Weak-type operator ($\mu \{|Sf|>\lambda \}\le (C\Vert f{\Vert }_p/\lambda {)}^q$)

An operator has weak type $(p,q)$, with finite $q$, if the displayed bound holds for every input and every $\lambda >0$, with one constant independent of both. It controls the distribution of its output rather than requiring its strong $L^q$ norm to be finite.

Let $S$ be a sublinear operator with positive constants and $\mu \{Sf>a\}\le C_1\Vert f{\Vert }_1/a$ and $\Vert Sf{\Vert }_{\infty }\le C_{\infty }\Vert f{\Vert }_{\infty }$. Split $f$ into its part above $a/(2C_{\infty })$ and its bounded remainder. The latter contributes at most $a/2$, so the weak bound controls the superlevel measure by $2C_1{a}^{-1}{\int }_{|f|>a/(2C_{\infty })}|f|$. Integrating this estimate with layer cake representation proves the displayed bound for $1<p<\infty$ and $q=p/(p-1)$. In particular it gives the strong $L^p$ estimate for the Hardy-Littlewood maximal function.

For a normed space $E$ and $0<q<\infty$, an operator $T:E\to L^0$ is of weak type $(E,q)$ when the displayed distribution bound holds uniformly over $f\in E$ and $a>0$. No completeness of $E$ is needed. A sublinear operator has pointwise triangle domination and absolute homogeneity; its outputs may be understood modulo almost everywhere equality.

Suppose linear operators $A_t$ and $A_0$ on a normed space satisfy $|A_{t}f|,|A_{0}f|\le Sf$, where $S$ has weak type with a normed domain, and all relevant suprema are measurable. If $A_{t}g\to A_{0}g$ almost everywhere on a dense subspace, the same holds for every input. For a fixed approximation $g$, the limiting error is at most $2S(f-g)$. The measure where it exceeds $a$ is at most $(2C\Vert f-g\Vert /a{)}^q$, which tends to zero as the approximation improves. This is a convergence-to-a-specified-limit version of almost-everywhere convergence from weak-type domination.

For a nonnegative sublinear operator, weak type $(1,1)$ is the displayed level-set bound for all integrable inputs and positive thresholds. It supplies an exceptional-set estimate for approximation errors; the uncentered maximal weak-type inequality is a central example.

Suppose a family of linear operators, including its limiting operator, is pointwise dominated by a sublinear operator of weak type (1,1). Convergence almost everywhere on an $L^1$-dense subspace then extends to every $L^1$ input. Approximate the input by errors of norm at most $2^{-2n}$. The weak estimate and the Borel-Cantelli first lemma make the corresponding domination errors tend to zero almost everywhere. Linearity bounds the difference between the full and approximating convergence errors by twice that domination error. This argument does not need an uncountable supremum to be measurable.