- Establish the inequality
_{r}≤_{p}_{q}whenever=+. - Let be counting measure on ℕ. Thus Ω = ℕ and S = Pwith μ= number of things in S. Let 1 ≤ p ≤ q. Show that in this case,
Hint: This is real easy if you consider what ∫

_{Ω}fdμ equals. How are the norms related? - Consider the function, f=+for x,y > 0 and+= 1. Show directly that f≥ 1 for all such x,y and show this implies xy ≤+.
- Give an example of a sequence of functions in L
^{p}which converges to zero in L^{p}but does not converge pointwise to 0. Does this contradict the proof of the theorem that L^{p}is complete? - Let ϕ : ℝ → ℝ be convex. This means
whenever λ ∈ [0,1]. Verify that if x < y < z, then

≤and that≤. Show if s ∈ ℝ there exists λ such that ϕ(s) ≤ ϕ(t) + λ(s−t) for all t. Show that if ϕ is convex, then ϕ is continuous. - ↑ Prove Jensen’s inequality. If ϕ : ℝ → ℝ is convex, μ(Ω) = 1, and f : Ω → ℝ is in L
^{1}(Ω), then ϕ(∫_{Ω}fdu) ≤∫_{Ω}ϕ(f)dμ. Hint: Let s = ∫_{Ω}fdμ and use Problem 5. - B(p,q) = ∫
_{0}^{1}x^{p−1}(1 − x)^{q−1}dx,Γ(p) = ∫_{0}^{∞}e^{−t}t^{p−1}dt for p,q > 0. The first of these is called the beta function, while the second is the gamma function. Show a.)Γ(p + 1) = pΓ(p);b.) Γ(p)Γ(q) = B(p,q)Γ(p + q). - Let f ∈ C
_{c}(0,∞) and define F(x) =∫_{0}^{x}f(t)dt. ShowHint: Argue there is no loss of generality in assuming f ≥ 0 and then assume this is so. Integrate ∫

_{0}^{∞}|F(x)|^{p}dx by parts as follows:Now show xF

^{′}= f − F and use this in the last integral. Complete the argument by using Holder’s inequality and p − 1 = p∕q. The measure is one dimensional Lebesgue measure in this problem. - ↑ Now suppose f ∈ L
^{p}(0,∞),p > 1, and f not necessarily in C_{c}(0,∞). Show that F(x) =∫_{0}^{x}f(t)dt still makes sense for each x > 0. Show the inequality of Problem 8 is still valid. This inequality is called Hardy’s inequality. Hint: To show this, use the above inequality along with the density of C_{c}in L^{p}. - Suppose f,g ≥ 0. When does equality hold in Holder’s inequality?
- Let α ∈ (0,1]. We define, for X a compact subset of ℝ
^{p},where

and

Show that

is a complete normed linear space. This is called a Holder space. What would this space consist of if α > 1? - Let {f
_{n}}_{n=1}^{∞}⊆ C^{α}where X is a compact subset of ℝ^{p}and supposefor all n. Show there exists a subsequence, n

_{k}, such that f_{nk}converges in C. We say the given sequence is precompact when this happens. (This also shows the embedding of C^{α}into Cis a compact embedding.) Hint: You might want to use the Ascoli Arzela theorem. - Let f :ℝ × ℝ
^{n}→ ℝ^{n}be continuous and bounded and let x_{0}∈ ℝ^{n}. Ifand h > 0, let

For t ∈

, letShow using the Ascoli Arzela theorem that there exists a sequence h → 0 such that

in C

. Next argueand conclude the following theorem. If f :ℝ × ℝ

^{n}→ ℝ^{n}is continuous and bounded, and if x_{0}∈ ℝ^{n}is given, there exists a solution to the following initial value problem. - Suppose f ∈ L
^{∞}∩ L^{1}. Show lim_{p→∞}||f||_{Lp}= ||f||_{∞}. Hint:Now raise both ends to the 1∕p power and take liminf and limsup as p →∞. You should get

_{∞}− ε ≤ liminf_{p}≤ limsup_{p}≤_{∞} - Suppose μ(Ω) < ∞. Show that if 1 ≤ p < q, then L
^{q}(Ω) ⊆ L^{p}(Ω). Hint Use Holder’s inequality. - Show L
^{1}(ℝ) ⊈ L^{2}(ℝ) and L^{2}(ℝ) ⊈ L^{1}(ℝ) if Lebesgue measure is used. Hint: Consider 1∕and 1 ∕x. - Suppose that θ ∈ [0,1] and r,s,q > 0 with
show that

If q,r,s ≥ 1 this says that

Using this, show that

Hint:

Now note that 1 =

+and use Holder’s inequality. - Suppose f is a function in L
^{1}and f is infinitely differentiable. Is f^{′}∈ L^{1}? Hint: What if ϕ ∈ C_{c}^{∞}and f= ϕfor x ∈, f= 0 if x < 0?

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