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I saw the Hardy-Littlewood maximal inequality described as "weak-type (1, 1)". What is meant by a "weak-type" inequality in general, and what does the "(1, 1)" mean?

Apparently the Marcinkiewicz interpolation theorem gives a "strong-type" inequality. Does the weak-type inequality follow from the strong-type? If so, what is the weak type inequality good for?

user901823
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An operator from measurable functions on one measure space to measurable functions on another measure space is called weak type $(p,p)$ if for some constant $C>0$ we have $$ \text{measure}(|Tf| > \lambda) \le C \lambda^{-p} \|f\|^p_p ,$$ and strong type $(p,p)$ if for some constant $C>0$ we have $$ \|Tf\|_p \le C \|f\|_p .$$ (Actually the operator only needs to be defined on a dense subset of the space of measurable functions.)

Now it is easy to show that strong type $(p,p)$ implies weak type $(p,p)$, and it is also easy to generate examples that show the converse is not true.

The Marcinkiewicz interpolation theorem shows how to obtain strong type inequalities from weak type inequalities. This makes it extremely useful.

For example, the Hilbert transform is known to be weak type $(1,1)$, and strong type $(2,2)$. So from the Marcinkiewicz interpolation theorem one can show that the Hilbert transform is strong type $(p,p)$ for any $1 < p \le 2$.

J.R.
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Stephen Montgomery-Smith
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    Relevant: http://en.wikipedia.org/wiki/Marcinkiewicz_interpolation_theorem.

    An operator (say, sublinear) is of weak type (p,q) if the weak $L^q$ norm of $Tf$ is bounded by the $L^p$ norm of $f$.

     –  Apr 26 '15 at 05:57