→ doubleN :1.f 是 weak L^2 05/08 01:19
→ doubleN :2.用 f(x) = f(y) + ∫f' 與 Holder 05/08 01:21
n
1. Let f :R →R be a Lebesgue measurable function s.t.
n -2
m(λ) = μ({x in R : |f(x)| > λ}) ≦ Cλ , λ > 0
Prove that there is a constant C such that for any Borel set
n 1
E in R of finite and positive measure
1/2
∫ |f(x)|dμ(x) ≦ C (μ(E)) .
E 1
1 -n
2. Assume that f_n in C ([0,1]) is a sequence s.t. ║f_n║ ≦ 2
n/4 ∞
and ║f'_n║ ≦ 2 . Let f =Σf_n , prove that there exists a
∞ 1/2
constant K < ∞ s.t. |f(x)-f(y)| ≦ K|x-y| , for all x,y in [0,1]
請問這兩題要怎麼做呢?
第一題從結果來看，我覺得是用Holder ineq.做，所以只要
想辦法去証明f是L^2就可以了。但是我証不出它是L^2...
我這樣想有問題嗎??
第二題的話，不知道怎麼做...想用均值定理好像也不太對
請指教 謝謝!! <(_ _)>
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