Find the necessary and sufficient conditions on a and b

Math 555a Homework 4
Due: Monday, November 29, 2021 in class There will be no homework accepted after Wednesday, December 1, 2021, in class

1. [Evans, p. 307 #14] Verify that if n ≥ 2, the unbounded function u = loglog(1+1/|x|) belongs to W1,n(U), for U = B0(0,1).
2. [Evans, p. 309 #20] Use the Fourier transform to prove that if u ∈ Hs(Rn) for s > n/2, then u ∈ L∞(Rn), with the bound kukL∞(Rn) ≤ CkukHs(Rn) the constant C depending only on s and n.
3. Let p ∈ [1,∞). Prove that
   u =
   1 |x|a belongs to W1,p(Rn) if and only if (a + 1)p < n. For the ‘if’ part, do it two ways, using an approximation as in the class and using a cut-off as in Evans.
4. Let p ∈ [1,∞). Find the necessary and sufficient conditions on a and b such that
   u =
   1 |x|a(−log|x|)b
   belongs to W1,p(B1/2). Provide full justifications.
5. Evans, p. 307 #10 Integrate by parts to prove ZU |Du|p ≤ CZU |u|p dx1/2ZU |D2u|p dx1/2
   for 2 ≤ p < ∞ and all u ∈ C∞ 0 (U). (Hint: RU |Du|p dx =Pn i=1RU uxiuxi|Du|p−2 dx.)(b) Prove kDukL2p ≤ Ckuk1/2 L∞kD2uk1/2 Lp for 1 ≤ p < ∞ and all u ∈ C∞ 0 (U).