= Solution
A nonnegative <sublinear operator> satisfies $S(f+g)\leq S(f)+S(g)$ and $S(cf)=|c|S(f)$ <almost everywhere>. It has <weak type (1,1)> if there is a finite constant $C$, independent of the function and threshold, such that
$$
\boxed{\lambda\{x:S(f)(x)>\alpha\}\leq
\frac{C}{\alpha}\|f\|_{L^1(\mathbb R)}
\quad\text{for every }f\in L^1(\mathbb R),\ \alpha>0.}
$$
For a signed or complex output use $|S(f)|$ in the <level set>. The bound controls the distribution of large values; it does not assert that $S(f)$ is integrable.
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