Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator $S^{R,\lambda}$ may be defined as

$S^{R,\lambda}f(x) = \mathcal{F}^{-1}\left[\left(1-\frac{\rho}{R}\right)^{\lambda}_+ \widehat{f}\right](x)$

where $\rho$ is an appropriate distance function and $\mathcal F^{-1}$ is the inverse Fourier transform. The behavior of $S^{R,\lambda}$ on $L^p(\mathbf{R}^d\times\mathbf{R})$ is described for $\rho(\xi',\xi_{d+1})=\max\{\vert\xi'\vert,\vert\xi_{d+1}\vert\}$, a rough distance function. We conjecture that this operator is bounded on $\mathbf{R}^{d}\times\mathbf{R}$ when $\lambda>\max\{d(\frac{1}{2}-\frac{1}{p})-\frac{1}{2},0\}$ and $p<2+\frac{6}{d-3}$, and unbounded when $p \geq 2+\frac{6}{d-3}$. This conjecture is verified for large ranges of $p$.