课程名称︰实分析一
课程性质︰数学研究所必修课；应用数学研究所必选课
课程教师︰沈俊严
开课学院：理学院
开课系所︰数学研究所
考试日期︰2018年01月10日(三)
考试时限：10:20-12:10，计110分钟。
试题 :
Do the following problems and write your arguments as detail as possible.
s r
1. (10%) Suppose f satisfies ∫|f| < ∞ and ∫|f| < ∞ for some 0 < s < r < ∞.
p
Does f also satisfy ∫|f| < ∞ for s < p < r ? Prove or disprove it.
2. (15%) Prove the simple Vitali Lemma for measurable sets. Given a measurable
n
set E⊂R with |E| < ∞. Let K be a collection of cubes Q covering E. Then
there exists a positive constant β, depending only on n, and a finite
N
number of disjoint cubes Q , ... , Q in K such that Σ|Q |≧β|E|.
1 N j=1 j
3. (10%) Let
/ 1 if x≧0
f(x) =
\ 0 if x＜0
Find the value of its maximal function at x = -2, i.e. what is the value of
f*(-2).
4. (15%) Given two statements (a) f∈L(R) and (b) lim ∫ |f| = 0.
λ->∞ {|f|>λ}
(1) Does (a) imply (b)? Prove or disprove it.
(2) Does (b) imply (a)? Prove or disprvoe it.
p
5. (15%) Suppose ∫ |f - f| → 0 for some p > 0 when k→∞. Does there exist
E k
a subsequence f → f a.e. in E? Prove or disprove it.
k_j
p kp k
6. (10%) If f≧0, show that f∈L if and only if Σ 2 ω(2 ) < ∞
k∈Z
n
7. (10%) Given an integrable function f∈L(R ), and let f*(x) be its
Hardy-Littlewood maximal function. Show that f*(x) is lower semicontinuous.
8. (15%) Prove that ψ(x) is convex on (a,b) if and only if ψ is continuous
and ψ[(x+y)/2] ≦ [ψ(x)+ψ(y)]/2 for any x,y∈(a,b).