Winter 2008 Problem 5: Let 1 ≤ p < ∞ and let I = (−1, 1) denote the open interval in ℝ. Find the values of α as a function of p for which the function |x|^{α} ∈ W^{1,p}(I).

Fall 2009 Problem 1: For ϵ > 0, let η_{ϵ} denote the family of standard mollifiers on ℝ^{2}. Given u ∈ L^{2}(ℝ^{2}), define the functions

u_{ϵ} = η_{ϵ} ⋆ u in ℝ^{2}.

Prove that

C∥Du_{ϵ}∥_{L^{2}(ℝ^{2})} ≤ ∥u∥_{L^{2}(ℝ^{2})},

where the constant C depends on the mollifying function but not on u.

Fall 2012 Problem 4: The heat kernel on ℝ^{3} is given by H_{t}(x) = (4πt)^{− \frac{3}{2}} e^{− \frac{|x|^{2}}{4t}} where |x| denotes the Euclidean norm of x ∈ ℝ^{3}. Prove that if u ∈ L^{3}(ℝ^{3}), then t^{\frac{1}{2}} ∥H_{t} ∗ u∥_{L^{∞}(ℝ^{3})} → 0 as t → 0^{+}. (Note that ∗ denotex convolution.)

Fall 2013 Problem 3: For δ > 0 small, let u ∈ L^{\frac{3}{2} +δ}(ℝ^{3}) ∩ L^{\frac{3}{2} −δ}(ℝ^{3}). Prove that v = u ∗ \frac{1}{|x|} ∈ L^{∞}(ℝ^{3}) and provide a bound for ∥v∥_{L^{∞}(ℝ^{3})} which depends only on ∥u∥_{L^{\frac{3}{2} +δ}(ℝ^{3})}.

Spring 2010 Problem 5: Let H^{s}(ℝ) denote the Sobolev space of order s on the real line ℝ, and let

∥u∥_{s} = (∫ℝ(1 + |ξ|^{2})^{s}|û(ξ)|^{2}dξ)^{\frac{1}{2}}

denote the norm on H^{s}(ℝ) where û(ξ) = \frac{1}{2π} ∫ u(x)e^{−ixξ}dx denotes the Fourier transform of u. Suppose that r < s < t, all real, and ϵ > 0 is given. Show that there exists a constant C > 0 such that

∥u∥_{s} ≤ ϵ∥u∥_{t} + C∥u∥_{r} ∀u ∈ H^{t}(ℝ).

Spring 2013 Problem 6: Let I be the interval (0, 1) and q ≥ p ≥ 1. Show that there exists a constant C = C(p, q, I) such that

∥u∥_{L^{q}(I)} ≤ C∥u∥_{W^{1,p}(I)}

for all u ∈ W_{0}^{1,p}(I). (Hint: First show that ∥u∥_{L^{∞}(I)} ≤ C∥u∥_{W^{1,p}(I)} for all u ∈ W_{0}^{1,p}(I).)

Fall 2015 Problem 4: Let f : ℝ^{3} → ℝ with f, ∇f ∈ L^{1}(ℝ^{3}). Show that

∫_{ℝ_{3}} |f(x)|^{\frac{3}{2}} dx ≤ (∫_{ℝ_{3}} |∇f(x)|dx)^{\frac{3}{2}} .