Homework 15, Real Analysis

Due June 8

Problem 1

Consider the Koch-type curve K^l, for 1/4 < l < 1/2, described by the diagram

Koch_l.png

  1. The function t\mapsto K^l(t) satisfies a Hölder condition of exponent \gamma = -\log l/\log 4.
  2. t\mapsto K^l(t) is a simple curve.
  3. t\mapsto K^l(t) is continuous and nowhere differentiable.
  4. \dim K^l = 1/\gamma.

Problem 2

If we take l=1/2 in the previous definition, we obtain a space-filling curve.

Problem 3

On \R^d, define the functions, for \alpha > 0,

f_0(x) = \begin{cases} 1/|x|^{\alpha} & |x|<1\\0 & |x|\ge 1;\end{cases}\qquad f_\infty(x) = \begin{cases} 0 & |x|<1\\1/|x|^{\alpha} & |x|\ge 1.\end{cases}

  1. f_0\in L^p if and only if p\alpha < d.
  2. f_\infty \in L^p if and only if p\alpha > d.
  3. What happens if we replace |x|^\alpha with |x|^\alpha \big|\log 2|x|\big| in the previous definitions?

Problem 4

Suppose X is a measure space and 0 < p < 1.

  1. ||fg||_{L^1} \ge ||f||_{L^p}||g||_{L^q}, where q<0 is the conjugate exponent of p.
  2. If f_1, f_2 are non-negative, ||f_1 + f_2||_{L^p} \ge ||f_1||_{L^p} + ||f_2||_{L^p}.
  3. d(f,g) = ||f - g||_{L^p}^p defines a metric on L^p(X).
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