(a) Consider a measure space $(X, \mathcal{A}, \mu)$ and a complex-valued measurable function $F$ on $X$. Prove that for any $\varphi:[0,+\infty) \rightarrow[0,+\infty)$ differentiable and increasing such that $\varphi(0)=0$, then

$$\int_{X} \varphi(|F(x)|) \mathrm{d} \mu(x)=\int_{0}^{+\infty} \varphi^{\prime}(s) \mu(\{|F|>s\}) \mathrm{d} \lambda(s)$$

where $\lambda$ is the Lebesgue measure.

(b) Consider a complex-valued measurable function $f \in L^{1}\left(\mathbb{R}^{n}\right) \cap L^{\infty}\left(\mathbb{R}^{n}\right)$ and its maximal function $M f(x)=\sup _{r>0} \frac{1}{|B(x, r)|} \int_{B(x, r)}|f| \mathrm{d} \lambda$. Prove that for $p \in(1,+\infty)$ there is a constant $c_{p}>0$ such that $\|M f\|_{L^{p}\left(\mathbb{R}^{n}\right)} \leqslant c_{p}\|f\|_{L^{p}\left(\mathbb{R}^{n}\right)}$.

[Hint: Split $f=f_{0}+f_{1}$ with $f_{0}=f \chi_{\{|f|>s / 2\}}$ and $f_{1}=f \chi_{\{|f| \leqslant s / 2\}}$ and prove that $\lambda(\{M f>s\}) \leqslant \lambda\left(\left\{M f_{0}>s / 2\right\}\right)$. Then use the maximal inequality $\lambda(\{M f>s\}) \leqslant$ $\frac{C_{1}}{s}\|f\|_{L^{1}\left(\mathbb{R}^{n}\right)}$ for some constant $\left.C_{1}>0 .\right]$

(c) Consider $p, q \in(1,+\infty)$ with $p<q$ and $\alpha \in(0, n)$ such that $1 / q=1 / p-\alpha / n$. Define $I_{\alpha}|f|(x):=\int_{\mathbb{R}^{n}} \frac{|f(y)|}{|x-y|^{n-\alpha}} \mathrm{d} \lambda(y)$ and prove $I_{\alpha}|f|(x) \leqslant\|f\|_{L^{p}\left(\mathbb{R}^{n}\right)}^{\alpha p / n} M f(x)^{1-\alpha p / n}$.

[Hint: Split the integral into $|x-y| \geqslant r$ and $|x-y| \in\left[2^{-k-1} r, 2^{-k} r\right)$ for all $k \geqslant 0$, given some suitable $r>0 .]$