128 1. Measure theory (ii) Show that if P is a good kernel, then cd ∞ n=−∞ 2dn ˜(2n) P ≤ Cd for some constants 0 cd Cd depending only on d. (Hint: Compare P with such “horizontal wedding cake” functions as ∑∞ n=−∞ 12n−1 |x|≤2n ˜(2n).) P (iii) Establish the quantitative upper bound | Rd f(y)Pt(x − y) dy| ≤ Cd sup r0 1 |B(x, r)| B(x,r) |f(y)| dy for any absolutely integrable function f and some constant Cd 0 depending only on d. (iv) Show that if f : Rd → C is absolutely integrable and x is a Lebesgue point of f, then the convolution f ∗ Pt(x) := Rd f(y)Pt(x − y) dy converges to f(x) as t → 0. (Hint: Split f(y) as the sum of f(x) and f(y) − f(x).) In particular, f ∗ Pt converges pointwise almost everywhere to f. 1.6.3. Almost everywhere differentiability. As we see in undergradu- ate real analysis, not every continuous function f : R → R is differentiable, with the standard example being the absolute value function f(x) := |x|, which is continuous not differentiable at the origin x = 0. Of course, this function is still almost everywhere differentiable. With a bit more effort, one can construct continuous functions that are in fact nowhere differentiable: Exercise 1.6.28 (Weierstrass function). Let F : R → R be the function F (x) := ∞ n=1 4−n sin(8nπx). (i) Show that F is well defined (in the sense that the series is absolutely convergent) and that F is a bounded continuous function. (ii) Show that for every 8-dyadic interval [ j 8n , j+1 8n ] with n ≥ 1, one has |F ( j+1) 8n − F ( j 8n )| ≥ c4−n for some absolute constant c 0. (iii) Show that F is not differentiable at any point x ∈ R. (Hint: Argue by contradiction and use the previous part of this exercise.) Note that it is not enough to formally differentiate the series term by term and observe that the resulting series is divergent—why not?

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