An estimate for the Bochner-Riesz operator on functions of product type in \(\mathbf R^2\) (Q1598368)

The author obtains boundedness for the Bochner-Riesz operator \[ (S^\delta f)\widehat{ (\xi)} = (1 - |\xi|^2)^\delta_+ \widehat f(\xi) \] on \(L^p(\mathbb R^2)\) for all \(1 < p < \infty\) and \(\delta > 0\), when the function \(f\) is a tensor product of two one-dimensional functions, \(f(\xi_1,\xi_2) = f_1(\xi_1) f_2(\xi_2)\). Note that without this restriction, the boundedness is only possible when \(\delta > 2|1/2-1/p|- 1/2\). The estimate is almost sharp (the same result should also hold when \(\delta = 0\), since Fefferman's example of the disc multiplier would not be applicable to tensor product counterexamples).