Let $ f $ be in $ L^1(\Omega) $ where $ \Omega $ is an open subset of $ \mathbb{R}^n $. Also, assume that $ B(x_i,r_i) $ is a collection of disjoint open balls in $ \Omega $ such that $ B(x_i,2r_i) \subseteq \Omega $. I was wondering if there is a way that we can find a function $ g \in L^1(\Omega) $ and possibly a constant $ C $ such that

$ \int_{B(x_i,2r_i)} | f | \le C\int_{B(x_i,r_i)} | g | $, for all $ i \in \mathbb{N} $

(If $ f \in L^p(\Omega) $ with $ p > 1 $, then I can find such $ g $ via the Hardy-Littlewood maximal function.)

Thank you so much.