The limiting case of the Marcinkiewicz integral: growth for convex sets

The Marcinkiewicz integral

$I_{\lambda }\left( x\right) =\underset{\Omega }{\int } \frac{\lef... ...} {\left\vert x-y\right\vert ^{n+\lambda }}dy\text{, where }\lambda >0\text{,}$

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case $\lambda \rightarrow 0$. Throughout, we assume that $\Omega$ is convex; it is interesting that this condition cannot be dropped.