Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions

Abstract:

The relation between approach regions and singularities of nonnegative kernels $K_t(x,y)$ is studied, where $t\in (0,\infty )$, $x$, $y\in X$, and $X$ is a homogeneous space. For $1\le p<q<\infty$, a sufficient condition on approach regions $\Omega _a$ ($a\in X$) is given so that the maximal function \begin{equation*} \sup _{(x,t)\in \Omega _{a}} \int _X K_t(x,y)f(y) d\sigma (y) \end{equation*} is weak-type $(p,q)$ with respect to a pair of measures $\sigma$ and $\omega$. It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813–874).