# Hardy-Littlewood maximal function: the relation between $\mu_{\lambda}^{Mf}$ and $\mu_{\lambda}^f$

Denote $M_f$ as uncentered Hardy-Littlewood maximal function. It is well known that $M_f\in L^{1,\infty}$, i.e. the distribution $\mu_{\lambda}^{Mf}$ is closely related to $||f||_{1}$, thus closely related to $\mu_{\lambda}^f$. Actually, from Grafakos's book, we have $$\mu_{2\lambda}^{Mf}\leq \frac{3^n}{\alpha}\int_{|f|>\alpha}|f|$$ $$\mu_{c_n\lambda}^{Mf}\geq \frac{2^{-n}}{\alpha}\int_{|f|>\alpha}|f|$$ where $c_n=(\frac{n}{2})^{\frac{n}{2}}\omega_n$, $\omega_n$ is the volume of unit ball.

Does anyone have some reference introducing the relation for these stuff ?

• Grafakos - Modern Fourier Analysis, or Grafakos - Classical Fourier Analysis? – Chill2Macht Jan 24 '17 at 22:05