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

Abstract:

The Marcinkiewicz integral \begin{equation*} I_{\lambda }\left ( x\right ) =\underset {\Omega }{\int } \frac {\left ( dist\left ( y,\mathbb {R}^{n}\backslash \Omega \right ) \right ) ^{\lambda }} {\left \vert x-y\right \vert ^{n+\lambda }}dy\text {, where }\lambda >0\text {,} \end{equation*} 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.