A note on the “hyperbolic” Bochner-Riesz means

Abstract

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