Calderón–Zygmund decomposition ($f=g+{\sum }_Q{b}_Q$)

For $f\in L^1({\mathbb{R}}^d)$ and a positive threshold $\lambda$, take maximal dyadic cubes with mean of $|f|$ exceeding $\lambda$. They are disjoint, have total volume at most $\Vert f{\Vert }_1/\lambda$, and their means are at most $2^d\lambda$. Replace $f$ inside each selected cube by its mean to obtain $g$; write $b_Q=(f-f_Q)1_Q$. Then $\Vert g{\Vert }_{\infty }\le 2^d\lambda$, $\Vert g{\Vert }_1\le \Vert f{\Vert }_1$, $\int b_Q=0$, and ${\sum }_Q\Vert b_Q{\Vert }_1\le 2\Vert f{\Vert }_1$. The outside bound follows from the Lebesgue differentiation theorem. This separates a bounded part from localized cancellation errors.