On the hyperbolic Riesz means

Abstract:

We define the hyperbolic Riesz means in ${{\mathbf {R}}^2}$ by ${H_\lambda }f = {({m_\lambda }\hat f)^\upsilon }$ where ${m_\lambda }({\xi _1},{\xi _2}) = (1 - {({\xi _1}{\xi _2})^2})_ + ^\lambda ,\lambda \geqslant 0$, and show that ${H_\lambda }$ is bounded on ${L^p}({{\mathbf {R}}^2})$ for $\tfrac {4}{3} \leqslant p \leqslant 4$ and $\lambda > \tfrac {1}{2}$ or $1 < p < \infty$ and $\lambda \geqslant 1$.