= Calderón–Zygmund weak type criterion for convolution
{c}
Suppose a <convolution> operator has $L^2$ operator norm at most $A$, and its kernel satisfies the <Hörmander integral kernel condition> with constant $C$. Under hypotheses making its integrals well-defined, the <Calderón–Zygmund decomposition> gives <weak type (1,1)>. Indeed, the bounded part has squared $L^2$ norm at most $2^d\lambda\|f\|_1$. The bad cubes occupy at most $\|f\|_1/\lambda$, while outside their $4\sqrt d$ dilations the <cancellation estimate outside a dilated cube> bounds the bad image in $L^1$. Applying distributional estimates to these three regions gives $|\{|Tf|>\lambda\}|\leq((4\sqrt d)^d+2^{d+2}A^2+4C)\|f\|_1/\lambda$.
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