Math 524
Homework due 03/07/01
Problem 1. Let 1 ≤p < ∞, and α∈(0,1). Let u∈Lp(Rn) satisfy
ZB(x,r)
|u−ux,r|pdy ≤rn+pα ,
where B(x, r)⊂Rndenotes the ball of center xand radius r > 0, and ux,r denotes the
average of uon B(x, r).
1.1 Let ρ∈(0,1), and for j≥1 let rj=ρj. Show that (after possibly redefining uin a set
of measure 0) for x∈Rn
|u(x)−ux,rj| ≤ Crα
j,
where C > 0 is a constant that depends on nand ρbut not on j.
1.2. Show that for r∈(0,1), and x∈Rn
|u(x)−ux,r| ≤ C rα,
where C > 0 is a constant that depends on nand ρbut not on r.
1.3 Show that u∈C0,α
loc . More precisely prove that there is a constant C > 0 depending only
on nsuch that for x, y ∈Rn, with |x−y| ≤ 1
|u(x)−u(y)| ≤ C|x−y|α.
Problems from Folland
Chapter 4, Section 1: problems 3, 5, 6, 7, 10.
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