Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$,
$$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$
where $C_q$ is a positive constant depending only on $q$ and $d$.
Here the function $f^*(x)=\sup_{x\in B}\frac{1}{|B|}\int_B |f(y)|dy$ is the Hardy-Littlewood maximal function.
Notes
It seems to me the weak type estimate $\forall \alpha>0,\enspace |\{x: f^*(x)>\alpha\}|\leq \frac{3^d}{\alpha}||f||_{L^1(\mathbb{R}^d)}$ is of great use but I am having trouble putting this to any use. Any help is appreciated.
Best Answer
Indeed, the weak type estimate is useful. Using Fubini's theorem, we have $$\int_E|f^{*}(x)|\mathrm dx=q\int_0^\infty t^{q-1}\lambda\{|f^*(x)|\chi_E\geqslant t\}\mathrm dt.$$ Notice that $$\lambda\{|f^*(x)|\chi_E\geqslant t\}\leqslant \min\left\{|E|;\frac{3^d}t\lVert f\rVert_{\mathbb L^1}\right\},$$ hence cut the integrals and conclude.