Logaritmic convolution condition for homogeneous Calderón-Zygmund kernel

Petr Honzík

University of Missouri

Homogeneous Calderón-Zygmund kernel is a convolution kernel of the type

K_{\Omega}(x)=\frac{\Omega(x/\vert x\vert)}{\vert x\vert^n},

where $\Omega$ is an integrable function with mean zero on the unit sphere ${\mathcal {S}}^{n-1}$. It is well known that the corresponding convolution operator is bounded on $L^2$ if and only if the convolution
\int_{S^{n-1}} \Omega(\theta)\log \frac 1{\vert\xi \cdot \theta\vert} d\theta
\end{displaymath} (1)

is essentially bounded. We provide an example that shows that on $L^p$, $p\neq 2$, the condition ([*]) is no longer sufficient.