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MATH 519 Homework Fall 2013 51. If f, g ∈ L1 (Rn ) , show that supp (f ∗ g) ⊂ supp f + supp g Here the sum of two sets is defined in the usual way, A + B = {z = x + y : x ∈ A, y ∈ B} 52. Let Ω be a bounded open set and K ⊂⊂ Ω. Show that there exists φ ∈ C0∞ (Ω) such that 0 ≤ φ(x) ≤ 1 and φ(x) ≡ 1 for x ∈ K. (Hint: approximate the characteristic function of Σ by convolution, where Σ satisfies K ⊂⊂ Σ ⊂⊂ Ω. Use problem #51 to prove the needed support property.) 1 53. If Ω is a bounded open set in R3 , and u(x) = |x| , show that u ∈ 3 W 1,p (Ω) for 1 ≤ p < 2 . Along the way, you should show carefully that a ∂u distributional first derivative ∂x agrees with the corresponding pointwise i derivative. 54. If u ∈ W 1,2 (0, 1), show that |u(x) − u(y)| ≤ C p |x − y| x, y ∈ [0, 1] where C = ||u||W 1,2 (0,1) . 55. Evaluate the Fourier series ∞ X (−1)n n sin (nx) n=1 in D0 (R) (The result of problem 38 can be used here.) If possible, plot some partial sums of this series.