Homework 15, Real Analysis

Due June 8

Problem 1

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

1. The function $latex t\mapsto K^l(t)$ satisfies a Hölder condition of exponent $latex \gamma = -\log l/\log 4$.


2. $latex t\mapsto K^l(t)$ is a simple curve.


3. $latex t\mapsto K^l(t)$ is continuous and nowhere differentiable.


4. $latex \dim K^l = 1/\gamma$.

Problem 2

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

Problem 3

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

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

1. $latex f_0\in L^p$ if and only if $latex p\alpha < d$.


2. $latex f_\infty \in L^p$ if and only if $latex p\alpha > d$.


3. What happens if we replace $latex |x|^\alpha$ with $latex |x|^\alpha \big|\log 2|x|\big|$ in the previous definitions?

Problem 4

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

1. $latex ||fg||_{L^1} \ge ||f||_{L^p}||g||_{L^q}$, where $latex q<0$ is the conjugate exponent of p.


2. If $latex f_1, f_2$ are non-negative, $latex ||f_1 + f_2||_{L^p} \ge ||f_1||_{L^p} + ||f_2||_{L^p}.$


3. $latex d(f,g) = ||f - g||_{L^p}^p$ defines a metric on $latex L^p(X)$.