(a) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $1 \leqslant p<\infty$. What does it mean to say that $f$ belongs to $L^{p}(E, \mathcal{E}, \mu)$ ?

(b) State Hölder's inequality.

(c) Consider the measure space of the unit interval endowed with Lebesgue measure. Suppose $f \in L^{2}(0,1)$ and let $0<\alpha<1 / 2$.

(i) Show that for all $x \in \mathbb{R}$,

$$\int_{0}^{1}|f(y)||x-y|^{-\alpha} d y<\infty$$

(ii) For $x \in \mathbb{R}$, define

$$g(x)=\int_{0}^{1} f(y)|x-y|^{-\alpha} d y$$

Show that for $x \in \mathbb{R}$ fixed, the function $g$ satisfies

$$|g(x+h)-g(x)| \leqslant\|f\|_{2} \cdot(I(h))^{1 / 2},$$

where

$$I(h)=\int_{0}^{1}\left(|x+h-y|^{-\alpha}-|x-y|^{-\alpha}\right)^{2} d y .$$

(iii) Prove that $g$ is a continuous function. [Hint: You may find it helpful to split the integral defining $I(h)$ into several parts.]