Sublinear operators and weak or strong type $(p,q)$ bounds

Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be measure spaces, and let $T$ assign to each measurable function on $X$ a measurable function on $Y$.

The operator $T$ is sublinear if for all scalars $a,b$ and measurable functions $f,g$ one has $\mathrm{\mid }T(af+bg)\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }Tf\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }\mathrm{\mid }Tg\mathrm{\mid }\mathrm{.}$

For $1\le p\le \mathrm{\infty }$ and $1\le q\le \mathrm{\infty }$, we say that $T$ is of strong type $(p,q)$ if there is a constant $C\ge 0$ such that $\mathrm{\parallel }Tf{\mathrm{\parallel }}_{L^q(\nu )}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mu )}$ for every $f\in L^p(\mu )$ (The space $L^p(\mu )$ as the quotient by null functions, The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions).

For $1\le p<\mathrm{\infty }$ and $1\le q<\mathrm{\infty }$, we say that $T$ is of weak type $(p,q)$ if there is a constant $C\ge 0$ such that for every $f\in L^p(\mu )$ and every $t>0$, $\nu (\{\mathrm{\mid }Tf\mathrm{\mid }>t\})\le {\left(\frac{C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mu )}}{t}\right)}^q\mathrm{.}$ Equivalently, the distribution function of $Tf$ (The distribution function of absolute value) obeys $A_{Tf}(t)\le {\left(\frac{C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mu )}}{t}\right)}^q(t>0)\mathrm{.}$