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

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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