A note on the ``hyperbolic'' Bochner-Riesz means

We consider the ${L^p}({{\mathbf{R}}^2})$ boundedness properties of the Fourier multiplier $m({\xi _1},{\xi _2}) = (1 - \xi _1^2\xi _2^2)_ + ^\alpha {\text{ for }}\alpha > 0$. We prove that if $\alpha \geqslant \frac{1}{2}$, then $m$ is bounded on ${L^p}$, $1 < p < \infty$, and that if $\alpha > 0$, then $m$ is bounded on ${L^p}$, $\frac{4}{3} \leqslant p \leqslant 4$.