Sobolev Spaces on Metric Spaces

Abstract

We have seen that if u is a smooth function defined on a ball B in ℝⁿ (possibly with infinite radius so that B = ℝⁿ), then the inequality

$$ \left| {u(x) - u(y)} \right| \leqslant C (n)(I_1 \left| {\nabla u} \right|(x) + I_1 \left| {\nabla u} \right|(y)) $$

(5.1)

holds for each pair of points x, y in B, where I₁ is the Riesz potential. It is easily seen that in the definition of I₁ we can integrate ‖∇u‖ against the Riesz kernel ‖z‖¹⁻ⁿ over a ball whose radius is roughly ‖x − y‖ and still retain inequality (5.1); then, I₁‖∇u‖(x) is controlled by a constant C(n) times

$$ \left| {x - y} \right|M\left| {\nabla u} \right|(x), $$

as is easily seen by dividing the ball over which the integration occurs into annuli as in the proof of Proposition 3.19. By symmetry, there is a similar bound for I₁‖∇u‖(y), and we therefore conclude that

$$ \left| {u(x) - u(y)} \right| \leqslant C(n)\left| {x - y} \right|(M\left| {\nabla u} \right|(x) + M\left| {\nabla u} \right|(y)) $$

(5.2)

for each pair of points x, y in B. If u belongs to W^1,p(B), so that its gradient is in L^p(B), and if p > 1, we conclude from the maximal function theorem that

$$ \left| {u(x) - u(y)} \right| \leqslant \left| {x - y} \right|(g(x) + g(y)) $$

(5.3)

for each pair of points x, y in B, where g ε L^p(B); in fact, we can choose g in inequality (5.3) to be a constant times the maximal function M‖∇u‖. By the density of smooth functions in W^1,p(B), we thus obtain that inequality (5.3) continues to hold almost everywhere in B (in the sense that by ruling out a set of measure zero, inequality (5.3) holds for all points x and y outside this set).