Journal of the
Korean Mathematical Society JKMS

In this note we give the mapping properties of the Marcinkiewicz integral $\mu_\Omega$ at some end spaces. More precisely, we first prove that $\mu_\Omega$ is a bounded operator from $H^{1,\infty}(\Bbb R^n)$ to $L^{1,\infty}$ $(\Bbb R^n)$. As a corollary of the results above, we obtain again the weak type (1,1) boundedness of $\mu_\Omega,$ but the condition assumed on $\Omega$ is weaker than Stein's condition. Finally, we show that $\mu_\Omega$ is bounded from BMO$(\Bbb R^n)$ to BMO$(\Bbb R^n)$. The results in this note are the extensions of the results obtained by Lee and Rim recently.

Keywords: Marcinkiewicz integral, weak Hardy space, BMO

MSC numbers: 42B25, 42B30