλ need not belong to Lp (X) and we set uλ = sgn vλ min (|vλ |, λ). Then uλ is again a 2λ–lipschitz function on X and uλ = u on Eλ . Moreover, uλ − u p → 0 as λ → ∞, since uλ − u p p |uλ − u|p dµ ≤ 2p−1 ≤ X\Eλ |uλ |p dµ X\Eλ |u|p dµ → 0 as λ → ∞.