# Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that $$\int_{R^n}\sum_{i}a_i\chi_{B(x_i,3r_i)}(x)|g(x)|dx\leq C_0 \int_{R^n}\sum_{i}a_i\chi_{B(x_i,r_i)}(x)g^*(x)dx,\ \ \ \ (**)$$for some constant $C_0$ only depending on n. Here $C_0$ can be chosen like $3^n$.

Then I want to use $(**)$ to prove the following inequality:$$\left\lVert\sum_{i}a_i\chi_{B(x_i,3r_i)}\right\rVert_{L^p}\leq C_1\left\lVert\sum_{i}a_i\chi_{B(x_i,r_i)}\right\rVert_{L^p}\ \ \ \ \ \ (***)$$ for some constant $C_1$ is independent of $a_i$'s. I don't know how to prove $(***)$

• () is not hard to prove. How to use () to prove (***) – Shine Jun 24 '14 at 16:14
• – user357151 Nov 27 '17 at 21:16