= Calderón–Zygmund decomposition
{c}
{title2=$f=g+\sum_Qb_Q$}
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 $\|f\|_1/\lambda$, and their means are at most $2^d\lambda$. Replace $f$ inside each selected cube by its mean to obtain $g$; write $b_Q=(f-f_Q)\mathbf1_Q$. Then $\|g\|_\infty\leq2^d\lambda$, $\|g\|_1\leq\|f\|_1$, $\int b_Q=0$, and $\sum_Q\|b_Q\|_1\leq2\|f\|_1$. The outside bound follows from the <Lebesgue differentiation theorem>. This separates a bounded part from localized cancellation errors.
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